//  Copyright (C) 2001-2010 Roberto Bagnara <bagnara@cs.unipr.it>
//  Copyright (C) 2010-2011 BUGSENG srl (http://bugseng.com)
//
//  This document describes the Parma Polyhedra Library (PPL).
//
//  Permission is granted to copy, distribute and/or modify this document
//  under the terms of the GNU Free Documentation License, Version 1.1 or
//  any later version published by the Free Software Foundation; with no
//  Invariant Sections, with no Front-Cover Texts, and with no Back-Cover
//  Texts.
//
//  The PPL is free software; you can redistribute it and/or modify it
//  under the terms of the GNU General Public License as published by
//  the Free Software Foundation; either version 3 of the License, or
//  (at your option) any later version.
//
//  The PPL is distributed in the hope that it will be useful, but WITHOUT
//  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
//  FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
//  for more details.
//
//  For the most up-to-date information see the Parma Polyhedra Library
//  site: http://www.cs.unipr.it/ppl/

/*!
  \defgroup PPL_CXX_interface C++ Language Interface
  \brief
  The core implementation of the Parma Polyhedra Library
  is written in C++.

  See Namespace, Hierarchical and Compound indexes
  for additional information about each single data type.
*/

/*! \mainpage General Information on the PPL
\section preamble The Main Features

The Parma Polyhedra Library (PPL) is a modern C++ library for the
manipulation of numerical information that can be represented by
points in some \f$n\f$-dimensional vector space. For instance, one of
the key domains the PPL supports is that of rational convex polyhedra
(Section \ref convex_polys). Such domains
are employed in several systems for the analysis and verification of
hardware and software components, with applications spanning
imperative, functional and logic programming languages, synchronous
languages and synchronization protocols, real-time and hybrid systems.
Even though the PPL library is not meant to target a particular
problem, the design of its interface has been largely influenced by
the needs of the above class of applications.  That is the reason why
the library implements a few operators that are more or less specific
to static analysis applications, while lacking some other operators
that might be useful when working, e.g., in the field of computational
geometry.

The main features of the library are the following:
- it is user friendly:
  you write <CODE>x + 2*y + 5*z \<= 7</CODE> when you mean it;
- it is fully dynamic:
  available virtual memory is the only limitation to the
  dimension of anything;
- it provides full support for the manipulation of convex polyhedra
  that are not topologically closed;
- it is written in standard C++: meant to be portable;
- it is exception-safe: never leaks resources or leaves invalid object
  fragments around;
- it is rather efficient: and we hope to make it even more so;
- it is thoroughly documented:
  perhaps not literate programming but close enough;
- it has interfaces to other programming languages: including C, Java, OCaml
  and a number of Prolog systems;
- it is free software: distributed under the terms of the
  GNU General Public License.

In the following section we describe all the domains available to the PPL user.
More detailed descriptions of these domains and the operations provided
will be found in subsequent sections.

In the final section of this chapter (Section \ref use_of_library),
we provide some additional advice on the use of the library.

\subsection Semantic_Geometric_Descriptors Semantic Geometric Descriptors

A <EM>semantic geometric descriptor</EM> is a subset of \f$ \Rset^n \f$.
The PPL provides several classes of semantic GDs.  These are
identified by their C++ class name, together with the class template
parameters, if any. These classes include the <EM>simple classes</EM>:
  - \link Parma_Polyhedra_Library::C_Polyhedron \c C_Polyhedron \endlink,
  - \link Parma_Polyhedra_Library::NNC_Polyhedron \c NNC_Polyhedron \endlink,
  - \link Parma_Polyhedra_Library::BD_Shape \c BD_Shape\<T\> \endlink,
  - \link Parma_Polyhedra_Library::Octagonal_Shape \c Octagonal_Shape\<T\> \endlink,
  - \link Parma_Polyhedra_Library::Box \c Box\<ITV\> \endlink, and
  - \link Parma_Polyhedra_Library::Grid \c Grid \endlink,

where:
  - \c T is a numeric type chosen among
    \c mpz_class, \c mpq_class,
    <CODE>signed char</CODE>, \c short, \c int, \c long, \c long long
    (or any of the C99 exact width integer equivalents \c int8_t, int16_t,
    and so forth); and
  - \c ITV is an instance of the
    \link Parma_Polyhedra_Library::Interval \c Interval \endlink
    template class.

Other semantic GDs, the <EM>compound classes</EM>, can be constructed
(also recursively) from all the GDs classes.
These include:
  - \link Parma_Polyhedra_Library::Pointset_Powerset \c Pointset_Powerset\<PSET\> \endlink,
  - \link Parma_Polyhedra_Library::Partially_Reduced_Product <CODE>Partially_Reduced_Product\<D1, D2, R\></CODE> \endlink,
  .
where \c PSET, \c D1 and \c D2 can be any semantic GD classes and \c R is the
reduction operation to be applied to the component domains of the
product class.

A uniform set of operations is provided for creating, testing and
maintaining each of the semantic GDs.  However, as many of these
depend on one or more syntactic GDs, we first describe the syntactic GDs.


\subsection Syntactic_Geometric_Descriptors Syntactic Geometric Descriptors

A <EM>syntactic geometric descriptor</EM> is for defining, modifying
and inspecting a semantic GD.  There are three kinds of
<EM>syntactic GDs</EM>: <EM>basic GDs</EM>, <EM>constraint GDs</EM> and
<EM>generator GDs</EM>.
Some of these are <EM>generic</EM> and some <EM>specific</EM>.
A generic syntactic GD can be used (in the appropriate context)
with any semantic GD; clearly, different semantic GDs will usually
provide different levels of support for the different subclasses
of generic GDs. In contrast, the use of a specific GD may be
restricted to apply to a given subset of the semantic GDs
(i.e., some semantic GDs provide no support at all for them).


\subsubsection Basic_Geometric_Descriptors Basic Geometric Descriptors

The following basic GDs currently supported by the PPL are:
  - space dimension;
  - variable and variable set;
  - coefficient;
  - linear expression;
  - relation symbol;
  - vector point.

These classes, which are all generic syntactic GDs, are used
to build the constraint and generator GDs as
well as support many generic operations on the semantic GDs.


\subsubsection Constraint_Geometric_Descriptors Constraint Geometric Descriptors

The PPL currently supports the following classes of \e generic constraint GDs:
  - linear constraint;
  - linear congruence.

Each linear constraint can be further classified to belong to one or more
of the following syntactic subclasses:
  - inconsistent constraints (e.g., \f$0 \geq 2\f$);
  - tautological constraints (e.g., \f$0 \leq 2\f$);
  - interval constraints (e.g., \f$x \leq 2\f$);
  - bounded-difference constraints (e.g., \f$x - y \leq 2\f$);
  - octagonal constraints (e.g., \f$x + y \leq 2\f$);
  - linear equality constraints (e.g., \f$x = 2\f$);
  - non-strict linear inequality constraints (e.g., \f$x - 3y \leq 2\f$);
  - strict linear inequality constraints (e.g., \f$x - 3y < 2\f$).

Note that the subclasses are not disjoint.

Similarly, each linear congruence can be classified to belong
to one or more of the following syntactic subclasses:
  - inconsistent congruences (e.g., \f$0 \equiv_2 1\f$);
  - tautological congruences (e.g., \f$0 \equiv_2 2\f$);
  - linear equality, i.e., non-proper congruences
    (e.g., \f$x + 3y \equiv_0 0\f$);
  - proper congruences (e.g., \f$x + 3y \equiv_5 0\f$).

The library also supports systems, i.e., finite collections, of either
linear constraints or linear congruences (but see the note below).

Each semantic GD provides \e optimal support for some of the subclasses
of generic syntactic GDs listed above: here, the word "optimal" means that
the considered semantic GD computes the <em>best upward approximation</em>
of the exact meaning of the linear constraint or congruence.
When a semantic GD operation is applied to a syntactic GD that is not
optimally supported, it will either indicate its unsuitability
(e.g., by throwing an exception) or it will apply an upward approximation
semantics (possibly not the best one).

For instance, the semantic GD of topologically closed convex polyhedra
provides optimal support for non-strict linear inequality and equality
constraints, but it does not provide optimal support for strict inequalities.
Some of its operations (e.g., \c add_constraint and \c add_congruence)
will throw an exception if supplied with a non-trivial strict inequality
constraint or a proper congruence; some other operations
(e.g., \c refine_with_constraint or \c refine_with_congruence)
will compute an over-approximation.

Similarly, the semantic GD of rational boxes (i.e., multi-dimensional
intervals) having integral values as interval boundaries provides optimal
support for all interval constraints: even though the interval constraint
\f$2x \leq 5\f$ cannot be represented exactly, it will be optimally
approximated by the constraint \f$x \leq 3\f$.

\note
When providing an upward approximation for a constraint or congruence,
we consider it in isolation: in particular, the approximation of each
element of a system of GDs is independent from the other elements;
also, the approximation is independent from the current value of
the semantic GD.


\subsubsection Generator_Geometric_Descriptors Generator Geometric Descriptors

The PPL currently supports two classes of generator GDs:
  - polyhedra generator: these are polyhedra points, rays and lines;
  - grid generator: these are grid points, parameters and lines.

Rays, lines and parameters are specific of the mentioned semantic GDs
and, therefore, they cannot be used by other semantic GDs.
In contrast, as already mentioned above, points are basic geometric
descriptors since they are also used in <EM>generic</EM> PPL operations.


\subsection Generic_Operations_on_Semantic_Geometric_Descriptors Generic Operations on Semantic Geometric Descriptors

<OL>
<LI>
  Constructors of a universe or empty semantic GD with the given
  space dimension.
</LI>
<LI>
  Operations on a semantic GD that do not depend on the syntactic GDs.
  <UL>
  <LI>
    <CODE>is_empty()</CODE>, <CODE>is_universe()</CODE>,
    <CODE>is_topologically_closed()</CODE>, <CODE>is_discrete()</CODE>,
    <CODE>is_bounded()</CODE>, <CODE>contains_integer_point()</CODE>

    test for the named properties of the semantic GD.
  </LI>
  <LI>
    <CODE>%total_memory_in_bytes()</CODE>,
    <CODE>%external_memory_in_bytes()</CODE>

    return the total and external memory size in bytes.
  </LI>
  <LI>
    <CODE>OK()</CODE>

    checks that the semantic GD has a valid internal representation.
    (Some GDs provide this method with an optional Boolean argument
    that, when true, requires to also check for non-emptiness.)
  </LI>
  <LI>
    <CODE>space_dimension()</CODE>, <CODE>affine_dimension()</CODE>

    return, respectively, the space and affine dimensions of the GD.
  </LI>
  <LI>
    <CODE>add_space_dimensions_and_embed()</CODE>,
    <CODE>add_space_dimensions_and_project()</CODE>,
    <CODE>expand_space_dimension()</CODE>,
    <CODE>remove_space_dimensions()</CODE>,
    <CODE>fold_space_dimensions()</CODE>, <CODE>map_space_dimensions()</CODE>

    modify the space dimensions of the semantic GD; where, depending on the
    operation, the arguments can include the number of space dimensions
    to be added or removed a variable or set of variables denoting the
    actual dimensions to be used and a partial function defining a mapping
    between the dimensions.
  </LI>
  <LI>
    <CODE>contains()</CODE>, <CODE>strictly_contains()</CODE>,
    <CODE>is_disjoint_from()</CODE>

    compare the semantic GD with an argument semantic GD of the same class.
  </LI>
  <LI>
    <CODE>topological_closure_assign()</CODE>,
    <CODE>intersection_assign()</CODE>,
    <CODE>upper_bound_assign()</CODE>, <CODE>difference_assign()</CODE>,
    <CODE>time_elapse_assign()</CODE>, <CODE>widening_assign()</CODE>,
    <CODE>concatenate_assign()</CODE>, <CODE>%swap()</CODE>

    modify the semantic GD, possibly with an argument semantic GD of
    the same class.
  </LI>
  <LI>
    <CODE>constrains()</CODE>, <CODE>bounds_from_above()</CODE>,
    <CODE>bounds_from_below()</CODE>, <CODE>maximize()</CODE>,
    <CODE>minimize()</CODE>.

    These find information about the bounds of the semantic GD where
    the argument variable or linear expression define the direction
    of the bound.
  </LI>
  <LI>
    <CODE>affine_image()</CODE>, <CODE>affine_preimage()</CODE>,
    <CODE>generalized_affine_image()</CODE>,
    <CODE>generalized_affine_preimage()</CODE>,
    <CODE>bounded_affine_image()</CODE>,
    <CODE>bounded_affine_preimage()</CODE>.

    These perform several variations of the affine image and preimage
    operations where, depending on the operation, the arguments can
    include a variable representing the space dimension to which the
    transformation will be applied and linear expressions
    with possibly a relation symbol and denominator value
    that define the exact form of the transformation.
  </LI>
  <LI>
    <CODE>ascii_load()</CODE>, <CODE>ascii_dump()</CODE>

    are the ascii input and output operations.
  </UL>
</LI>
<LI>
  Constructors of a semantic GD of one class from a semantic GD of any
  other class.
  These constructors obey an <EM>upward approximation semantics</EM>,
  meaning that the constructed semantic GD is guaranteed to contain
  all the points of the source semantic GD, but possibly more.
  Some of these constructors provide a complexity parameter with which
  the application can control the complexity/precision trade-off
  for the construction operation: by using the complexity parameter,
  it is possible to keep the construction operation in the polynomial
  or the simplex worst-case complexity class, possibly incurring into
  a further upward approximation if the precise constructor is based
  on an algorithm having exponential complexity.
</LI>
<LI>
  Constructors of a semantic GD from a constraint GD;
  either a linear constraint system or a linear congruence system.
  These constructors assume that the given semantic GD provides optimal
  support for the argument syntactic GD: if that is not the case,
  an invalid argument exception is thrown.
</LI>
<LI>
  Other interaction between the semantic GDs and constraint GDs.
  <UL>
  <LI>
    <CODE>add_constraint()</CODE>, <CODE>add_constraints()</CODE>,
    <CODE>add_recycled_constraints()</CODE>,
    <CODE>add_congruence()</CODE>, <CODE>add_congruences()</CODE>,
    <CODE>add_recycled_congruences()</CODE>.

    These methods assume that the given semantic GD provides optimal
    support for the argument syntactic GD: if that is not the case,
    an invalid argument exception is thrown.

    For <CODE>add_recycled_constraints()</CODE> and
    <CODE>add_recycled_congruences()</CODE>,
    the only assumption that can be made on the constraint GD
    after return (successful or exceptional) is that
    it can be safely destroyed.
  </LI>
  <LI>
    <CODE>refine_with_constraint()</CODE>,
    <CODE>refine_with_constraints()</CODE>,
    <CODE>refine_with_congruence()</CODE>,
    <CODE>refine_with_congruences()</CODE>.

    If the argument constraint GD is optimally supported by the
    semantic GD, the methods behave the same as the corresponding
    \c add_* methods listed above.  Otherwise the constraint GD
    is used only to a limited extent to refine the semantic GD;
    possibly not at all.  Notice that, while repeating an add operation
    is pointless, this is not true for the refine operations.
    For example, in those cases where
\code
      Semantic_GD.add_constraint(c)
\endcode
    raises an exception, a fragment of the form
\code
      Semantic_GD.refine_with_constraint(c)
      // Other add_constraint(s) or refine_with_constraint(s) operations
      // on Semantic_GD.
      Semantic_GD.refine_with_constraint(c)
\endcode
    may give more precise results than a single
\code
      Semantic_GD.refine_with_constraint(c).
      // Other add_constraint(s) or refine_with_constraint(s) operations
      // on Semantic_GD.
\endcode
  </LI>
  <LI>
    <CODE>constraints()</CODE>, <CODE>minimized_constraints()</CODE>,
    <CODE>congruences()</CODE>, <CODE>minimized_congruences()</CODE>.

    Returns the indicated system of constraint GDs satisfied by
    the semantic GD.
  </LI>
  <LI>
    <CODE>can_recycle_constraint_systems()</CODE>,
    <CODE>can_recycle_congruence_systems()</CODE>.

    Return true if and only if the semantic GD can recycle the
    indicated constraint GD.
  </LI>
  <LI>
    <CODE>relation_with()</CODE>.

    This takes a constraint GD as an argument and returns the relations holding
    between the semantic GD and the constraint GD.
    The possible relations are:
    <CODE>IS_INCLUDED()</CODE>, <CODE>SATURATES()</CODE>,
    <CODE>STRICTLY_INTERSECTS()</CODE>, <CODE>IS_DISJOINT()</CODE>
    and <CODE>NOTHING()</CODE>.
    This operator also can take a polyhedron generator GD as an argument
    and returns the relation <CODE>SUBSUMES()</CODE> or <CODE>NOTHING()</CODE>
    that holds between the generator GD and the semantic GD.
  </LI>
  </UL>
</LI>
</OL>


\section Upward_Approximation Upward Approximation

The Parma Polyhedra Library, for those cases where an exact result
cannot be computed within the specified complexity limits, computes an
<EM>upward approximation</EM> of the exact result.  For semantic GDs this
means that the computed result is a possibly strict superset of the set of
points of \f$ \Rset^n \f$ that constitutes the exact result.  Notice
that the PPL does not provide direct support to compute <EM>downward
approximations</EM> (i.e., possibly strict subsets of the exact
results).  While downward approximations can often be computed from
upward ones, the required algorithms and the conditions upon which
they are correct are outside the current scope of the PPL.
Beware, in particular, of the following possible pitfall:
the library provides methods to compute upward approximations
of set-theoretic difference, which is antitone in its second
argument.  Applying a difference method to a second argument
that is not an exact representation or a downward approximation of
reality, would yield a result that, of course, is not an upward
approximation of reality.  It is the responsibility of the
library user to provide the PPL's method with approximations of reality
that are consistent with respect to the desired results.


\section Approximating_Integers Approximating Integers

The Parma Polyhedra Library provides support for approximating
integer computations using the geometric descriptors it provides.
In this section we briefly explain these facilities.


\subsection Dropping_Non_Integer_Points Dropping Non-Integer Points

When a geometric descriptor is used to approximate integer quantities,
all the points with non-integral coordinates represent an imprecision
of the description.  Of course, removing all these points may be
impossible (because of convexity) or too expensive.  The PPL provides
the operator <CODE>drop_some_non_integer_points</CODE> to possibly
tighten a descriptor by dropping some points with non-integer
coordinates, using algorithms whose complexity is bounded by
a parameter.  The set of dimensions that represent integer quantities
can be optionally specified.  It is worth to stress the role of
<EM>some</EM> in the operator name: in general no optimality guarantee
is provided.


\subsection Approximating_Bounded_Integers Approximating Bounded Integers

The Parma Polyhedra Library provides services that allow to compute
correct approximations of bounded arithmetic as available in widespread
programming languages.  Supported bit-widths are 8, 16, 32 and 64 bits,
with some limited support for 128 bits.
Supported representations are binary unsigned and two's complement signed.
Supported overflow behaviors are:
<DL>
  <DT>Wrapping:</DT>
  <DD>
    this means that, for a \f$w\textrm{-bit}\f$ bounded integer, the
    computation
    happens modulo \f$2^w\f$.  In turn, this signifies that the computation
    happens <EM>as if</EM> the unbounded arithmetic result was computed
    and then wrapped.  For unsigned integers, the wrapping function is
    simply \f$x \bmod 2^w\f$, most conveniently defined as
    \f[
      \mathrm{wrap}^\mathrm{u}_w(x)
        \defeq
          x - 2^w \lfloor x/2^w \rfloor.
    \f]
    For signed integers the wrapping function is, instead,
    \f[
      \mathrm{wrap}^\mathrm{s}_w(x)
        \defeq
          \begin{cases}
            \mathrm{wrap}^\mathrm{u}_w(x),
              &\text{if $\mathrm{wrap}^\mathrm{u}_w(x) < 2^{w-1}$;} \\
            \mathrm{wrap}^\mathrm{u}_w(x) - 2^w,
              &\text{otherwise.}
          \end{cases}
    \f]
  </DD>
  <DT>Undefined:</DT>
  <DD>
    this means that the result of the operation resulting in an
    overflow can take any value.  This is useful to partially
    model systems where overflow has unspecified effects on
    the computed result.
    Even though something more serious can happen in the system
    being analyzed ---due to, e.g., C's undefined behavior---, here we
    are only concerned with the results of arithmetic operations.
    It is the responsibility of the analyzer to ensure that other
    manifestations of undefined behavior are conservatively approximated.
  </DD>
  <DT>Impossible:</DT>
  <DD>
    this is for the analysis of languages where overflow is trapped
    before it affects the state, for which, thus, any indication that
    an overflow may have affected the state is necessarily due to
    the imprecision of the analysis.
  </DD>
</DL>


\subsubsection Wrapping_Operator Wrapping Operator

One possibility for precisely approximating the semantics of programs that
operate on bounded integer variables is to follow the approach described
in \ref SK07 "[SK07]".  The idea is to associate space dimensions to the
<EM>unwrapped values</EM> of bounded variables.  Suppose <CODE>j</CODE>
is a \f$w\textrm{-bit}\f$, unsigned program variable associated to a space 
dimension
labeled by the variable \f$x\f$.  If \f$x\f$ is constrained by some
numerical abstraction to take values in a set \f$S \sseq \Rset\f$, then
the program variable <CODE>j</CODE> can only take values in
\f$\bigl\{\, \mathrm{wrap}^\mathrm{u}_w(z) \bigm| z \in S \,\bigr\}\f$.
There are two reasons why this is interesting: firstly, this allows for
the retention of relational information by using a single numerical abstraction
tracking multiple program variables.  Secondly, the integers modulo
\f$2^w\f$ form a ring of equivalence classes on which addition
and multiplication are well defined.  This means, e.g., that assignments
with affine right-hand sides and involving only variables with the same
bit-width and representation can be safely modeled by affine images.
While upper bounds and widening can be used without any precaution,
anything that can be reconducted to intersection requires a preliminary
<EM>wrapping</EM> phase, where the dimensions corresponding to bounded
integer types are brought back to their natural domain.
This necessity arises naturally for the analysis of conditionals
and conversion operators, as well as in the realization of domain
combinations.

The PPL provides a general wrapping operator that is parametric with
respect to the set of space dimensions (variables) to be wrapped,
the width, representation and overflow behavior of all these variables.
An optional constraint system can, when given, improve the
precision.  This constraint system, which must only depend on variables
with respect to which wrapping is performed, is assumed to represent
the conditional or looping construct guard with respect to which
wrapping is performed.  Since wrapping requires the computation of
upper bounds and due to non-distributivity of constraint refinement
over upper bounds, passing a constraint system in this way can be more
precise than refining the result of the wrapping operation afterwards.
The general wrapping operator offered by the PPL also allows control of
the complexity/precision ratio by means of two additional parameters:
an unsigned integer encoding a complexity threshold, with higher values
resulting in possibly improved precision; and a Boolean controlling
whether space dimensions should be wrapped individually, something that
results in much greater efficiency to the detriment of precision,
or collectively.

Note that the PPL assumes that any space dimension subject to wrapping
is being used to capture the value of bounded integer values.  As a
consequence the library is free to drop, from the involved numerical
abstraction, any point having a non-integer coordinate that corresponds
to a space dimension subject to wrapping.  It must be stressed that
freedom to drop such points does not constitute an obligation to remove
all of them (especially because this would be extraordinarily expensive
on some numerical abstractions).
The PPL provides operators for the more systematic
\ref Dropping_Non_Integer_Points
"removal of points with non-integral coordinates".

The wrapping operator will only remove some of these points
as a by-product of its main task and only when this comes at a negligible
extra cost.


\section convex_polys Convex Polyhedra

In this section we introduce convex polyhedra,
as considered by the library, in more detail.
For more information about the definitions and results stated here see
\ref BRZH02b "[BRZH02b]",
\ref Fuk98 "[Fuk98]",
\ref NW88 "[NW88]", and
\ref Wil93 "[Wil93]".

\subsection Vectors_Matrices_and_Scalar_Products Vectors, Matrices and Scalar Products

We denote by \f$\Rset^n\f$ the \f$n\textrm{-dimensional}\f$ vector space on the field
of real numbers \f$\Rset\f$, endowed with the standard topology.
The set of all non-negative reals is denoted by \f$\nonnegRset\f$.
For each \f$i \in \{0, \ldots, n-1\}\f$, \f$v_i\f$ denotes
the \f$i\textrm{-th}\f$ component of the (column) vector
\f$\vect{v} = (v_0, \ldots, v_{n-1})^\transpose \in \Rset^n\f$.
We denote by \f$\vect{0}\f$ the vector of \f$\Rset^n\f$,
called <EM>the origin</EM>, having all components equal to zero.
A vector \f$\vect{v} \in \Rset^n\f$ can be also interpreted
as a matrix in \f$\Rset^{n \times 1}\f$ and manipulated accordingly
using the usual definitions for addition, multiplication
(both by a scalar and by another matrix), and transposition,
denoted by \f$\vect{v}^\transpose\f$.

The <EM>scalar product</EM> of \f$\vect{v},\vect{w} \in \Rset^n\f$,
denoted \f$\langle \vect{v}, \vect{w} \rangle\f$,
is the real number
\f[
  \vect{v}^\transpose \vect{w} = \sum_{i=0}^{n-1} v_i w_i.
\f]

For any \f$S_1, S_2 \sseq \Rset^n\f$, the <EM>Minkowski's sum</EM> of
\f$S_1\f$ and \f$S_2\f$ is:
\f$S_1 + S_2 = \{\, \vect{v}_1 + \vect{v}_2 \mid
                \vect{v}_1 \in S_1, \vect{v}_2 \in S_2 \,\}.\f$


\subsection Affine_Hyperplanes_and_Half_spaces Affine Hyperplanes and Half-spaces

For each vector \f$\vect{a} \in \Rset^n\f$ and scalar \f$b \in \Rset\f$,
where \f$\vect{a} \neq \vect{0}\f$,
and for each relation symbol \f$\mathord{\relsym} \in \{ =, \geq, > \}\f$,
the linear constraint \f$\langle \vect{a}, \vect{x} \rangle \relsym b\f$
defines:
- an affine hyperplane if it is an equality constraint,
  i.e., if \f$\mathord{\relsym} \in \{ = \}\f$;
- a topologically closed affine half-space if it is
  a non-strict inequality constraint,
  i.e., if \f$\mathord{\relsym} \in \{ \geq \}\f$;
- a topologically open affine half-space if it is
  a strict inequality constraint,
  i.e., if \f$\mathord{\relsym} \in \{ > \}\f$.

Note that each hyperplane \f$\langle \vect{a}, \vect{x} \rangle = b\f$
can be defined as the intersection of the two closed affine half-spaces
\f$\langle \vect{a}, \vect{x} \rangle \geq b\f$ and
\f$\langle -\vect{a}, \vect{x} \rangle \geq -b\f$.
Also note that, when \f$\vect{a} = \vect{0}\f$,
the constraint \f$\langle \vect{0}, \vect{x} \rangle \relsym b\f$
is either a tautology (i.e., always true)
or inconsistent (i.e., always false), so that it defines
either the whole vector space \f$\Rset^n\f$ or the empty set \f$\emptyset\f$.


\subsection Convex_Polyhedra Convex Polyhedra

The set \f$\cP \sseq \Rset^n\f$ is a
<EM>not necessarily closed convex polyhedron</EM>
(<EM>NNC polyhedron</EM>, for short) if and only if either
\f$\cP\f$ can be expressed as the intersection of a finite number
of (open or closed) affine half-spaces of \f$\Rset^n\f$
or \f$n = 0\f$ and \f$\cP = \emptyset\f$.
The set of all NNC polyhedra on the vector space \f$\Rset^n\f$
is denoted \f$\Pset_n\f$.

The set \f$\cP \in \Pset_n\f$ is a <EM>closed convex polyhedron</EM>
(<EM>closed polyhedron</EM>, for short) if and only if either
\f$\cP\f$ can be expressed as the intersection of a finite number
of closed affine half-spaces of \f$\Rset^n\f$
or \f$n = 0\f$ and \f$\cP = \emptyset\f$.
The set of all closed polyhedra on the vector space \f$\Rset^n\f$
is denoted \f$\CPset_n\f$.

When ordering NNC polyhedra by the set inclusion relation,
the empty set \f$\emptyset\f$ and the vector space \f$\Rset^n\f$
are, respectively, the smallest and the biggest elements of both
\f$\Pset_n\f$ and \f$\CPset_n\f$.
The vector space \f$\Rset^n\f$ is also called the <EM>universe</EM> polyhedron.

In theoretical terms,
\f$\Pset_n\f$ is a <EM>lattice</EM> under set inclusion and
\f$\CPset_n\f$ is a <EM>sub-lattice</EM> of \f$\Pset_n\f$.

\note
In the following, we will usually specify operators on the domain
\f$\Pset_n\f$ of NNC polyhedra. Unless an explicit distinction is
made, these operators are provided with the same specification when
applied to the domain \f$\CPset_n\f$ of topologically closed
polyhedra.  The implementation maintains a clearer separation between
the two domains of polyhedra
(see \ref Topologies_and_Topological_compatibility "Topologies and Topological-compatibility"):
while computing polyhedra in \f$\Pset_n\f$ may provide more precise
results, polyhedra in \f$\CPset_n\f$ can be represented and
manipulated more efficiently.  As a rule of thumb, if your application
will only manipulate polyhedra that are topologically closed, then it
should use the simpler domain \f$\CPset_n\f$. Using NNC polyhedra is
only recommended if you are going to actually benefit from the
increased accuracy.


\subsection Bounded_Polyhedra Bounded Polyhedra

An NNC polyhedron \f$\cP \in \Pset_n\f$ is <EM>bounded</EM>
if there exists a \f$\lambda \in \nonnegRset\f$ such that:
\f[
  \cP
    \sseq
      \bigl\{\,
        \vect{x} \in \Rset^n
      \bigm|
        - \lambda \leq x_j \leq \lambda \text{ for } j = 0, \ldots, n-1
      \,\bigr\}.
\f]
A bounded polyhedron is also called a <EM>polytope</EM>.


\section representation Representations of Convex Polyhedra

NNC polyhedra can be specified by using two possible representations,
the constraints (or implicit) representation and
the generators (or parametric) representation.


\subsection Constraints_Representation Constraints Representation

In the sequel, we will simply write ``equality'' and ``inequality''
to mean ``linear equality'' and ``linear inequality'', respectively;
also, we will refer to either an equality or an inequality
as a <EM>constraint</EM>.

By definition, each polyhedron \f$\cP \in \Pset_n\f$ is the set of solutions
to a <EM>constraint system</EM>, i.e., a finite number of constraints.
By using matrix notation, we have
\f[
  \cP
    \defeq
      \{\,
        \vect{x} \in \Rset^n
      \mid
        A_1 \vect{x} =    \vect{b}_1,
        A_2 \vect{x} \geq \vect{b}_2,
        A_3 \vect{x} >    \vect{b}_3
      \,\},
\f]
where, for all \f$i \in \{1, 2, 3\}\f$,
\f$A_i \in \Rset^{m_i} \times \Rset^n\f$ and \f$\vect{b}_i \in \Rset^{m_i}\f$,
and \f$m_1, m_2, m_3 \in \Nset\f$ are
the number of equalities,
the number of non-strict inequalities, and
the number of strict inequalities,
respectively.


\subsection Combinations_and_Hulls Combinations and Hulls

Let \f$S = \{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n\f$
be a finite set of vectors.
For all scalars \f$\lambda_1, \ldots, \lambda_k \in \Rset\f$,
the vector \f$\vect{v} = \sum_{j=1}^k \lambda_j \vect{x}_j\f$
is said to be a <EM>linear</EM> combination of the vectors in \f$S\f$.
Such a combination is said to be
- a <EM>positive</EM> (or <EM>conic</EM>) combination,
  if \f$\forall j \in \{ 1, \ldots, k \} \itc \lambda_j \in \nonnegRset\f$;
- an <EM>affine</EM> combination, if \f$\sum_{j = 1}^k \lambda_j = 1\f$;
- a <EM>convex</EM> combination, if it is both positive and affine.

We denote by \f$\linearhull(S)\f$
(resp., \f$\conichull(S)\f$, \f$\affinehull(S)\f$, \f$\convexhull(S)\f$)
the set of all the linear (resp., positive, affine, convex)
combinations of the vectors in \f$S\f$.

Let \f$P, C \sseq \Rset^n\f$, where \f$P \union C = S\f$.
We denote by  \f$\NNChull(P, C)\f$ the set of all convex combinations
of the vectors in \f$S\f$
such that \f$\lambda_j > 0\f$ for some \f$\vect{x}_j \in P\f$
(informally, we say that there exists a vector of \f$P\f$ that
plays an active role in the convex combination).
Note that \f$\NNChull(P, C) = \NNChull(P, P \union C)\f$
so that, if \f$C \sseq P\f$,
\f[
   \convexhull(P)
     = \NNChull(P, \emptyset)
     = \NNChull(P, P)
     = \NNChull(P, C).
\f]
It can be observed that
\f$\linearhull(S)\f$ is an affine space,
\f$\conichull(S)\f$ is a topologically closed convex cone,
\f$\convexhull(S)\f$ is a topologically closed polytope, and
\f$\NNChull(P, C)\f$ is an NNC polytope.


\subsection Points_Closure_Points_Rays_and_Lines Points, Closure Points, Rays and Lines

Let \f$\cP \in \Pset_n\f$ be an NNC polyhedron. Then
- a vector \f$\vect{p} \in \cP\f$ is called a <EM>point</EM> of \f$\cP\f$;
- a vector \f$\vect{c} \in \Rset^n\f$ is called a
  <EM>closure point</EM> of \f$\cP\f$ if it is a point
  of the topological closure of \f$\cP\f$;
- a vector \f$\vect{r} \in \Rset^n\f$, where \f$\vect{r} \neq \vect{0}\f$,
  is called a <EM>ray</EM> (or direction of infinity) of \f$\cP\f$
  if \f$\cP \neq \emptyset\f$ and \f$\vect{p} + \lambda \vect{r} \in \cP\f$,
  for all points \f$\vect{p} \in \cP\f$ and all \f$\lambda \in \nonnegRset\f$;
- a vector \f$\vect{l} \in \Rset^n\f$ is called a <EM>line</EM> of \f$\cP\f$
  if both \f$\vect{l}\f$ and \f$-\vect{l}\f$ are rays of \f$\cP\f$.

A point of an NNC polyhedron \f$\cP \in \Pset_n\f$ is a
<EM>vertex</EM> if and only if
it cannot be expressed as a convex combination of any other pair of
distinct points in \f$\cP\f$.
A ray \f$\vect{r}\f$ of a polyhedron \f$\cP\f$ is an <EM>extreme ray</EM>
if and only if it cannot be expressed as
a positive combination of any other pair
\f$\vect{r}_1\f$ and \f$\vect{r}_2\f$ of rays of \f$\cP\f$,
where
\f$\vect{r} \neq \lambda \vect{r}_1\f$,
\f$\vect{r} \neq \lambda \vect{r}_2\f$ and
\f$\vect{r}_1 \neq \lambda \vect{r}_2\f$
for all \f$\lambda \in \nonnegRset\f$
(i.e., rays differing by a positive scalar factor are considered
to be the same ray).


\subsection Generators_Representation Generators Representation

Each NNC polyhedron \f$\cP \in \Pset_n\f$ can be represented by
finite sets of lines \f$L\f$, rays \f$R\f$,
points \f$P\f$ and closure points \f$C\f$ of \f$\cP\f$.
The 4-tuple \f$\cG = (L, R, P, C)\f$ is said to be
a <EM>generator system</EM> for \f$\cP\f$,
in the sense that
\f[
  \cP = \linearhull(L) + \conichull(R) + \NNChull(P, C),
\f]
where the symbol '\f$+\f$' denotes the Minkowski's sum.

When \f$\cP \in \CPset_n\f$ is a closed polyhedron,
then it can be represented by
finite sets of lines \f$L\f$, rays \f$R\f$ and
points \f$P\f$ of \f$\cP\f$.
In this case, the 3-tuple \f$\cG = (L, R, P)\f$ is said to be
a <EM>generator system</EM> for \f$\cP\f$ since we have
\f[
  \cP = \linearhull(L) + \conichull(R) + \convexhull(P).
\f]
Thus, in this case,
every closure point of \f$\cP\f$ is a point of \f$\cP\f$.

For any \f$\cP \in \Pset_n\f$ and generator system \f$\cG = (L, R, P, C)\f$
for \f$\cP\f$, we have
 \f$\cP = \emptyset\f$ if and only if \f$P = \emptyset\f$.
Also \f$P\f$ must contain all the vertices of \f$\cP\f$
although \f$\cP\f$ can be non-empty and have no vertices.
In this case, as \f$P\f$ is necessarily non-empty, it must
contain points of \f$\cP\f$ that are <EM>not</EM> vertices.
For instance, the half-space of \f$\Rset^2\f$ corresponding to the single
constraint \f$y \geq 0\f$ can be represented by the generator system
\f$\cG = (L, R, P, C)\f$ such that
\f$L = \bigl\{ (1, 0)^\transpose \bigr\}\f$,
\f$R = \bigl\{ (0, 1)^\transpose \bigr\}\f$,
\f$P = \bigl\{ (0, 0)^\transpose \bigr\}\f$, and
\f$C = \emptyset\f$.
It is also worth noting that the only ray in \f$R\f$
is <EM>not</EM> an extreme ray of \f$\cP\f$.

\subsection Minimized_Representations Minimized Representations

A constraints system \f$\cC\f$ for an NNC polyhedron \f$\cP \in \Pset_n\f$
is said to be <EM>minimized</EM> if no proper subset of \f$\cC\f$ is
a constraint system for \f$\cP\f$.

Similarly, a generator system \f$\cG = (L, R, P, C)\f$
for an NNC polyhedron \f$\cP \in \Pset_n\f$ is said to be <EM>minimized</EM>
if there does not exist a generator system
\f$\cG' = (L', R', P', C') \neq \cG\f$ for \f$\cP\f$
such that \f$L' \sseq L\f$, \f$R' \sseq R\f$, \f$P' \sseq P\f$ and
\f$C' \sseq C\f$.


\subsection Double_Description Double Description

Any NNC polyhedron \f$\cP\f$ can be described by using
a constraint system \f$\cC\f$, a generator system \f$\cG\f$, or
both by means of the <EM>double description pair (DD pair)</EM>
\f$(\cC, \cG)\f$.
The <EM>double description method</EM> is a collection
of well-known as well as novel theoretical results showing that,
given one kind of representation, there are algorithms
for computing a representation of the other kind
and for minimizing both representations by removing
redundant constraints/generators.

Such changes of representation form a key step
in the implementation of many operators on NNC polyhedra:
this is because some operators, such as intersections and poly-hulls,
are provided with a natural and efficient implementation
when using one of the representations in a DD pair,
while being rather cumbersome when using the other.


\subsection Topologies_and_Topological_compatibility Topologies and Topological-compatibility

As indicated above, when an NNC polyhedron \f$\cP\f$ is necessarily closed,
we can ignore the closure points contained in its generator system
\f$\cG = (L, R, P, C)\f$ (as every closure point is also a point)
and represent \f$\cP\f$ by the triple \f$(L, R, P)\f$.
Similarly, \f$\cP\f$ can be represented by a constraint system
that has no strict inequalities.
Thus a necessarily closed polyhedron can have a smaller representation
than one that is not necessarily closed.
Moreover, operators restricted to work on closed polyhedra only
can be implemented more efficiently.
For this reason the library provides two alternative
``topological kinds'' for a polyhedron, <EM>NNC</EM> and <EM>C</EM>.
We shall abuse terminology by referring to the topological kind of
a polyhedron as its <EM>topology</EM>.

In the library, the topology of each polyhedron object is fixed
once for all at the time of its creation and must be respected
when performing operations on the polyhedron.

Unless it is otherwise stated,
all the polyhedra, constraints and/or generators in any library operation
must obey the following <EM>topological-compatibility</EM> rules:
- polyhedra are topologically-compatible if and only if they have
  the same topology;
- all constraints except for strict inequality constraints
  and all generators except for closure points are
  topologically-compatible with both C and NNC polyhedra;
- strict inequality constraints and closure points are
  topologically-compatible with a polyhedron if and only if it is NNC.

Wherever possible,
the library provides methods that, starting from a polyhedron of
a given topology, build the corresponding polyhedron having
the other topology.


\subsection Space_Dimensions_and_Dimension_Compatibility Space Dimensions and Dimension Compatibility

The <EM>space dimension</EM> of an NNC polyhedron \f$\cP \in \Pset_n\f$
(resp., a C polyhedron \f$\cP \in \CPset_n\f$) is the dimension
\f$n \in \Nset\f$ of the corresponding vector space \f$\Rset^n\f$.
The space dimension of constraints, generators and other objects
of the library is defined similarly.

Unless it is otherwise stated,
all the polyhedra, constraints and/or generators in any library operation
must obey the following (space) <EM>dimension-compatibility</EM> rules:
- polyhedra are dimension-compatible if and only if they have
  the same space dimension;
- the constraint
  \f$\langle \vect{a}, \vect{x} \rangle \relsym b\f$
  where \f$\mathord{\relsym} \in \{ =, \geq, > \}\f$
  and \f$\vect{a}, \vect{x} \in \Rset^m\f$, is dimension-compatible
  with a polyhedron having space dimension \f$n\f$
  if and only if \f$m \leq n\f$;
- the generator \f$\vect{x} \in \Rset^m\f$ is dimension-compatible
  with a polyhedron having space dimension \f$n\f$
  if and only if \f$m \leq n\f$;
- a system of constraints (resp., generators) is dimension-compatible with
  a polyhedron if and only if all the constraints (resp., generators)
  in the system are dimension-compatible with the polyhedron.

While the space dimension of a constraint, a generator or a system thereof
is automatically adjusted when needed, the space dimension of a polyhedron
can only be changed by explicit calls to operators provided for that
purpose.

\subsection Affine_Independence_and_Affine_Dimension Affine Independence and Affine Dimension

A finite set of points
\f$\{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n\f$ is
<EM>affinely independent</EM> if,
for all \f$\lambda_1, \ldots, \lambda_k \in \Rset\f$,
the system of equations
\f[
  \sum_{i = 1}^k \lambda_i \vect{x}_i = \vect{0},
    \quad
  \sum_{i = 1}^k \lambda_i = 0
\f]
implies that, for each \f$i = 1, \ldots, k\f$, \f$\lambda_i = 0\f$.

The maximum number of affinely independent points in \f$\Rset^n\f$
is \f$n + 1\f$.

A <EM>non-empty</EM> NNC polyhedron \f$\cP \in \Pset_n\f$ has
<EM>affine dimension</EM> \f$k \in \Nset\f$, denoted by \f$\pdim(\cP) = k\f$,
if the maximum number of affinely independent points in \f$\cP\f$ is
\f$k + 1\f$.

We remark that the above definition only applies to polyhedra
that are not empty, so that \f$0 \leq \pdim(\cP) \leq n\f$.
By convention, the affine dimension of an empty polyhedron is 0
(even though the ``natural'' generalization of the definition above
would imply that the affine dimension of an empty polyhedron is \f$-1\f$).

\note
The affine dimension \f$k \leq n\f$ of an NNC polyhedron \f$\cP \in \Pset_n\f$
must not be confused with the space dimension \f$n\f$ of \f$\cP\f$,
which is the dimension of the enclosing vector space \f$\Rset^n\f$.
In particular, we can have \f$\pdim(\cP) \neq \pdim(\cQ)\f$ even though
\f$\cP\f$ and \f$\cQ\f$ are dimension-compatible;
and vice versa, \f$\cP\f$ and \f$\cQ\f$ may be dimension-incompatible
polyhedra even though \f$\pdim(\cP) = \pdim(\cQ)\f$.


\subsection Rational_Polyhedra Rational Polyhedra

An NNC polyhedron is called <EM>rational</EM>
if it can be represented by a constraint system where
all the constraints have rational coefficients.
It has been shown that an NNC polyhedron is rational
if and only if it can be represented by a generator system where
all the generators have rational coefficients.

The library only supports rational polyhedra.
The restriction to rational numbers applies not only to polyhedra,
but also to the other numeric arguments that may be required
by the operators considered, such as the coefficients defining
(rational) affine transformations.


\section Operations_on_Convex_Polyhedra Operations on Convex Polyhedra

In this section we briefly describe operations on NNC polyhedra
that are provided by the library.

\subsection Intersection_and_Convex_Polyhedral_Hull Intersection and Convex Polyhedral Hull

For any pair of NNC polyhedra \f$\cP_1, \cP_2 \in \Pset_n\f$,
the <EM>intersection</EM> of \f$\cP_1\f$ and \f$\cP_2\f$,
defined as the set intersection \f$\cP_1 \inters \cP_2\f$,
is the biggest NNC polyhedron included in both \f$\cP_1\f$ and \f$\cP_2\f$;
similarly, the <EM>convex polyhedral hull</EM> (or <EM>poly-hull</EM>)
of \f$\cP_1\f$ and \f$\cP_2\f$, denoted by \f$\cP_1 \uplus \cP_2\f$,
is the smallest NNC polyhedron that includes both
\f$\cP_1\f$ and \f$\cP_2\f$.
The intersection and poly-hull of any pair of closed polyhedra
in \f$\CPset_n\f$ is also closed.

In theoretical terms,
the intersection and poly-hull operators defined above
are the binary <EM>meet</EM> and the binary <EM>join</EM>
operators on the lattices \f$\Pset_n\f$ and \f$\CPset_n\f$.

\subsection Convex_Polyhedral_Difference Convex Polyhedral Difference

For any pair of NNC polyhedra \f$\cP_1, \cP_2 \in \Pset_n\f$,
the <EM>convex polyhedral difference</EM> (or <EM>poly-difference</EM>)
of \f$\cP_1\f$ and \f$\cP_2\f$ is defined as the smallest
convex polyhedron containing the set-theoretic difference
of \f$\cP_1\f$ and \f$\cP_2\f$.

In general, even though \f$\cP_1, \cP_2 \in \CPset_n\f$
are topologically closed polyhedra, their poly-difference may be
a convex polyhedron that is not topologically closed.
For this reason, when computing the poly-difference of two
C polyhedra, the library will enforce the topological closure
of the result.

\subsection Concatenating_Polyhedra Concatenating Polyhedra

Viewing a polyhedron as a set of tuples (its points), it is sometimes
useful to consider the set of tuples obtained by concatenating
an ordered pair of polyhedra.
Formally, the <EM>concatenation</EM> of the polyhedra
\f$\cP \in \Pset_n\f$ and \f$\cQ \in \Pset_m\f$ (taken in this order)
is the polyhedron \f$\cR \in \Pset_{n+m}\f$ such that
\f[
  \cR
    \defeq
      \Bigl\{\,
        (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose
          \in \Rset^{n+m}
      \Bigm|
        (x_0, \ldots, x_{n-1})^\transpose \in \cP,
        (y_0, \ldots, y_{m-1})^\transpose \in \cQ
      \,\Bigl\}.
\f]
Another way of seeing it is as follows: first embed polyhedron \f$\cP\f$
into a vector space of dimension \f$n+m\f$ and then add a suitably
renamed-apart version of the constraints defining \f$\cQ\f$.

\subsection Adding_New_Dimensions_to_the_Vector_Space Adding New Dimensions to the Vector Space

The library provides two operators for adding a number \f$i\f$ of
space dimensions to an NNC polyhedron \f$\cP \in \Pset_n\f$,
therefore transforming it into a new NNC polyhedron
\f$\cQ \in \Pset_{n+i}\f$.
In both cases, the added dimensions of the vector space
are those having the highest indices.

The operator <CODE>add_space_dimensions_and_embed</CODE>
\e embeds the polyhedron \f$\cP\f$ into the new vector space of
dimension \f$i+n\f$ and returns the polyhedron \f$\cQ\f$
defined by all and only the constraints defining \f$\cP\f$
(the variables corresponding to the added dimensions are unconstrained).
For instance, when starting from a polyhedron \f$\cP \sseq \Rset^2\f$
and adding a third space dimension, the result will be the polyhedron
\f[
 \cQ = \bigl\{\,
         (x_0, x_1, x_2)^\transpose \in \Rset^3
       \bigm|
         (x_0, x_1)^\transpose \in \cP
       \,\bigr\}.
\f]

In contrast, the operator  <CODE>add_space_dimensions_and_project</CODE>
\e projects the polyhedron \f$\cP\f$ into the new vector space of
dimension \f$i+n\f$ and returns the polyhedron \f$\cQ\f$
whose constraint system, besides the constraints defining \f$\cP\f$,
will include additional constraints on the added dimensions.
Namely, the corresponding variables are all constrained to be equal to 0.
For instance, when starting from a polyhedron \f$\cP \sseq \Rset^2\f$
and adding a third space dimension, the result will be the polyhedron
\f[
  \cQ = \bigl\{\,
          (x_0, x_1, 0)^\transpose \in \Rset^3
        \bigm|
          (x_0, x_1)^\transpose \in \cP
        \,\bigr\}.
\f]

\subsection Removing_Dimensions_from_the_Vector_Space Removing Dimensions from the Vector Space

The library provides two operators for removing
space dimensions from an NNC polyhedron \f$\cP \in \Pset_n\f$,
therefore transforming it into a new NNC polyhedron
\f$\cQ \in \Pset_m\f$ where \f$m \leq n\f$.

Given a set of variables, the operator <CODE>remove_space_dimensions</CODE>
removes all the space dimensions specified by the variables in the set.
For instance, letting \f$\cP \in \Pset_4\f$ be the singleton set
\f$\bigl\{ (3, 1, 0, 2)^\transpose \bigr\} \sseq \Rset^4\f$,
then after invoking this operator with the set of variables
\f$\{x_1, x_2\}\f$ the resulting polyhedron is
\f[
  \cQ = \bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2.
\f]

Given a space dimension \f$m\f$ less than or equal to that of the polyhedron,
the operator <CODE>remove_higher_space_dimensions</CODE> removes
the space dimensions having indices greater than or equal to \f$m\f$.
For instance, letting \f$\cP \in \Pset_4\f$ defined as before,
by invoking this operator with \f$m = 2\f$ the resulting polyhedron
will be
\f[
  \cQ = \bigl\{ (3, 1)^\transpose \bigr\} \sseq \Rset^2.
\f]

\subsection Mapping_the_Dimensions_of_the_Vector_Space Mapping the Dimensions of the Vector Space

The operator <CODE>map_space_dimensions</CODE> provided by the library
maps the dimensions of the vector space \f$\Rset^n\f$ according
to a partial injective function \f$\pard{\rho}{\{0, \ldots, n-1\}}{\Nset}\f$
such that \f$\rho\bigl(\{0, \ldots, n-1\}\bigr) = \{0, \ldots, m-1\}\f$
with \f$m \leq n\f$.
Dimensions corresponding to indices that are not mapped
by \f$\rho\f$ are removed.

If \f$m = 0\f$, i.e., if the function \f$\rho\f$ is undefined everywhere,
then the operator projects the argument polyhedron \f$\cP \in \Pset_n\f$
onto the zero-dimension space \f$\Rset^0\f$;
otherwise the result is \f$\cQ \in \Pset_m\f$ given by
\f[
  \cQ
    \defeq
      \Bigl\{\,
        \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose
      \Bigm|
        (v_0, \ldots, v_{n-1})^\transpose \in \cP
      \,\Bigr\}.
\f]

\anchor expand_space_dimension
\subsection Expanding_One_Dimension_of_the_Vector_Space_to_Multiple_Dimensions Expanding One Dimension of the Vector Space to Multiple Dimensions

The operator <CODE>expand_space_dimension</CODE> provided by the library
adds \f$m\f$ new space dimensions to a polyhedron \f$\cP \in \Pset_n\f$,
with \f$n > 0\f$, so that dimensions
\f$n\f$, \f$n+1\f$, \f$\ldots\f$, \f$n+m-1\f$
of the result \f$\cQ\f$ are exact copies of the \f$i\f$-th space dimension
of \f$\cP\f$.  More formally,
\f[
  \cQ
    \defeq
      \sset{
        \vect{u} \in \Rset^{n+m}
      }{
        \exists \vect{v}, \vect{w} \in \cP
          \st u_i = v_i \\
          \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1
                                     \itc u_j = w_i \\
          \qquad \mathord{} \land \forall k = 0, \ldots, n-1
                                     \itc k \neq i \implies u_k = v_k = w_k
      }.
\f]
This operation has been proposed in \ref GDDetal04 "[GDDetal04]".


\anchor fold_space_dimensions
\subsection Folding_Multiple_Dimensions_of_the_Vector_Space_into_One_Dimension Folding Multiple Dimensions of the Vector Space into One Dimension

The operator <CODE>fold_space_dimensions</CODE> provided by the library,
given a polyhedron \f$\cP \in \Pset_n\f$, with \f$n > 0\f$,
folds a set of space dimensions \f$J = \{ j_0, \ldots, j_{m-1} \}\f$,
with \f$m < n\f$ and \f$j < n\f$ for each \f$j \in J\f$,
into space dimension \f$i < n\f$, where \f$i \notin J\f$.
The result is given by
\f[
  \cQ \defeq \biguplus_{d = 0}^m \cQ_d
\f]
where
\f[
  \cQ_m
    \defeq
      \sset{
        \vect{u} \in \Rset^{n-m}
      }{
        \exists \vect{v} \in \cP
          \st u_{i'} = v_i \\
          \qquad \mathord{} \land \forall k = 0, \ldots, n-1
                                     \itc k \neq i \implies u_{k'} = v_k
      }
\f]
and, for \f$ d = 0 \f$, \f$ \ldots \f$, \f$ m-1 \f$,
\f[
  \cQ_d
    \defeq
      \sset{
        \vect{u} \in \Rset^{n-m}
      }{
        \exists \vect{v} \in \cP
          \st u_{i'} = v_{j_d} \\
          \qquad \mathord{} \land \forall k = 0, \ldots, n-1
                                     \itc k \neq i \implies u_{k'} = v_k
      },
\f]
and, finally, for \f$ k = 0 \f$, \f$ \ldots \f$, \f$ n-1 \f$,
\f[
  k'
    \defeq
      k - \card \{\, j \in J \mid k > j \,\},
\f]
(\f$\card S\f$ denotes the cardinality of the finite set \f$S\f$).

This operation has been proposed in \ref GDDetal04 "[GDDetal04]".


\anchor affine_relation
\subsection Images_and_Preimages_of_Affine_Transfer_Relations Images and Preimages of Affine Transfer Relations

For each relation \f$\reld{\phi}{\Rset^n}{\Rset^m}\f$,
we denote by \f$\phi(S) \sseq \Rset^m\f$ the <EM>image</EM>
under \f$\phi\f$ of the set \f$S \sseq \Rset^n\f$; formally,
\f[
  \phi(S)
    \defeq
      \bigl\{\,
        \vect{w} \in \Rset^m
      \bigm|
        \exists \vect{v} \in S \st
          (\vect{v}, \vect{w}) \in \phi
      \,\bigr\}.
\f]
Similarly, we denote by \f$\phi^{-1}(S') \sseq \Rset^n\f$
the <EM>preimage</EM> under \f$\phi\f$ of \f$S' \sseq \Rset^m\f$,
that is
\f[
  \phi^{-1}(S')
    \defeq
      \bigl\{\,
        \vect{v} \in \Rset^n
      \bigm|
        \exists \vect{w} \in S' \st
          (\vect{v}, \vect{w}) \in \phi
      \,\bigr\}.
\f]
If \f$n = m\f$, then the relation \f$\phi\f$ is said to be
<EM>space dimension preserving</EM>.

The relation \f$\reld{\phi}{\Rset^n}{\Rset^m}\f$
is said to be an <EM>affine relation</EM>
if there exists \f$\ell \in \Nset\f$ such that
\f[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigland_{i=1}^{\ell}
            \bigl(
              \langle \vect{c}_i, \vect{w} \rangle
                \relsym_i
                  \langle \vect{a}_i, \vect{v} \rangle + b_i
            \bigr),
\f]
where
\f$\vect{a}_i \in \Rset^n\f$,
\f$\vect{c}_i \in \Rset^m\f$,
\f$b_i \in \Rset\f$ and
\f$\mathord{\relsym}_i \in \{ <, \leq, =, \geq, > \}\f$,
for each \f$i = 1, \ldots, \ell\f$.

As a special case, the relation \f$\reld{\phi}{\Rset^n}{\Rset^m}\f$
is an <EM>affine function</EM> if and only if there exist
a matrix \f$A \in \Rset^m \times \Rset^n\f$ and
a vector \f$\vect{b} \in \Rset^m\f$ such that,
\f[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^m
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \vect{w} = A\vect{v} + \vect{b}.
\f]

The set \f$\Pset_n\f$ of NNC polyhedra is closed under the application
of images and preimages of any space dimension preserving affine relation.
The same property holds for the set \f$\CPset_n\f$ of closed polyhedra,
provided the affine relation makes no use of the strict relation symbols
\f$<\f$ and \f$>\f$.
Images and preimages of affine relations can be used to model
several kinds of transition relations, including
deterministic assignments of affine expressions,
(affinely constrained) nondeterministic assignments and
affine conditional guards.

A space dimension preserving relation
\f$\reld{\phi}{\Rset^n}{\Rset^n}\f$
can be specified by means of a shorthand notation:
 - the vector \f$\vect{x} = (x_0, \ldots, x_{n-1})^\transpose\f$
   of <EM>unprimed</EM> variables is used to represent the
   space dimensions of the domain of \f$\phi\f$;
 - the vector \f$\vect{x}' = (x'_0, \ldots, x'_{n-1})^\transpose\f$
   of <EM>primed</EM> variables is used to represent the
   space dimensions of the range of \f$\phi\f$;
 - any primed variable that ``does not occur'' in the shorthand
   specification is meant to be <EM>unaffected</EM> by the relation;
   namely, for each index \f$i \in \{0, \ldots, n-1\}\f$,
   if in the syntactic specification of the relation
   the primed variable \f$x'_i\f$ only occurs (if ever) with
   coefficient 0, then it is assumed that the specification also
   contains the constraint \f$x'_i = x_i\f$.

As an example, assuming \f$\reld{\phi}{\Rset^3}{\Rset^3}\f$,
the notation \f$x'_0 - x'_2 \geq 2 x_0 - x_1\f$, where the primed
variable \f$x'_1\f$ does not occur, is meant to specify
the affine relation defined by
\f[
  \forall \vect{v} \in \Rset^3, \vect{w} \in \Rset^3
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          (w_0 - w_2 \geq 2 v_0 - v_1) \land (w_1 = v_1).
\f]
The same relation is specified by
\f$x'_0 + 0 \cdot x'_1 - x'_2 \geq 2 x_0 - x_1\f$,
since \f$x'_1\f$ occurs with coefficient 0.

The library allows for the computation of images and preimages
of polyhedra under restricted subclasses of space dimension preserving
affine relations, as described in the following.

\subsection Single_Update_Affine_Functions Single-Update Affine Functions.
Given a primed variable \f$x'_k\f$ and an unprimed affine expression
\f$\langle \vect{a}, \vect{x} \rangle + b\f$,
the <EM>affine function</EM>
\f$\fund{\phi = \bigl(x'_k = \langle \vect{a}, \vect{x} \rangle + b\bigr)}
        {\Rset^n}{\Rset^n}\f$
is defined by
\f[
  \forall \vect{v} \in \Rset^n \itc
    \phi(\vect{v}) = A\vect{v} + \vect{b},
\f]
where
\f[
 A =
   \begin{pmatrix}
      1      &        &    0    &   0    & \cdots  & \cdots &    0    \\
             & \ddots &         & \vdots &         &        & \vdots  \\
      0      &        &    1    &   0    & \cdots  & \cdots &    0    \\
     a_0     & \cdots & a_{k-1} &  a_k   & a_{k+1} & \cdots & a_{n-1} \\
      0      & \cdots & \cdots  &   0    &   1     &        &    0    \\
    \vdots   &        &         & \vdots &         & \ddots &         \\
      0      & \cdots & \cdots  &   0    &   0     &        &    1
   \end{pmatrix}, \qquad
 \vect{b} =
   \begin{pmatrix}
     0      \\
     \vdots \\
     0      \\
     b      \\
     0      \\
     \vdots \\
     0
   \end{pmatrix}
\f]
and the \f$a_i\f$ (resp., \f$b\f$) occur in the \f$(k+1)\f$st row
in \f$A\f$ (resp., position in \f$\vect{b}\f$).
Thus function \f$\phi\f$ maps any vector
\f$(v_0, \ldots, v_{n-1})^\transpose\f$
to
\f[
   \Bigl(v_0, \ldots,
           \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i v_i + b\bigr),
             \ldots, v_{n-1}\Bigr)^\transpose.
\f]

The <EM>affine image</EM> operator computes the affine image
of a polyhedron \f$\cP\f$ under
\f$x'_k = \langle \vect{a}, \vect{x} \rangle + b\f$.
For instance, suppose the polyhedron \f$\cP\f$
to be transformed is the square in \f$\Rset^2\f$
generated by the set of points
\f$\bigl\{
     (0, 0)^\transpose,
     (0, 3)^\transpose,
     (3, 0)^\transpose,
     (3, 3)^\transpose
   \bigr\}\f$.
Then, if the primed variable is \f$x_0\f$ and
the affine expression is \f$x_0 + 2 x_1 + 4\f$
(so that \f$k = 0\f$, \f$a_0 = 1, a_1 = 2, b = 4\f$),
the affine image operator will translate \f$\cP\f$
to the parallelogram \f$\cP_1\f$ generated by the set of points
\f$\bigl\{
     (4, 0)^\transpose,
     (10, 3)^\transpose,
     (7, 0)^\transpose,
     (13, 3)^\transpose
   \bigr\}\f$
with height equal to the side of the square
and oblique sides parallel to the line \f$x_0 - 2 x_1\f$.
If the primed variable is as before (i.e., \f$k = 0\f$)
but the affine expression is \f$x_1\f$
(so that \f$a_0 = 0, a_1 = 1, b = 0\f$),
then the resulting polyhedron \f$\cP_2\f$
is the positive diagonal of the square.

The <EM>affine preimage</EM> operator computes the affine preimage
of a polyhedron \f$\cP\f$ under
\f$x'_k = \langle \vect{a}, \vect{x} \rangle + b\f$.
For instance,
suppose now that we apply the affine
preimage operator as given in the first example
using primed variable \f$x_0\f$
and affine expression \f$x_0 + 2 x_1 + 4\f$
to the parallelogram \f$\cP_1\f$;
then we get the original square \f$\cP\f$ back.
If, on the other hand, we apply the affine
preimage operator as given in the second example
using primed variable \f$x_0\f$ and affine expression \f$x_1\f$
to \f$\cP_2\f$, then the resulting polyhedron is the stripe obtained
by adding the line \f$(1, 0)^\transpose\f$ to polyhedron \f$\cP_2\f$.

Observe that provided the coefficient \f$a_k\f$ of the considered variable
in the affine expression is non-zero, the affine function is invertible.

\subsection Single_Update_Bounded_Affine_Relations Single-Update Bounded Affine Relations.
Given a primed variable \f$x'_k\f$ and
two unprimed affine expressions
\f$\mathrm{lb} = \langle \vect{a}, \vect{x} \rangle + b\f$ and
\f$\mathrm{ub} = \langle \vect{c}, \vect{x} \rangle + d\f$,
the <EM>bounded affine relation</EM>
\f$\phi = (\mathrm{lb} \leq x'_k \leq \mathrm{ub})\f$
is defined as
\f[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigl(
            \langle \vect{a}, \vect{v} \rangle + b
              \leq w_k
                \leq \langle \vect{c}, \vect{v} \rangle + d
          \bigr)
            \land
          \Bigl(
            \bigland_{0 \leq i < n, i \neq k}
              w_i = v_i
          \Bigr).
\f]

\subsection Generalized_Affine_Relations Generalized Affine Relations.
Similarly, the <EM>generalized affine relation</EM>
\f$\phi = (\mathrm{lhs}' \relsym \mathrm{rhs})\f$, where
\f$\mathrm{lhs} = \langle \vect{c}, \vect{x} \rangle + d\f$ and
\f$\mathrm{rhs} = \langle \vect{a}, \vect{x} \rangle + b\f$
are affine expressions and
\f$\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}\f$
is a relation symbol, is defined as
\f[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigl(
            \langle \vect{c}, \vect{w} \rangle + d
              \relsym \langle \vect{a}, \vect{v} \rangle + b
          \bigr)
            \land
          \Bigl(
            \bigland_{0 \leq i < n, c_i = 0}
              w_i = v_i
          \Bigr).
\f]
When \f$\mathrm{lhs} = x_k\f$ and \f$\mathord{\relsym} \in \{ = \}\f$,
then the above affine relation becomes equivalent to the
single-update affine function \f$x'_k = \mathrm{rhs}\f$
(hence the name given to this operator).
It is worth stressing that the notation is not symmetric, because
the variables occurring in expression \f$\mathrm{lhs}\f$ are interpreted
as primed variables, whereas those occurring in \f$\mathrm{rhs}\f$
are unprimed; for instance, the transfer relations
\f$\mathrm{lhs}' \leq \mathrm{rhs}\f$ and
\f$\mathrm{rhs}' \geq \mathrm{lhs}\f$
are not equivalent in general.

\subsection Cylindrification Cylindrification Operator

The operator <CODE>unconstrain</CODE> computes
the <EM>cylindrification</EM> \ref HMT71 "[HMT71]"
of a polyhedron with respect to one of its variables.
Formally, the cylindrification \f$\cQ \in \Pset_n\f$
of an NNC polyhedron \f$\cP \in \Pset_n\f$ with respect to
variable index \f$i \in \{ 0, \ldots, n-1 \}\f$ is defined as follows:
\f[
  \cQ
    = \bigl\{\,
        \vect{w} \in \Rset^n
      \bigm|
        \exists \vect{v} \in \cP \st
          \forall j \in \{0, \ldots, n-1\} \itc
            j \neq i \implies w_j = v_j
      \,\bigr\}.
\f]
Cylindrification is an idempotent operation; in particular, note that
the computed result has the same space dimension of the original polyhedron.
A variant of the operator above allows for the cylindrification of
a polyhedron with respect to a finite set of variables.

\subsection Time_Elapse_Operator Time-Elapse Operator

The <EM>time-elapse</EM> operator has been defined in
\ref HPR97 "[HPR97]".
Actually, the time-elapse operator provided by the library
is a slight generalization of that one,
since it also works on NNC polyhedra.
For any two NNC polyhedra \f$\cP, \cQ \in \Pset_n\f$,
the time-elapse between \f$\cP\f$ and \f$\cQ\f$,
denoted \f$ \cP \nearrow \cQ\f$,
is the smallest NNC polyhedron containing the set
\f[
  \bigl\{\,
    \vect{p} + \lambda \vect{q} \in \Rset^n
  \bigm|
    \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset
  \,\bigr\}.
\f]
Note that, if \f$\cP,\cQ \in \CPset_n\f$ are closed polyhedra,
the above set is also a closed polyhedron.
In contrast, when \f$\cQ\f$ is not topologically closed,
the above set might not be an NNC polyhedron.

\subsection Meet_Preserving_Simplification Meet-Preserving Enlargement and Simplification

Let \f$\cP, \cQ, \cR \in \Pset_n\f$ be NNC polyhedra. Then:

 - \f$\cR\f$ is <EM>meet-preserving</EM> with respect to \f$\cP\f$
   using context \f$\cQ\f$ if \f$\cR \inters \cQ = \cP \inters \cQ\f$;

 - \f$\cR\f$ is an <EM>enlargement</EM> of \f$\cP\f$ if \f$\cR \Sseq \cP\f$.

 - \f$\cR\f$ is a <EM>simplification</EM> with respect to \f$\cP\f$
   if \f$r \leq p\f$, where \f$r\f$ and \f$p\f$ are the cardinalities
   of minimized constraint representations for \f$\cR\f$ and \f$\cP\f$,
   respectively.

Notice that an enlargement need not be a simplification, and vice versa;
moreover, the identity function is (trivially)
a meet-preserving enlargement and simplification.

The library provides a binary operator (<CODE>simplify_using_context</CODE>)
for the domain of NNC polyhedra that returns a polyhedron which is a
meet-preserving enlargement simplification of its first argument
using the second argument as context.

The concept of meet-preserving enlargement and simplification also
applies to the other basic domains (boxes, grids, BD and octagonal shapes).
See below for a definition of the concept of
\ref Powerset_Meet_Preserving_Simplification "meet-preserving simplification for powerset domains".

\anchor relation_with
\subsection Relation_With_Operators Relation-With Operators

The library provides operators for checking the relation holding
between an NNC polyhedron  and
either a constraint or a generator.

Suppose \f$\cP\f$ is an NNC polyhedron
and \f$\cC\f$ an arbitrary constraint system representing \f$\cP\f$.
Suppose also that
\f$
  c = \bigl(
        \langle \vect{a}, \vect{x} \rangle \relsym b
      \bigr)
\f$
is a constraint
with \f$\mathord{\relsym} \in \{ =, \geq, > \}\f$
and \f$\cQ\f$ the set of points
that satisfy \f$c\f$.
The possible relations between \f$\cP\f$ and \f$c\f$ are as follows.
- \f$\cP\f$ <EM>is disjoint</EM> from \f$c\f$ if
  \f$\cP \inters \cQ = \emptyset\f$; that is, adding \f$c\f$
  to \f$\cC\f$ gives us the empty polyhedron.
- \f$\cP\f$ <EM>strictly intersects</EM> \f$c\f$ if
  \f$\cP \inters \cQ \neq \emptyset\f$ and \f$\cP \inters \cQ \subset \cP\f$;
  that is, adding \f$c\f$ to \f$\cC\f$
  gives us a non-empty polyhedron strictly smaller than \f$\cP\f$.
- \f$\cP\f$ <EM>is included</EM> in \f$c\f$ if \f$\cP \sseq \cQ\f$;
  that is, adding \f$c\f$ to \f$\cC\f$
  leaves \f$\cP\f$ unchanged.
- \f$\cP\f$ <EM>saturates</EM> \f$c\f$ if \f$\cP \sseq \cH\f$,
  where \f$\cH\f$ is the hyperplane induced by constraint \f$c\f$,
  i.e., the set of points satisfying the equality constraint
  \f$\langle \vect{a}, \vect{x} \rangle = b\f$;
  that is, adding the constraint \f$\langle \vect{a}, \vect{x} \rangle = b\f$
  to \f$\cC\f$ leaves \f$\cP\f$ unchanged.

The polyhedron \f$\cP\f$ <EM>subsumes</EM> the generator \f$g\f$ if
adding \f$g\f$ to any generator system representing \f$\cP\f$
does not change \f$\cP\f$.

\subsection Widening_Operators Widening Operators

The library provides two widening operators for the domain of polyhedra.
\anchor H79_widening
The first one, that we call <EM>H79-widening</EM>, mainly
follows the specification provided in the PhD thesis of N. Halbwachs
\ref Hal79 "[Hal79]", also described in \ref HPR97 "[HPR97]".
Note that in the computation of the H79-widening \f$\cP \widen \cQ\f$
of two polyhedra \f$\cP, \cQ \in \CPset_n\f$ it is required as a
precondition that \f$\cP \sseq \cQ\f$ (the same assumption was implicitly
present in the cited papers).

\anchor BHRZ03_widening
The second widening operator, that we call <EM>BHRZ03-widening</EM>,
is an instance of the specification provided in \ref BHRZ03a "[BHRZ03a]".
This operator also requires as a precondition that \f$\cP \sseq \cQ\f$
and it is guaranteed to provide a result which is at least as precise as
the H79-widening.

Both widening operators can be applied to NNC polyhedra.
The user is warned that, in such a case, the results may not closely
match the geometric intuition which is at the base of the
specification of the two widenings. The reason is that, in the current
implementation, the widenings are not directly applied to the NNC
polyhedra, but rather to their internal representations.
Implementation work is in progress and future versions of the library
may provide an even better integration of the two widenings with the
domain of NNC polyhedra.

\note
As is the case for the other operators on polyhedra, the
implementation overwrites one of the two polyhedra arguments with the
result of the widening application. To avoid trivial
misunderstandings, it is worth stressing that if polyhedra \f$\cP\f$
and \f$\cQ\f$ (where \f$\cP \sseq \cQ\f$) are identified by program
variables <CODE>p</CODE> and <CODE>q</CODE>, respectively, then the
call <CODE>q.H79_widening_assign(p)</CODE> will assign the polyhedron
\f$\cP \widen \cQ\f$ to variable <CODE>q</CODE>.  Namely, it is the
bigger polyhedron \f$\cQ\f$ which is overwritten by the result of the
widening. The smaller polyhedron is not modified, so as to lead to an
easier coding of the usual convergence test
(\f$\cP \Sseq \cP \widen \cQ\f$ can be coded as <CODE>p.contains(q)</CODE>).
Note that, in the above context, a call such as
<CODE>p.H79_widening_assign(q)</CODE> is likely to result in undefined
behavior, since the precondition \f$\cQ \sseq \cP\f$ will be missed
(unless it happens that \f$\cP = \cQ\f$).
The same observation holds for all flavors of widenings and extrapolation
operators that are implemented in the library and for all the language
interfaces.


\subsection Widening_with_Tokens Widening with Tokens

When approximating a fixpoint computation using widening operators,
a common tactic to improve the precision of the final result is to delay
the application of widening operators. The usual approach is to fix
a parameter \f$k\f$ and only apply widenings starting from the
\f$k\f$-th iteration.

The library also supports an improved widening delay strategy,
that we call <EM>widening with tokens</EM> \ref BHRZ03a "[BHRZ03a]".
A token is a sort of wild card allowing for the replacement of the widening
application by the exact upper bound computation: the token is used
(and thus consumed) only when the widening would have resulted in an
actual precision loss (as opposed to the <EM>potential</EM> precision loss
of the classical delay strategy).
Thus, all widening operators can be supplied with an optional argument,
recording the number of available tokens, which is decremented when
tokens are used. The approximated fixpoint computation will start with
a fixed number \f$k\f$ of tokens, which will be used if and when needed.
When there are no tokens left, the widening is always applied.

\subsection Extrapolation_Operators Extrapolation Operators

Besides the two widening operators, the library also implements several
<EM>extrapolation</EM> operators, which differ from widenings in that
their use along an upper iteration sequence does not ensure convergence
in a finite number of steps.

\anchor limited_extrapolation
In particular, for each of the two widenings there is a corresponding
<EM>limited</EM> extrapolation operator, which can be used to implement
the <EM>widening ``up to''</EM> technique as described in
\ref HPR97 "[HPR97]".
Each limited extrapolation operator takes a constraint system as an
additional parameter and uses it to improve the approximation yielded
by the corresponding widening operator. Note that a convergence guarantee
can only be obtained by suitably restricting the set of constraints that
can occur in this additional parameter. For instance, in \ref HPR97 "[HPR97]"
this set is fixed once and for all before starting the computation of
the upward iteration sequence.

\anchor bounded_extrapolation
The <EM>bounded</EM> extrapolation operators further enhance each one
of the limited extrapolation operators described above by
intersecting the result of the limited extrapolation operation
with the box obtained as a result of applying the
\ref CC76_interval_widening "CC76-widening"
to the smallest \ref Intervals_and_Boxes "boxes"
enclosing the two argument polyhedra.

\section Intervals_and_Boxes Intervals and Boxes

The PPL provides support for computations on non-relational domains,
called boxes, and also the interval domains used for their representation.

\anchor intervals
An <EM>interval</EM> in \f$\Rset\f$ is a pair of <EM>bounds</EM>,
called <EM>lower</EM> and <EM>upper</EM>.
Each bound can be either (1) <EM>closed and bounded</EM>,
(2) <EM>open and bounded</EM>, or
(3) <EM>open and unbounded</EM>.
If the bound is <EM>bounded</EM>, then it has a value in \f$\Rset\f$.
For each vector \f$\vect{a} \in \Rset^n\f$ and scalar \f$b \in \Rset\f$,
and for each relation symbol \f$\mathord{\relsym} \in \{ =, \geq, >\}\f$,
the constraint \f$\langle \vect{a}, \vect{x} \rangle \relsym b\f$
is said to be a <EM>interval constraint</EM> if there exist an index
\f$i \in \{ 0, \ldots, n-1 \}\f$ such that, for all
\f$k \in \{ 0, \ldots, i-1, i+1, \ldots, n-1 \}\f$, \f$a_k = 0\f$.
Thus each interval constraint that is not a tautology or
inconsistent has the form
\f$x = r\f$, \f$x \leq r\f$, \f$x \geq r\f$,
\f$x < r\f$ or \f$x > r\f$, with \f$r \in \Rset\f$.

Letting  \f$\cB\f$ be a sequence of
\f$n\f$ intervals and \f$\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose\f$
be the vector in \f$\Rset^n\f$ with 1 in the \f$i\f$'th position
and zeroes in every other position;
if the lower bound of the \f$i\f$'th interval in \f$\cB\f$
is bounded, the corresponding interval constraint is defined as
\f$\langle \vect{e}_i, \vect{x} \rangle \relsym b\f$,
where \f$b\f$ is the value of the bound and
\f$\mathord{\relsym}\f$ is \f$\mathord{\geq}\f$ if it is a closed bound and
\f$\mathord{>}\f$ if it is an open bound.
Similarly,
if the upper bound of the \f$i\f$'th interval in \f$\cB\f$ is bounded,
the corresponding interval constraint is defined as
\f$\langle\vect{e}_i,\vect{x}\rangle \relsym b\f$,
where \f$b\f$ is the value of the bound and
\f$\mathord{\relsym}\f$ is \f$\mathord{\leq}\f$ if it is a closed bound and
\f$\mathord{<}\f$ if it is an open bound.

A convex polyhedron \f$\cP \in \CPset_n\f$ is said to be a
<EM>box</EM> if and only if
either \f$\cP\f$ is the set of solutions to a finite set
of interval constraints or \f$n = 0\f$ and \f$\cP = \emptyset\f$.
Therefore any \f$n\f$-dimensional <EM>box</EM> \f$\cP\f$ in \f$\Rset^n\f$
where \f$n > 0\f$ can be represented by a sequence of \f$n\f$ intervals
\f$\cB\f$ in \f$\Rset\f$ and  \f$\cP\f$ is a closed polyhedron
if every bound in the intervals in \f$\cB\f$
is either closed and bounded or open and unbounded.

\anchor CC76_interval_widening
\subsection Widening_and_Extrapolation_Operators_on_Boxes Widening and Extrapolation Operators on Boxes

The library provides a widening operator for boxes.
Given two sequences of intervals defining two \f$n\f$-dimensional boxes, the
<EM>CC76-widening</EM> applies, for each corresponding interval and bound, the
interval constraint widening defined in \ref CC76 "[CC76]".
For extra precision, this incorporates the widening with thresholds
as defined in \ref BCCetal02 "[BCCetal02]" with
\f$\{-2, -1, 0, 1, 2\}\f$ as the set of default threshold values.

\section Weakly_Relational_Shapes Weakly-Relational Shapes

The PPL provides support for computations on numerical domains that,
in selected contexts, can achieve a better precision/efficiency ratio
with respect to the corresponding computations on a ``fully
relational'' domain of convex polyhedra.  This is achieved by
restricting the syntactic form of the constraints that can be used to
describe the domain elements.

\subsection Bounded_Difference_Shapes Bounded Difference Shapes

For each vector \f$\vect{a} \in \Rset^n\f$ and scalar \f$b \in \Rset\f$,
and for each relation symbol \f$\mathord{\relsym} \in \{ =, \geq\}\f$,
the linear constraint \f$\langle \vect{a}, \vect{x} \rangle \relsym b\f$
is said to be a <EM>bounded difference</EM> if there exist two indices
\f$i, j \in \{ 0, \ldots, n-1 \}\f$ such that:
  - \f$a_i, a_j \in \{ -1, 0, 1 \}\f$ and \f$a_i \neq a_j\f$;
  - \f$a_k = 0\f$, for all \f$k \notin \{ i, j \}\f$.

A convex polyhedron \f$\cP \in \CPset_n\f$ is said to be a
<EM>bounded difference shape</EM> (BDS, for short) if and only if
either \f$\cP\f$ can be expressed as the intersection of a finite number
of bounded difference constraints or \f$n = 0\f$ and \f$\cP = \emptyset\f$.

\subsection Octagonal_Shapes Octagonal Shapes

For each vector \f$\vect{a} \in \Rset^n\f$ and scalar \f$b \in \Rset\f$,
and for each relation symbol \f$\mathord{\relsym} \in \{ =, \geq\}\f$,
the linear constraint \f$\langle \vect{a}, \vect{x} \rangle \relsym b\f$
is said to be an <EM>octagonal</EM> if there exist two indices
\f$i, j \in \{ 0, \ldots, n-1 \}\f$ such that:
  - \f$a_i, a_j \in \{ -1, 0, 1 \}\f$;
  - \f$a_k = 0\f$, for all \f$k \notin \{ i, j \}\f$.

A convex polyhedron \f$\cP \in \CPset_n\f$ is said to be an
<EM>octagonal shape</EM> (OS, for short) if and only if
either \f$\cP\f$ can be expressed as the intersection of a finite number
of octagonal constraints or \f$n = 0\f$ and \f$\cP = \emptyset\f$.

Note that, since any bounded difference is also an octagonal constraint,
any BDS is also an OS. The name ``octagonal'' comes from the fact that,
in a vector space of dimension 2, a bounded OS can have eight sides
at most.

\subsection Weakly_Relational_Shape_Interface Weakly-Relational Shapes Interface
By construction, any BDS or OS is always topologically closed.
Under the usual set inclusion ordering, the set of all BDSs (resp., OSs)
on the vector space \f$\Rset^n\f$ is a lattice having the empty set
\f$\emptyset\f$ and the universe \f$\Rset^n\f$ as the smallest and
the biggest elements, respectively.
In theoretical terms, it is a meet sub-lattice of \f$\CPset_n\f$;
moreover, the lattice of BDSs is a meet sublattice of the lattice of OSs.
The least upper bound of a finite set of BDSs (resp., OSs) is said to be
their <EM>bds-hull</EM> (resp., <EM>oct-hull</EM>).

As far as the representation of the rational inhomogeneous term of
each bounded difference or octagonal constraint is concerned, several
<EM>rounding-aware</EM> implementation choices are available, including:
  - bounded precision integer types;
  - bounded precision floating point types;
  - unbounded precision integer and rational types, as provided by GMP.

The user interface for BDSs and OSs is meant to be as similar as
possible to the one developed for the domain of closed polyhedra: in
particular, all operators on polyhedra are also available for the
domains of BDSs and OSs, even though they are typically characterized
by a lower degree of precision.
For instance, the <EM>bds-difference</EM> and <EM>oct-difference</EM>
operators return (the smallest) over-approximations of the set-theoretical
difference operator on the corresponding domains.
In the case of (generalized) images and preimages of affine relations,
suitable (possibly not-optimal) over-approximations are computed when
the considered relations cannot be precisely modeled by only using
bounded differences or octagonal constraints.

\subsection Widening_and_Extrapolation_Operators_on_WR_Shapes Widening and Extrapolation Operators on Weakly-Relational Shapes

\anchor BHMZ05_widening
For the domains of BDSs and OSs, the library provides a variant of
the widening operator for convex polyhedra defined in \ref CH78 "[CH78]".
The implementation follows the specification
in \ref BHMZ05a "[BHMZ05a,BHMZ05b]",
resulting in an operator which is well-defined on the corresponding domain
(i.e., it does not depend on the internal representation of BDSs or OSs),
while still ensuring convergence in a finite number of steps.

\anchor CC76_extrapolation
The library also implements an extension of the widening operator for
intervals as defined in \ref CC76 "[CC76]". The reader is warned that
such an extension, even though being well-defined on the domain of BDSs
and OSs, is not provided with a convergence guarantee and is therefore an
extrapolation operator.


\section sect_rational_grids Rational Grids

In this section we introduce rational grids
as provided by the library.
See also \ref BDHetal05 "[BDHetal05]" for a detailed description of this domain.

The library supports two representations for the grids domain;
<EM>congruence systems</EM> and <EM>grid generator systems</EM>.
We first describe <EM>linear congruence relations</EM> which form the elements
of a congruence system.

\subsection Congruence_Relations Congruences and Congruence Relations

For any \f$a, b, f \in \Rset\f$,
\f$a \equiv_f b\f$ denotes the <EM>congruence</EM>
\f$\exists \mu \in \Zset \st a - b = \mu f\f$.

Let \f$\Sset \in \{ \Qset, \Rset \}\f$.
For each vector
\f$\vect{a} \in \Sset^n \setdiff \{\vect{0}\}\f$ and scalars
\f$b, f \in \Sset\f$,
the notation \f$\langle \vect{a}, \vect{x} \rangle \equiv_f b\f$
stands for the <EM>linear congruence relation in \f$\Sset^n\f$</EM>
defined by the set of vectors
\f[
  \bigl\{\,
    \vect{v} \in \Rset^n
  \bigm|
    \exists \mu \in \Zset \st
      \langle \vect{a}, \vect{v} \rangle = b + \mu f
  \,\bigr\};
\f]
when \f$f \neq 0\f$, the relation is said to be
<EM>proper</EM>;
\f$\langle \vect{a}, \vect{x} \rangle \equiv_0 b\f$ (i.e., when \f$f = 0\f$)
denotes the equality \f$\langle \vect{a}, \vect{x} \rangle = b\f$.
\f$f\f$ is called the <EM>frequency</EM> or <EM>modulus</EM> and
\f$b\f$ the <EM>base value</EM> of the relation.
Thus, provided \f$\vect{a} \neq \vect{0}\f$,
the relation \f$\langle \vect{a}, \vect{x} \rangle \equiv_f b\f$
defines the set of affine hyperplanes
\f[
  \big\{\,
    \bigl(\langle \vect{a}, \vect{x} \rangle = b + \mu f\bigr)
  \bigm|
    \mu \in \Zset
  \,\bigr\};
\f]
if \f$b \equiv_f 0\f$, \f$\langle \vect{0}, \vect{x} \rangle \equiv_f b\f$
defines the universe \f$\Rset^n\f$ and the empty set, otherwise.

\subsection Rational_Grids Rational Grids

The set \f$\cL  \sseq \Rset^n\f$ is a <EM>rational grid</EM>
if and only if either \f$\cL\f$ is the set of vectors in
\f$\Rset^n\f$ that satisfy
a finite system \f$\cC\f$ of congruence relations in \f$\Qset^n\f$
or \f$n = 0\f$ and \f$\cL = \emptyset\f$.

We also say that  <EM>\f$\cL\f$ is described by \f$\cC\f$</EM>
and that <EM>\f$\cC\f$ is a congruence system for \f$\cL\f$</EM>.

The <EM>grid domain</EM> \f$\Gset_{n}\f$ is the set of all rational grids
described by finite sets of congruence relations in \f$\Qset^n\f$.

If the congruence system \f$\cC\f$ describes the \f$\emptyset\f$,
the <EM>empty</EM> grid,
then we say that \f$\cC\f$ is <EM>inconsistent</EM>.
For example, the congruence systems
\f$\bigl\{\langle\vect{0}, \vect{x}\rangle \equiv_0 1\bigr\}\f$
meaning that \f$0 = 1\f$ and
\f$\bigl\{\langle\vect{a}, \vect{x}\rangle \equiv_2 0,
     \langle\vect{a}, \vect{x}\rangle \equiv_2 1\bigr\}\f$,
for any \f$\vect{a} \in \Rset^n\f$,
meaning that the value of an expression must be both even and odd
are both inconsistent since both describe the empty grid.

When ordering grids by the set inclusion relation,
the empty set \f$\emptyset\f$ and the vector space \f$\Rset^n\f$
(which is described by the empty set of congruence relations)
are, respectively, the smallest and the biggest elements of
\f$\Gset_n\f$.
The vector space \f$\Rset^n\f$ is also called the <EM>universe</EM> grid.

In set theoretical terms,
\f$\Gset_n\f$ is a <EM>lattice</EM> under set inclusion.

\subsection Integer_Combinations Integer Combinations

Let \f$S = \{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n\f$
be a finite set of vectors.
For all scalars \f$\mu_1, \ldots, \mu_k \in \Zset\f$,
the vector \f$\vect{v} = \sum_{j=1}^k \mu_j \vect{x}_j\f$
is said to be a <EM>integer</EM> combination of the vectors in \f$S\f$.

We denote by \f$\inthull(S)\f$
(resp., \f$\intaffinehull(S)\f$)
the set of all the integer (resp., integer and affine)
combinations of the vectors in \f$S\f$.

\subsection Points_Parameters_Lines Points, Parameters and Lines

Let \f$\cL\f$ be a grid.
Then
- a vector \f$\vect{p} \in \cL\f$ is called a <EM>grid point</EM> of \f$\cL\f$;
- a vector \f$\vect{q} \in \Rset^n\f$, where \f$\vect{q} \neq \vect{0}\f$,
  is called a <EM>parameter</EM> of \f$\cL\f$
  if \f$\cL \neq \emptyset\f$ and \f$\vect{p} + \mu \vect{q} \in \cL\f$,
  for all points \f$\vect{p} \in \cL\f$ and all \f$\mu \in \Zset\f$;
- a vector \f$\vect{l} \in \Rset^n\f$ is called a <EM>grid line</EM> of
  \f$\cL\f$ if \f$\cL \neq \emptyset\f$ and
  \f$\vect{p} + \lambda \vect{l} \in \cL\f$,
  for all points \f$\vect{p} \in \cL\f$ and all \f$\lambda \in \Rset\f$.


\subsection Grid_Generator_Representation The Grid Generator Representation

We can generate any rational grid in \f$\Gset_n\f$
from a finite subset of its points, parameters and lines;
each point in a grid is obtained by adding a linear combination of
its generating lines to an integral combination of its parameters
and an integral affine combination of its generating points.

If \f$L, Q, P\f$ are each finite subsets of \f$\Qset^n\f$ and
\f[
  \cL = \linearhull(L) + \inthull(Q) + \intaffinehull(P)
\f]
where the symbol '\f$+\f$' denotes the Minkowski's sum,
then \f$\cL \in \Gset_n\f$ is a rational grid
(see Section 4.4 in \ref Sch99 "[Sch99]" and
also Proposition 8 in \ref BDHetal05 "[BDHetal05]").
The 3-tuple \f$(L, Q, P)\f$
is said to be a <EM>grid generator system</EM> for \f$\cL\f$
and we write \f$\cL = \ggen(L, Q, P)\f$.

Note that the grid \f$\cL = \ggen(L, Q, P) = \emptyset\f$
if and only if the set of grid points \f$P = \emptyset\f$.
If \f$P \neq \emptyset\f$, then
\f$\cL = \ggen(L, \emptyset, Q_{\vect{p}} \union P)\f$
where, for some \f$\vect{p} \in P\f$,
\f$Q_{\vect{p}} = \{\, \vect{p} + \vect{q} \mid \vect{q} \in Q \,\}\f$.

\subsection Grid_Minimized_Representations Minimized Grid Representations

A <EM>minimized</EM> congruence system \f$\cC\f$ for \f$\cL\f$
is such that, if \f$\cC'\f$ is another congruence system for \f$\cL\f$,
then \f$\card \cC \leq \card \cC'\f$.
Note that a minimized congruence system for a non-empty grid
has at most \f$n\f$ congruence relations.

Similarly, a <EM>minimized</EM> grid generator system
\f$\cG = (L, Q, P)\f$ for \f$\cL\f$
is such that, if \f$\cG' = (L', Q', P')\f$ is
another grid generator system for \f$\cL\f$,
then \f$\card L \leq \card L'\f$ and
\f$\card Q + \card P \leq \card Q' + \card P'\f$.
Note that a minimized grid generator system for a grid
has no more than a total of \f$n+1\f$ grid lines, parameters and points.

\subsection Grids_Double_Description_Grids Double Description for Grids

As for convex polyhedra, any grid \f$\cL\f$ can be described by using
a congruence system \f$\cC\f$ for \f$\cL\f$,
a grid generator system \f$\cG\f$ for \f$\cL\f$, or
both by means of the <EM>double description pair (DD pair)</EM>
\f$(\cC, \cG)\f$.
The <EM>double description method</EM> for grids is a collection of
theoretical results very similar to those for convex polyhedra showing that,
given one kind of representation, there are algorithms
for computing a representation of the other kind
and for minimizing both representations.

As for convex polyhedra, such changes of representation form a key step
in the implementation of many operators on grids such as, for example,
intersection and grid join.

\subsection Grid_Space_Dimensions Space Dimensions and Dimension-compatibility for Grids

The <EM>space dimension</EM> of a grid \f$\cL \in \Gset_n\f$
is the dimension
\f$n \in \Nset\f$ of the corresponding vector space \f$\Rset^n\f$.
The space dimension of congruence relations, grid generators and other objects
of the library is defined similarly.

\subsection Grid_Affine_Dimension Affine Independence and Affine Dimension for Grids

A <EM>non-empty</EM> grid \f$\cL \in \Gset_n\f$ has
<EM>affine dimension</EM> \f$k \in \Nset\f$, denoted by \f$\pdim(\cG) = k\f$,
if the maximum number of affinely independent points in \f$\cG\f$ is
\f$k + 1\f$.
The affine dimension of an empty grid is defined to be 0.
Thus we have \f$0 \leq \pdim(\cG) \leq n\f$.

\section rational_grid_operations Operations on Rational Grids

In general, the operations on rational grids are the same as those for the other
PPL domains and the definitions of these can be found in
Section \ref Operations_on_Convex_Polyhedra.
Below we just describe those operations that have features or behavior
that is in some way special to the grid domain.

\subsection Grid_Affine_Transformation Affine Images and Preimages

As for convex polyhedra (see \ref Single_Update_Affine_Functions
"Single-Update Affine Functions"),
the library provides affine image and preimage operators for grids:
given a variable \f$x_k\f$ and
linear expression \f$\mathrm{expr} = \langle \vect{a}, \vect{x} \rangle + b\f$,
these determine the affine transformation
\f$\fund{\phi = \bigl(x'_k = \langle \vect{a}, \vect{x} \rangle + b\bigr)}
        {\Rset^n}{\Rset^n}\f$
that transforms any point
\f$(v_0, \ldots, v_{n-1})^\transpose\f$
in a grid \f$\cL\f$ to
\f[
   \Bigl(v_0, \ldots, \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i v_i + b\bigr),
                   \ldots, v_{n-1}\Bigr)^\transpose.
\f]

The <EM>affine image</EM> operator computes the affine image
of a grid \f$\cL\f$ under
\f$x'_k = \langle \vect{a}, \vect{x} \rangle + b\f$.
For instance,
suppose the grid \f$\cL\f$
to be transformed is the non-relational grid in \f$\Rset^2\f$
generated by the set of grid points
\f$\bigl\{
     (0, 0)^\transpose,
     (0, 3)^\transpose,
     (3, 0)^\transpose
   \bigr\}\f$.
Then, if the considered variable is \f$x_0\f$ and
the linear expression is \f$3x_0 + 2 x_1 + 1\f$
(so that \f$k = 0\f$, \f$a_0 = 3, a_1 = 2, b = 1\f$),
the affine image operator will translate \f$\cL\f$
to the grid \f$\cL_1\f$ generated by the set of grid points
\f$\bigl\{
     (1, 0)^\transpose,
     (7, 3)^\transpose,
     (10, 0)^\transpose
   \bigr\}\f$
which is the grid generated by the grid point \f$(1, 0)\f$
and parameters \f$(3, -3), (0, 9)\f$; or, alternatively defined by the
congruence system \f$\{x \equiv_3 1, x + y \equiv_9 1\}\f$.
If the considered variable is as before (i.e., \f$k = 0\f$)
but the linear expression is  \f$x_1\f$
(so that \f$a_0 = 0, a_1 = 1, b = 0\f$),
then the resulting grid \f$\cL_2\f$
is the grid containing all the points
whose coordinates are integral multiples of 3 and lie on
line \f$x = y\f$.

The affine preimage operator computes the affine preimage
of a grid \f$\cL\f$ under \f$\phi\f$.
For instance,
suppose now that we apply the affine
preimage operator as given in the first example
using variable \f$x_0\f$
and linear expression \f$3x_0 + 2 x_1 + 1\f$
to the grid \f$\cL_1\f$;
then we get the original grid \f$\cL\f$ back.
If, on the other hand, we apply the affine
preimage operator as given in the second example
using variable \f$x_0\f$ and linear expression \f$x_1\f$ to \f$\cL_2\f$,
then the resulting grid will consist of
all the points in \f$\Rset^2\f$ where the \f$y\f$
coordinate is an integral multiple of 3.

Observe that provided the coefficient \f$a_k\f$ of the considered variable
in the linear expression is non-zero, the affine transformation is invertible.

\subsection Grid_Generalized_Image Generalized Affine Images

Similarly to convex polyhedra (see \ref Generalized_Affine_Relations
"Generalized Affine Relations"), the library provides two other grid operators
that are generalizations of the single update affine image and
preimage operators for grids.
The <EM>generalized affine image</EM> operator
\f$\fund{\phi = (\mathrm{lhs}', \mathrm{rhs}, f)}{\Rset^n}{\Rset^n}\f$, where
\f$\mathrm{lhs} = \langle \vect{c}, \vect{x} \rangle + d\f$ and
\f$\mathrm{rhs} = \langle \vect{a}, \vect{x} \rangle + b\f$
are affine expressions and \f$f \in \Qset\f$, is defined as
\f[
  \forall \vect{v} \in \Rset^n, \vect{w} \in \Rset^n
    \itc
      (\vect{v}, \vect{w}) \in \phi
        \iff
          \bigl(
            \langle \vect{c}, \vect{w} \rangle + d
              \equiv_f \langle \vect{a}, \vect{v} \rangle + b
          \bigr)
            \land
          \Bigl(
            \bigland_{0 \leq i < n, c_i = 0}
              w_i = v_i
          \Bigr).
\f]
Note that, when \f$\mathrm{lhs} = x_k\f$ and \f$f = 0\f$,
so that the transfer function is an equality,
then the above operator
is equivalent to the application of the standard affine image
of \f$\cL\f$ with respect to the variable \f$x_k\f$ and the
affine expression \f$\mathrm{rhs}\f$.

\subsection Grid_Frequency Frequency Operator

Let \f$\cL \in \Gset_n\f$ be any non-empty grid and
\f$\mathrm{expr} = \bigl(\langle \vect{a}, \vect{x} \rangle + b\bigr)\f$
be a linear expression. Then if, for
some \f$c, f \in \Rset\f$, all the points in \f$\cL\f$ satisfy the
congruence \f$\cg = ( \mathrm{expr} \equiv_f c )\f$, then the maximum
\f$f\f$ such that this holds is called the <EM>frequency</EM> of
\f$\cL\f$ with respect to \f$\mathrm{expr}\f$.

The frequency operator provided by the library returns both
the frequency \f$f\f$ and a value
\f$\mathrm{val} = \langle \vect{a}, \vect{w} \rangle + b\f$
where \f$\vect{w} \in \cL\f$ and
\f[
  \lvert\mathrm{val}\rvert
     = \min\Bigl\{\,
             \bigl\lvert\langle \vect{a}, \vect{v} \rangle + b \bigr\rvert
           \Bigm|
             \vect{v} \in \cL
           \,\Bigr\}.
\f]

Observe that the above definition is also applied to other simple objects in
the library like polyhedra, octagonal shapes, bd-shapes and boxes
and in such cases the definition of frequency can be simplified.
For instance, the frequency for an object \f$\cP \in \Pset_n\f$ is
defined if and only if there is a unique value \f$c\f$ such that
\f$\cP\f$ saturates the equality \f$( \mathrm{expr} = c )\f$;
in this case the frequency is \f$0\f$ and the value returned is \f$c\f$.

\subsection Grid_Time_Elapse Time-Elapse Operator

For any two grids \f$\cL_1, \cL_2 \in \Gset_n\f$,
the <EM>time-elapse</EM> between \f$\cL_1\f$ and \f$\cL_2\f$,
denoted \f$ \cL_1 \nearrow \cL_2\f$,
is the grid
\f[
  \bigl\{\,
    \vect{p} + \mu \vect{q} \in \Rset^n
  \bigm|
    \vect{p} \in \cL_1, \vect{q} \in \cL_2, \mu \in \Zset
  \,\bigr\}.
\f]

\subsection Grid_Relation_With Relation-with Operators

The library provides operators for checking the relation holding
between a grid  and
a congruence, a grid generator, a constraint or a (polyhedron) generator.

Suppose \f$\cL\f$ is a grid
and \f$\cC\f$ an arbitrary congruence system representing \f$\cL\f$.
Suppose also that
\f$
  \cg = \bigl(
          \langle \vect{a}, \vect{x} \rangle \equiv_f b
        \bigr)
\f$
is a congruence relation
with \f$\cL_{\cg} = \gcon\bigl(\{\cg\}\bigr)\f$.
The possible relations between \f$\cL\f$ and \f$\cg\f$ are as follows.
- \f$\cL\f$ <EM>is disjoint</EM> from \f$\cg\f$ if
  \f$\cL \inters \cL_{\cg} = \emptyset\f$; that is, adding \f$\cg\f$
  to \f$\cC\f$ gives us the empty grid.
- \f$\cL\f$ <EM>strictly intersects</EM> \f$\cg\f$ if
  \f$\cL \inters \cL_{\cg} \neq \emptyset\f$ and
  \f$\cL \inters \cL_{\cg} \subset \cL\f$;
  that is, adding \f$\cg\f$ to \f$\cC\f$
  gives us a non-empty grid strictly smaller than \f$\cL\f$.
- \f$\cL\f$ <EM>is included</EM> in \f$\cg\f$ if \f$\cL \sseq \cL_{\cg}\f$;
  that is, adding \f$\cg\f$ to \f$\cC\f$ leaves \f$\cL\f$ unchanged.
- \f$\cL\f$ <EM>saturates</EM> \f$\cg\f$ if \f$\cL\f$ <EM>is included</EM>
  in \f$\cg\f$ and \f$f = 0\f$, i.e., \f$\cg\f$ is an equality congruence.

For the relation between \f$\cL\f$ and a constraint, suppose that
\f$
  c = \bigl(
        \langle \vect{a}, \vect{x} \rangle \relsym b
      \bigr)
\f$
is a constraint
with \f$\mathord{\relsym} \in \{ =, \geq, > \}\f$
and \f$\cQ\f$ the set of points that satisfy \f$c\f$.
The possible relations between \f$\cL\f$ and \f$c\f$ are as follows.
- \f$\cL\f$ <EM>is disjoint</EM> from \f$c\f$ if
  \f$\cL \inters \cQ = \emptyset\f$.
- \f$\cL\f$ <EM>strictly intersects</EM> \f$c\f$ if
  \f$\cL \inters \cQ \neq \emptyset\f$ and \f$\cL \inters \cQ \subset \cL\f$.
- \f$\cL\f$ <EM>is included</EM> in \f$c\f$ if \f$\cL \sseq \cQ\f$.
- \f$\cL\f$ <EM>saturates</EM> \f$c\f$ if \f$\cL\f$ <EM>is included</EM>
  in \f$c\f$ and \f$\mathord{\relsym}\f$ is \f$=\f$.

A grid \f$\cL\f$ <EM>subsumes</EM> a grid generator \f$g\f$
if adding \f$g\f$ to any grid generator system representing \f$\cL\f$
does not change \f$\cL\f$.

A grid \f$\cL\f$ <EM>subsumes</EM> a (polyhedron) point or closure point
\f$g\f$ if adding the corresponding grid point to any grid generator system
representing \f$\cL\f$ does not change \f$\cL\f$.
A grid \f$\cL\f$ <EM>subsumes</EM> a (polyhedron) ray or line
\f$g\f$ if adding the corresponding grid line to any grid generator system
representing \f$\cL\f$ does not change \f$\cL\f$.

\subsection Grid_Wrapping_Operator Wrapping Operator

The operator <CODE>wrap_assign</CODE> provided by the library, allows
for the \ref Wrapping_Operator "wrapping" of a subset of the set of
space dimensions so as to fit the given bounded integer type and have
the specified overflow behavior.  In order to maximize the precision
of this operator for grids, the exact behavior differs in some
respects from the other simple classes of geometric descriptors.

Suppose \f$\cL \in \Gset_n\f$ is a grid and \f$J\f$ a subset of the
set of space dimensions \f$\{0, \ldots, n-1\}\f$.
Suppose also that the width of the bounded integer type is \f$w\f$ so that
the range of values \f$R = \{r \in \Rset \mid 0 \leq r < 2^w\}\f$
if the type is unsigned
and \f$R = \{r \in \Rset \mid -2^{w-1} \leq r < 2^{w-1}\}\f$ otherwise.
Consider a space dimension \f$j \in J\f$ and a variable \f$v_j\f$
for dimension \f$j\f$.

If the value in \f$\cL\f$ for the variable \f$v_j\f$ is a constant in
the range \f$R\f$, then it is unchanged.  Otherwise the result
\f$\cL'\f$ of the operation on \f$\cL\f$ will depend on the specified
overflow behavior.

- Overflow impossible. In this case, it is known that no wrapping can
  occur. If the grid \f$\cL\f$ has no value for the variable \f$v_j\f$ in the
  range \f$R\f$, then \f$\cL\f$ is set empty. If \f$v_j\f$ has exactly
  one value \f$a \in R\f$ in \f$\cL\f$, then \f$v_j\f$ is set
  equal to \f$a\f$. Otherwise, \f$\cL' = \cL\f$.

- Overflow undefined. In this case, for each value \f$a\f$ for
  \f$v_j\f$ in the grid \f$\cL\f$, the wrapped value can be any value
  \f$a + z \in R\f$ where \f$z \in \Zset\f$.
  Therefore \f$\cL'\f$ is obtained by adding the parameter
  \f$(0, \ldots, 0, v_j, 0, \ldots, 0)\f$,
  where \f$v_j = 1\f$, to the generator system for \f$\cL\f$.

- Overflow wraps. In this case, if \f$\cL\f$ already satisfies the
  congruence \f$v_j = a \mod 2^w\f$, for some \f$a \in \Rset\f$, then
  \f$v_j\f$ is set equal to \f$a'\f$ where \f$a' = a \mod 2^w\f$ and
  \f$a'\in R\f$.  Otherwise, \f$\cL'\f$ is obtained by adding the
  parameter \f$(0, \ldots, 0, v_j, 0, \ldots, 0)\f$, where
  \f$v_j = 2^w\f$, to the generator system for \f$\cL\f$.

\subsection Grid_Widening Widening Operators

The library provides <EM>grid widening</EM> operators for the domain of grids.
The congruence widening and generator widening
follow the specifications provided in \ref BDHetal05 "[BDHetal05]".
The third widening uses either
the congruence or the generator widening, the exact rule governing this choice
at the time of the call is left to the implementation.
Note that, as for the widenings provided for convex polyhedra,
all the operations provided by the library
for computing a widening \f$\cL_1 \widen \cL_2\f$ of
grids \f$\cL_1, \cL_2 \in \Gset_n\f$ require as a precondition that
\f$\cL_1 \sseq \cL_2\f$.

\note
As is the case for the other operators on grids, the
implementation overwrites one of the two grid arguments with the
result of the widening application.
It is worth stressing that, in any widening operation that computes the
widening \f$\cL_1 \widen \cL_2\f$, the resulting grid will be assigned
to overwrite the store containing the bigger grid \f$\cL_2\f$.
The smaller grid \f$\cL_1\f$ is not modified.
The same observation holds for all flavors of widenings and extrapolation
operators that are implemented in the library and for all the language
interfaces.

\subsection Grid_Widening_with_Tokens Widening with Tokens

This is as for \ref Widening_with_Tokens "widening with tokens"
for convex polyhedra.

\subsection Grid_Extrapolation Extrapolation Operators

Besides the widening operators, the library also implements several
<EM>extrapolation</EM> operators, which differ from widenings in that
their use along an upper iteration sequence does not ensure convergence
in a finite number of steps.

In particular, for each grid widening that is provided,
there is a corresponding
<EM>limited</EM> extrapolation operator, which can be used to implement
the <EM>widening ``up to''</EM> technique as described in
\ref HPR97 "[HPR97]".
Each limited extrapolation operator takes a congruence system as an
additional parameter and uses it to improve the approximation yielded
by the corresponding widening operator. Note that,
as in the case for convex polyhedra, a convergence guarantee
can only be obtained by suitably restricting the set of
congruence relations that can occur in this additional parameter.

\section powerset The Powerset Construction

The PPL provides the finite powerset construction; this takes a
pre-existing domain and upgrades it to one that can represent
disjunctive information (by using a <EM>finite</EM> number of
disjuncts).  The construction follows the approach described in
\ref Bag98 "[Bag98]", also summarized in \ref BHZ04 "[BHZ04]" where there
is an account of generic widenings for the powerset domain (some of
which are supported in the pointset powerset domain instantiation
of this construction described in Section \ref pointset_powerset).

\anchor powerset_domain
\subsection The_Powerset_Domain The Powerset Domain

The domain is built from a pre-existing base-level domain
\f$D\f$ which must include an entailment relation `\f$\mathord{\entails}\f$',
meet operation `\f$\mathord{\meet}\f$', a top element `\f$\true\f$' and
bottom element `\f$\false\f$'.

A set \f$\cS \in \wp(D)\f$ is called <EM>non-redundant</EM>
with respect to `\f$\mathord{\entails}\f$' if and only if
\f$\false \notin \cS\f$
and \f$\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2\f$.
The set of finite non-redundant subsets of \f$D\f$
(with respect to `\f$\mathord{\entails}\f$')
is denoted by \f$\wpfn{D}{\entails}\f$.
The function
\f$\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}\f$,
called <EM>Omega-reduction</EM>,
maps a finite set into its non-redundant counterpart;
it is defined, for each \f$\cS \in \wpf(D)\f$, by
\f[
  \nonredmap(\cS)
    \defeq
      \cS \setdiff \{\,
                   d \in \cS
                 \mid
                   d = \false \text{ or } \exists d' \in \cS \st d \sentails d'
                 \,\}.
\f]
where \f$d \sentails d'\f$ denotes \f$d \entails d' \land d \ne d'\f$.

As the intended semantics of a powerset domain element \f$\cS \in \wpf(D)\f$
is that of disjunction of the semantics of \f$D\f$,
the finite set \f$\cS\f$ is semantically equivalent
to the non-redundant set \f$\nonredmap(\cS)\f$;
and elements of \f$\cS\f$ will be called <EM>disjuncts</EM>.
The restriction to the finite subsets reflects the fact that here
disjunctions are implemented by explicit collections of disjuncts.
As a consequence of this restriction, for any \f$\cS \in \wpf(D)\f$
such that \f$\cS \neq \{ \false \}\f$, \f$\nonredmap(\cS)\f$ is the (finite)
set of the maximal elements of \f$\cS\f$.

The <EM>finite powerset domain</EM> over a domain \f$D\f$ is the set
of all finite non-redundant sets of \f$D\f$ and denoted by \f$D_{\smallP}\f$.
The domain includes an approximation ordering
`\f$\mathord{\entailsP}\f$' defined so that,
for any \f$\cS_1\f$ and \f$\cS_2 \in D_{\smallP}\f$,
\f$\cS_1 \entailsP \cS_2\f$ if and only if
\f[
  \forall d_1 \in \cS_1 \itc
    \exists d_2 \in \cS_2 \st
      d_1 \entails d_2.
\f]
Therefore the top element is \f$\{\true\}\f$ and the bottom element is the
emptyset.

\note
As far as Omega-reduction is concerned, the library adopts a
<EM>lazy</EM> approach: an element of the powerset domain is
represented by a potentially redundant sequence of disjuncts.
Redundancies can be eliminated by explicitly invoking the operator
<CODE>omega_reduce()</CODE>, e.g., before performing the output of a
powerset element. Note that all the documented operators automatically
perform Omega-reductions on their arguments, when needed or appropriate.

\section ps_operations Operations on the Powerset Construction

In this section we briefly describe the generic operations on Powerset Domains
that are provided by the library for any given base-level domain \f$D\f$.

\anchor ps_meet_upper_bound
\subsection Meet_and_Upper_Bound Meet and Upper Bound

Given the sets \f$\cS_1\f$ and \f$\cS_2 \in D_{\smallP}\f$, the
<EM>meet</EM> and <EM>upper bound</EM> operators provided by the
library returns the set
\f$
  \nonredmap
    \bigl(
      \{\,
        d_1 \meet d_2
      \mid
        d_1 \in \cS_1, d_2 \in \cS_2
      \,\}
    \bigr)
\f$
and Omega-reduced set union \f$\nonredmap(\cS_1 \union \cS_2)\f$ respectively.

\anchor ps_add_disjunct
\subsection Adding_a_Disjunct Adding a Disjunct

Given the powerset element \f$\cS \in D_{\smallP}\f$ and the base-level
element \f$d \in D\f$, the <EM>add disjunct</EM> operator provided by
the library returns the powerset element
\f$\nonredmap\bigl(\cS \union \{d\}\bigr)\f$.

\anchor ps_collapse
\subsection Collapsing_a_Powerset_Element Collapsing a Powerset Element

If the given powerset element is not empty, then the
<EM>collapse</EM> operator returns the
singleton powerset consisting of an upper-bound of all the disjuncts.

\section pointset_powerset The Pointset Powerset Domain

The pointset powerset domain provided by the PPL is the finite
powerset domain (defined in Section \ref powerset) whose base-level domain
\f$D\f$ is one of the classes of semantic geometric descriptors listed in
Section \ref Semantic_Geometric_Descriptors.

In addition to the operations described for the generic powerset
domain in Section \ref ps_operations, the PPL provides
all the generic operations
listed in \ref Generic_Operations_on_Semantic_Geometric_Descriptors.
Here we just describe those operations that
are particular to the pointset powerset domain.

\subsection Powerset_Meet_Preserving_Simplification Meet-Preserving Simplification

Let \f$\cS_1 = \{ d_1, \ldots, d_m \}\f$,
\f$\cS_2 = \{ c_1, \ldots, c_n \}\f$ and
\f$\cS = \{ s_1, \ldots, s_q \}\f$
be Omega-reduced elements of a pointset powerset domain
over the same base-level domain.
Then:

 - \f$\cS\f$ is <EM>powerset meet-preserving</EM>
   with respect to \f$\cS_1\f$ using context \f$\cS_2\f$ if
   the meet of \f$\cS\f$ and \f$\cS_2\f$
   is equal to
   the meet of \f$\cS_1\f$ and \f$\cS_2\f$;

 - \f$\cS\f$ is a <EM>powerset simplification</EM> with respect to
   \f$\cS_1\f$ if \f$q \leq m\f$.

 - \f$\cS\f$ is a <EM>disjunct meet-preserving simplification</EM>
    with respect to \f$\cS_1\f$
    if, for each \f$s_k \in \cS\f$,
    there exists \f$d_i \in \cS_1\f$ such that, for each \f$c_j \in \cS_2\f$,
    \f$s_k\f$ is a meet-preserving enlargement and simplification
    of \f$d_i\f$ using context \f$c_j\f$.

The library provides a binary operator (<CODE>simplify_using_context</CODE>)
for the pointset powerset domain that returns a powerset which is a
powerset meet-preserving, powerset simplification and
disjunct meet-preserving simplification
of its first argument using the second argument as context.

Notice that, due to the powerset simplification property, in general
a meet-preserving powerset simplification is <em>not</em> an enlargement
with respect to the ordering defined on the powerset lattice.
Because of this, the operator provided by the library
is only well-defined when the base-level domain
is not itself a powerset domain.


\anchor pps_geometric
\subsection Geometric_Comparisons Geometric Comparisons

Given the pointset powersets \f$\cS_1, \cS_2\f$ over the same base-level domain
and with the same space dimension, then
we say that \f$\cS_1\f$ <EM>geometrically covers</EM> \f$\cS_2\f$
if every point (in some disjunct) of \f$\cS_2\f$ is also
a point in a disjunct of \f$\cS_1\f$.
If \f$\cS_1\f$ geometrically covers \f$\cS_2\f$
and \f$\cS_2\f$ geometrically covers \f$\cS_1\f$,
then we say that they are <EM>geometrically equal</EM>.


\anchor pps_pairwise_merge
\subsection Pairwise_Merge Pairwise Merge

Given the pointset powerset \f$\cS\f$ over a base-level semantic GD domain
\f$D\f$, then the
<EM>pairwise merge</EM> operator takes pairs of distinct elements in
\f$\cS\f$ whose upper bound (denoted here by \f$\uplus\f$) in \f$D\f$
(using the PPL operator <CODE>upper_bound_assign()</CODE> for \f$D\f$)
is the same as their set-theoretical union
and replaces them by their union.  This replacement is done
recursively so that, for each pair \f$c, d\f$ of distinct
disjuncts in the result set, we have
\f$c \uplus d \neq c \union d\f$.

\anchor pps_bgp99_extrapolation
\subsection Powerset_Extrapolation_Operators Powerset Extrapolation Operators

The library implements a generalization of the extrapolation operator
for powerset domains proposed in \ref BGP99 "[BGP99]".
The operator <CODE>BGP99_extrapolation_assign</CODE> is made parametric
by allowing for the specification of any PPL extrapolation operator
for the base-level domain.
Note that, even when the extrapolation operator for the
base-level domain \f$D\f$ is known to be a widening on \f$D\f$,
the <CODE>BGP99_extrapolation_assign</CODE> operator cannot guarantee the
convergence of the iteration sequence in a finite number of steps
(for a counter-example, see \ref BHZ04 "[BHZ04]").

\anchor pps_certificate_widening
\subsection Certificate_Based_Widenings Certificate-Based Widenings

The PPL library provides support for the specification of proper widening
operators on the pointset powerset domain. In particular,
this version of the library implements an instance of the
<EM>certificate-based widening framework</EM> proposed in
\ref BHZ03b "[BHZ03b]".

A <EM>finite convergence certificate</EM> for an extrapolation operator
is a formal way of ensuring that such an operator is indeed a widening
on the considered domain. Given a widening operator on the base-level
domain \f$D\f$, together with the corresponding convergence certificate,
the BHZ03 framework is able to lift this widening on \f$D\f$
to a widening on the pointset powerset domain; ensuring
convergence in a finite number of iterations.

Being highly parametric, the BHZ03 widening framework can be
instantiated in many ways. The current implementation provides
the templatic operator
<CODE>BHZ03_widening_assign\<Certificate, Widening\></CODE>
which only exploits a fraction of this generality, by allowing the
user to specify the base-level widening function and the corresponding
certificate. The widening strategy is fixed and uses two extrapolation
heuristics: first, the upper bound operator for the base-level domain
is tried; second,
the \ref pps_bgp99_extrapolation "BGP99 extrapolation operator" is tried,
possibly applying \ref pps_pairwise_merge "pairwise merging".
If both heuristics fail to converge according to the convergence
certificate, then an attempt is made to apply the base-level widening
to the upper bound of the two arguments, possibly improving the result
obtained by means of the difference operator for the base-level domain.
For more details and a justification of the overall approach,
see \ref BHZ03b "[BHZ03b]" and \ref BHZ04 "[BHZ04]".

The library provides several convergence certificates.
Note that, for the domain of Polyhedra, while
\ref Parma_Polyhedra_Library::BHRZ03_Certificate the "BHRZ03_Certificate"
is compatible with both the BHRZ03 and the H79 widenings,
\ref Parma_Polyhedra_Library::H79_Certificate "H79_Certificate"
is only compatible with the latter.
Note that using different certificates will change the results obtained,
even when using the same base-level widening operator.
It is also worth stressing that it is up to the user
to see that the widening operator is actually compatible with
a given convergence certificate. If such a requirement is not met,
then an extrapolation operator will be obtained.

\section use_of_library Using the Library

\subsection A_Note_on_the_Implementation_of_the_Operators A Note on the Implementation of the Operators

When adopting the double description method for the representation of
convex polyhedra, the implementation of most of the operators may
require an explicit conversion from one of the two representations
into the other one, leading to algorithms having a worst-case
exponential complexity.  However, thanks to the adoption of lazy and
incremental computation techniques, the library turns out to be rather
efficient in many practical cases.

In earlier versions of the library, a number of operators were introduced
in two flavors: a <EM>lazy</EM> version and an <EM>eager</EM> version,
the latter having the operator name ending with <CODE>_and_minimize</CODE>.
In principle, only the lazy versions should be used.
The eager versions were added to help a knowledgeable user obtain better
performance in particular cases. Basically, by invoking the eager version
of an operator, the user is trading laziness to better exploit the
incrementality of the inner library computations.
Starting from version 0.5, the lazy and incremental computation techniques
have been refined to achieve a better integration: as a consequence,
the lazy versions of the operators are now almost always more efficient
than the eager versions.

One of the cases when an eager computation might still make sense
is when the well-known <EM>fail-first</EM> principle comes into play.
For instance, if you have to compute the intersection of several
polyhedra and you strongly suspect that the result will become empty
after a few of these intersections, then you may obtain a better
performance by calling the eager version of the intersection operator,
since the minimization process also enforces an emptiness check.
Note anyway that the same effect can be obtained by interleaving
the calls of the lazy operator with explicit emptiness checks.

\warning
For the reasons mentioned above, starting from version 0.10 of the
library, the usage of the eager versions (i.e., the ones having a
name ending with <CODE>_and_minimize</CODE>) of these operators is
\em deprecated; this is in preparation of their complete removal,
which will occur starting from version 0.11.

\subsection On_Pointset_Powerset_and_Partially_Reduced_Product_Domains_A_Warning On Pointset_Powerset and Partially_Reduced_Product Domains: A Warning

  For future versions of the PPL library all practical instantiations
for the disjuncts for a pointset_powerset and component domains for
the partially_reduced_product domains will be fully
supported. However, for version 0.10, these compound domains should
not themselves occur as one of their argument domains.  Therefore their
use comes with the following warning.

\warning
The <CODE>Pointset_Powerset<PSET></CODE> and
<CODE>Partially_Reduced_Product<D1, D2, R></CODE>
should only be used with the following instantiations
for the disjunct domain template \p PSET and component domain
templates \p D1 and \p D2:
<CODE>C_Polyhedron</CODE>,
<CODE>NNC_Polyhedron</CODE>,
<CODE>Grid</CODE>,
<CODE>Octagonal_Shape<T></CODE>,
<CODE>BD_Shape<T></CODE>,
<CODE>Box<T></CODE>.

\subsection On_Object_Orientation_and_Polymorphism_A_Disclaimer On Object-Orientation and Polymorphism: A Disclaimer

The PPL library is mainly a collection of so-called ``concrete data
types'': while providing the user with a clean and friendly interface,
these types are not meant to --- i.e., they should not --- be used
polymorphically (since, e.g., most of the destructors are not declared
<CODE>virtual</CODE>). In practice, this restriction means that the
library types should not be used as <EM>public base classes</EM> to be
derived from. A user willing to extend the library types, adding new
functionalities, often can do so by using <EM>containment</EM> instead
of inheritance; even when there is the need to override a
<CODE>protected</CODE> method, non-public inheritance should suffice.

\subsection On_Const_Correctness_A_Warning_about_the_Use_of_References_and_Iterators On Const-Correctness: A Warning about the Use of References and Iterators

Most operators of the library depend on one or more parameters
that are declared ``const'', meaning that they will not be changed
by the application of the considered operator.
Due to the adoption of lazy computation techniques, in many cases
such a const-correctness guarantee only holds at the semantic level,
whereas it does not necessarily hold at the implementation level.
For a typical example, consider the extraction from a polyhedron
of its constraint system representation.
While this operation is not going to change the polyhedron,
it might actually invoke the internal conversion algorithm
and modify the generators representation of the polyhedron object,
e.g., by reordering the generators and removing those that are
detected as redundant. Thus, any previously computed reference
to the generators of the polyhedron (be it a direct reference object
or an indirect one, such as an iterator) will no longer be valid.
For this reason, code fragments such as the following should be
avoided, as they may result in undefined behavior:
\code
// Find a reference to the first point of the non-empty polyhedron `ph'.
const Generator_System& gs = ph.generators();
Generator_System::const_iterator i = gs.begin();
for (Generator_System::const_iterator gs_end = gs.end(); i != gs_end; ++i)
  if (i->is_point())
    break;
const Generator& p = *i;
// Get the constraints of `ph'.
const Constraint_System& cs = ph.constraints();
// Both the const iterator `i' and the reference `p'
// are no longer valid at this point.
cout << p.divisor() << endl;  // Undefined behavior!
++i;                          // Undefined behavior!
\endcode
As a rule of thumb, if a polyhedron plays any role in a computation
(even as a const parameter), then any previously computed reference
to parts of the polyhedron may have been invalidated.
Note that, in the example above, the computation of the constraint system
could have been placed after the uses of the iterator <CODE>i</CODE>
and the reference <CODE>p</CODE>.
Anyway, if really needed, it is always possible to take a copy of,
instead of a reference to, the parts of interest of the polyhedron;
in the case above, one may have taken a copy of the generator system
by replacing the second line of code with the following:
\code
Generator_System gs = ph.generators();
\endcode
The same observations, modulo syntactic sugar, apply to the operators
defined in the C interface of the library.

\section bibliography Bibliography

<DL>

<DT>[Anc91]</DT>
<DD>
\anchor Anc91
C.&nbsp;Ancourt.
 <em>G&eacute;n&eacute;ration automatique de codes de transfert pour
  multiprocesseurs &agrave; m&eacute;moires locales</em>.
 PhD thesis, Universit&eacute; de Paris VI, Paris, France, March 1991.

</DD>


<DT>[BA05]</DT>
<DD>
\anchor BA05
J.&nbsp;M. Bjorndalen and O.&nbsp;Anshus.
 Lessons learned in benchmarking - Floating point benchmarks: Can
  you trust them?
 In <em>Proceedings of the <em>Norsk informatikkonferanse 2005</em>
  (NIK 2005)</em>, pages 89-100, Bergen, Norway, 2005. Tapir Akademisk Forlag.


</DD>


<DT>[Bag97]</DT>
<DD>
\anchor Bag97
R.&nbsp;Bagnara.
 <em>Data-Flow Analysis for Constraint Logic-Based Languages</em>.
 PhD thesis, Dipartimento di Informatica, Universit&agrave; di Pisa, Pisa,
  Italy, March 1997.
 Printed as Report TD-1/97.

</DD>


<DT>[Bag98]</DT>
<DD>
\anchor Bag98
R.&nbsp;Bagnara.
 A hierarchy of constraint systems for data-flow analysis of
  constraint logic-based languages.
 <em>Science of Computer Programming</em>, 30(1-2):119-155, 1998.


</DD>


<DT>[BCC<sup>+</sup>02]</DT>
<DD>
\anchor BCCetal02
B.&nbsp;Blanchet, P.&nbsp;Cousot, R.&nbsp;Cousot, J.&nbsp;Feret, L.&nbsp;Mauborgne, A.&nbsp;Min&eacute;,
  D.&nbsp;Monniaux, and X.&nbsp;Rival.
 Design and implementation of a special-purpose static program
  analyzer for safety-critical real-time embedded software.
 In T.&nbsp;&AElig;. Mogensen, D.&nbsp;A. Schmidt, and I.&nbsp;Hal Sudborough,
  editors, <em>The Essence of Computation, Complexity, Analysis,
  Transformation. Essays Dedicated to Neil D. Jones [on occasion of his 60th
  birthday]</em>, volume 2566 of <em>Lecture Notes in Computer Science</em>, pages
  85-108. Springer-Verlag, Berlin, 2002.

</DD>


<DT>[BDH<sup>+</sup>05]</DT>
<DD>
\anchor BDHetal05
R.&nbsp;Bagnara, K.&nbsp;Dobson, P.&nbsp;M. Hill, M.&nbsp;Mundell, and E.&nbsp;Zaffanella.
 A linear domain for analyzing the distribution of numerical values.
 Report 2005.06, School of Computing, University of Leeds, UK, 2005.
 Available at
  <a href="http://www.comp.leeds.ac.uk/research/pubs/reports.shtml">http://www.comp.leeds.ac.uk/research/pubs/reports.shtml</a>.


</DD>


<DT>[BDH<sup>+</sup>06]</DT>
<DD>
\anchor BDHetal06
R.&nbsp;Bagnara, K.&nbsp;Dobson, P.&nbsp;M. Hill, M.&nbsp;Mundell, and E.&nbsp;Zaffanella.
 A practical tool for analyzing the distribution of numerical values,
  2006.
 Available at
  <a href="http://www.comp.leeds.ac.uk/hill/Papers/papers.html">http://www.comp.leeds.ac.uk/hill/Papers/papers.html</a>.

</DD>


<DT>[BDH<sup>+</sup>07]</DT>
<DD>
\anchor BDHetal07
R.&nbsp;Bagnara, K.&nbsp;Dobson, P.&nbsp;M. Hill, M.&nbsp;Mundell, and E.&nbsp;Zaffanella.
 Grids: A domain for analyzing the distribution of numerical values.
 In G.&nbsp;Puebla, editor, <em>Logic-based Program Synthesis and
  Transformation, 16th International Symposium</em>, volume 4407 of <em>Lecture
  Notes in Computer Science</em>, pages 219-235, Venice, Italy, 2007.
  Springer-Verlag, Berlin.


</DD>


<DT>[BFT00]</DT>
<DD>
\anchor BFT00
A.&nbsp;Bemporad, K.&nbsp;Fukuda, and F.&nbsp;D. Torrisi.
 Convexity recognition of the union of polyhedra.
 Report AUT00-13, Automatic Control Laboratory, ETHZ, Zurich,
  Switzerland, 2000.


</DD>


<DT>[BFT01]</DT>
<DD>
\anchor BFT01
A.&nbsp;Bemporad, K.&nbsp;Fukuda, and F.&nbsp;D. Torrisi.
 Convexity recognition of the union of polyhedra.
 <em>Computational Geometry: Theory and Applications</em>,
  18(3):141-154, 2001.

</DD>


<DT>[BGP99]</DT>
<DD>
\anchor BGP99
T.&nbsp;Bultan, R.&nbsp;Gerber, and W.&nbsp;Pugh.
 Model-checking concurrent systems with unbounded integer variables:
  Symbolic representations, approximations, and experimental results.
 <em>ACM Transactions on Programming Languages and Systems</em>,
  21(4):747-789, 1999.


</DD>


<DT>[BHMZ04]</DT>
<DD>
\anchor BHMZ04
R.&nbsp;Bagnara, P.&nbsp;M. Hill, E.&nbsp;Mazzi, and E.&nbsp;Zaffanella.
 Widening operators for weakly-relational numeric abstractions.
 Report <tt>arXiv:cs.PL/0412043</tt>, 2004.
 Extended abstract. Contribution to the <em>International workshop
  on &ldquo;Numerical &amp; Symbolic Abstract Domains&rdquo;</em> (NSAD'05, Paris, January 21,
  2005). Available at <a href="http://arxiv.org/">http://arxiv.org/</a> and
  <a href="http://www.cs.unipr.it/ppl/">http://www.cs.unipr.it/ppl/</a>.


</DD>


<DT>[BHMZ05a]</DT>
<DD>
\anchor BHMZ05a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, E.&nbsp;Mazzi, and E.&nbsp;Zaffanella.
 Widening operators for weakly-relational numeric abstractions.
 Quaderno 399, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2005.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>.


</DD>


<DT>[BHMZ05b]</DT>
<DD>
\anchor BHMZ05b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, E.&nbsp;Mazzi, and E.&nbsp;Zaffanella.
 Widening operators for weakly-relational numeric abstractions.
 In C.&nbsp;Hankin and I.&nbsp;Siveroni, editors, <em>Static Analysis:
  Proceedings of the 12th International Symposium</em>, volume 3672 of <em>Lecture
  Notes in Computer Science</em>, pages 3-18, London, UK, 2005. Springer-Verlag,
  Berlin.

</DD>


<DT>[BHRZ03a]</DT>
<DD>
\anchor BHRZ03a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, E.&nbsp;Ricci, and E.&nbsp;Zaffanella.
 Precise widening operators for convex polyhedra.
 In R.&nbsp;Cousot, editor, <em>Static Analysis: Proceedings of the 10th
  International Symposium</em>, volume 2694 of <em>Lecture Notes in Computer
  Science</em>, pages 337-354, San Diego, California, USA, 2003. Springer-Verlag,
  Berlin.


</DD>


<DT>[BHRZ03b]</DT>
<DD>
\anchor BHRZ03b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, E.&nbsp;Ricci, and E.&nbsp;Zaffanella.
 Precise widening operators for convex polyhedra.
 Quaderno 312, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2003.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>.

</DD>


<DT>[BHRZ05]</DT>
<DD>
\anchor BHRZ05
R.&nbsp;Bagnara, P.&nbsp;M. Hill, E.&nbsp;Ricci, and E.&nbsp;Zaffanella.
 Precise widening operators for convex polyhedra.
 <em>Science of Computer Programming</em>, 58(1-2):28-56, 2005.


</DD>


<DT>[BHZ02a]</DT>
<DD>
\anchor BHZ02a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 A new encoding and implementation of not necessarily closed convex
  polyhedra.
 Quaderno 305, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2002.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>.

</DD>


<DT>[BHZ02b]</DT>
<DD>
\anchor BHZ02b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 A new encoding of not necessarily closed convex polyhedra.
 In M.&nbsp;Carro, C.&nbsp;Vacheret, and K.-K. Lau, editors, <em>Proceedings of
  the 1st CoLogNet Workshop on Component-based Software Development and
  Implementation Technology for Computational Logic Systems</em>, pages 147-153,
  Madrid, Spain, 2002.
 Published as TR Number CLIP4/02.0, Universidad Polit&eacute;cnica de
  Madrid, Facultad de Inform&aacute;tica.


</DD>


<DT>[BHZ03a]</DT>
<DD>
\anchor BHZ03a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 A new encoding and implementation of not necessarily closed convex
  polyhedra.
 In M.&nbsp;Leuschel, S.&nbsp;Gruner, and S.&nbsp;Lo Presti, editors, <em>
  Proceedings of the 3rd Workshop on Automated Verification of Critical
  Systems</em>, pages 161-176, Southampton, UK, 2003.
 Published as TR Number DSSE-TR-2003-2, University of Southampton.


</DD>


<DT>[BHZ03b]</DT>
<DD>
\anchor BHZ03b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Widening operators for powerset domains.
 In B.&nbsp;Steffen and G.&nbsp;Levi, editors, <em>Verification, Model Checking
  and Abstract Interpretation: Proceedings of the 5th International Conference
  (VMCAI 2004)</em>, volume 2937 of <em>Lecture Notes in Computer Science</em>, pages
  135-148, Venice, Italy, 2003. Springer-Verlag, Berlin.


</DD>


<DT>[BHZ04]</DT>
<DD>
\anchor BHZ04
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Widening operators for powerset domains.
 Quaderno 349, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2004.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>.

</DD>


<DT>[BHZ05]</DT>
<DD>
\anchor BHZ05
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Not necessarily closed convex polyhedra and the double description
  method.
 <em>Formal Aspects of Computing</em>, 17(2):222-257, 2005.

</DD>


<DT>[BHZ06a]</DT>
<DD>
\anchor BHZ06a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 The Parma Polyhedra Library: Toward a complete set of numerical
  abstractions for the analysis and verification of hardware and software
  systems.
 Quaderno 457, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2006.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>. Also
  published as <tt>arXiv:cs.MS/0612085</tt>, available from
  <a href="http://arxiv.org/">http://arxiv.org/</a>.


</DD>


<DT>[BHZ06b]</DT>
<DD>
\anchor BHZ06b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Widening operators for powerset domains.
 <em>Software Tools for Technology Transfer</em>, 8(4/5):449-466, 2006.
 In the printed version of this article, all the figures have been
  improperly printed (rendering them useless). See
  \ref BHZ07c "[BHZ07c]".


</DD>


<DT>[BHZ07a]</DT>
<DD>
\anchor BHZ07a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Applications of polyhedral computations to the analysis and
  verification of hardware and software systems.
 Quaderno 458, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2007.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>. Also
  published as <tt>arXiv:cs.CG/0701122</tt>, available from
  <a href="http://arxiv.org/">http://arxiv.org/</a>.


</DD>


<DT>[BHZ07b]</DT>
<DD>
\anchor BHZ07b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 An improved tight closure algorithm for integer octagonal
  constraints.
 Quaderno 467, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2007.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>. Also
  published as <tt>arXiv:0705.4618v2 [cs.DS]</tt>, available from
  <a href="http://arxiv.org/">http://arxiv.org/</a>.


</DD>


<DT>[BHZ07c]</DT>
<DD>
\anchor BHZ07c
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Widening operators for powerset domains.
 <em>Software Tools for Technology Transfer</em>, 9(3/4):413-414, 2007.
 Erratum to \ref BHZ06b "[BHZ06b]" containing all the figures properly
  printed.

</DD>


<DT>[BHZ08a]</DT>
<DD>
\anchor BHZ08a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 An improved tight closure algorithm for integer octagonal
  constraints.
 In F.&nbsp;Logozzo, D.&nbsp;Peled, and L.&nbsp;Zuck, editors, <em>Verification,
  Model Checking and Abstract Interpretation: Proceedings of the 9th
  International Conference (VMCAI 2008)</em>, volume 4905 of <em>Lecture Notes in
  Computer Science</em>, pages 8-21, San Francisco, USA, 2008. Springer-Verlag,
  Berlin.


</DD>


<DT>[BHZ08b]</DT>
<DD>
\anchor BHZ08b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 The Parma Polyhedra Library: Toward a complete set of numerical
  abstractions for the analysis and verification of hardware and software
  systems.
 <em>Science of Computer Programming</em>, 72(1-2):3-21, 2008.


</DD>


<DT>[BHZ09a]</DT>
<DD>
\anchor BHZ09a
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Applications of polyhedral computations to the analysis and
  verification of hardware and software systems.
 <em>Theoretical Computer Science</em>, 410(46):4672-4691, 2009.


</DD>


<DT>[BHZ09b]</DT>
<DD>
\anchor BHZ09b
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Exact join detection for convex polyhedra and other numerical
  abstractions.
 Quaderno 492, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2009.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>. A corrected
  and improved version (corrected an error in the statement of condition (3) of
  Theorem&nbsp;3.6, typos corrected in statement and proof of Theorem&nbsp;6.8) has been
  published in \ref BHZ09c "[BHZ09c]".


</DD>


<DT>[BHZ09c]</DT>
<DD>
\anchor BHZ09c
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Exact join detection for convex polyhedra and other numerical
  abstractions.
 Report <tt>arXiv:cs.CG/0904.1783</tt>, 2009.
 Available at <a href="http://arxiv.org/">http://arxiv.org/</a> and
  <a href="http://www.cs.unipr.it/ppl/">http://www.cs.unipr.it/ppl/</a>.


</DD>


<DT>[BHZ09d]</DT>
<DD>
\anchor BHZ09d
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Weakly-relational shapes for numeric abstractions: Improved
  algorithms and proofs of correctness.
 <em>Formal Methods in System Design</em>, 35(3):279-323, 2009.


</DD>


<DT>[BHZ10]</DT>
<DD>
\anchor BHZ10
R.&nbsp;Bagnara, P.&nbsp;M. Hill, and E.&nbsp;Zaffanella.
 Exact join detection for convex polyhedra and other numerical
  abstractions.
 <em>Computational Geometry: Theory and Applications</em>,
  43(5):453-473, 2010.
 To appear in print. Available online at
  <a href="http://dx.doi.org/10.1016/j.comgeo.2009.09.002">http://dx.doi.org/10.1016/j.comgeo.2009.09.002</a>.


</DD>


<DT>[BJT99]</DT>
<DD>
\anchor BJT99
F.&nbsp;Besson, T.&nbsp;P. Jensen, and J.-P. Talpin.
 Polyhedral analysis for synchronous languages.
 In A.&nbsp;Cortesi and G.&nbsp;Fil&eacute;, editors, <em>Static Analysis:
  Proceedings of the 6th International Symposium</em>, volume 1694 of <em>Lecture
  Notes in Computer Science</em>, pages 51-68, Venice, Italy, 1999.
  Springer-Verlag, Berlin.

</DD>


<DT>[BK89]</DT>
<DD>
\anchor BK89
V.&nbsp;Balasundaram and K.&nbsp;Kennedy.
 A technique for summarizing data access and its use in parallelism
  enhancing transformations.
 In B.&nbsp;Knobe, editor, <em>Proceedings of the ACM SIGPLAN'89
  Conference on Programming Language Design and Implementation (PLDI)</em>, volume
  24(7) of <em>ACM SIGPLAN Notices</em>, pages 41-53, Portland, Oregon, USA,
  1989. ACM Press.

</DD>


<DT>[BMPZ10]</DT>
<DD>
\anchor BMPZ10
R.&nbsp;Bagnara, F.&nbsp;Mesnard, A.&nbsp;Pescetti, and E.&nbsp;Zaffanella.
 The automatic synthesis of linear ranking functions: The complete
  unabridged version.
 Quaderno 498, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2010.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>. Also
  published as <tt>arXiv:cs.PL/1004.0944</tt>, available from
  <a href="http://arxiv.org/">http://arxiv.org/</a>.

</DD>


<DT>[BRZH02a]</DT>
<DD>
\anchor BRZH02a
R.&nbsp;Bagnara, E.&nbsp;Ricci, E.&nbsp;Zaffanella, and P.&nbsp;M. Hill.
 Possibly not closed convex polyhedra and the Parma Polyhedra
  Library.
 In M.&nbsp;V. Hermenegildo and G.&nbsp;Puebla, editors, <em>Static Analysis:
  Proceedings of the 9th International Symposium</em>, volume 2477 of <em>Lecture
  Notes in Computer Science</em>, pages 213-229, Madrid, Spain, 2002.
  Springer-Verlag, Berlin.


</DD>


<DT>[BRZH02b]</DT>
<DD>
\anchor BRZH02b
R.&nbsp;Bagnara, E.&nbsp;Ricci, E.&nbsp;Zaffanella, and P.&nbsp;M. Hill.
 Possibly not closed convex polyhedra and the Parma Polyhedra
  Library.
 Quaderno 286, Dipartimento di Matematica, Universit&agrave; di Parma,
  Italy, 2002.
 See also \ref BRZH02c "[BRZH02c]". Available at
  <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>.

</DD>


<DT>[BRZH02c]</DT>
<DD>
\anchor BRZH02c
R.&nbsp;Bagnara, E.&nbsp;Ricci, E.&nbsp;Zaffanella, and P.&nbsp;M. Hill.
 Errata for technical report &ldquo;Quaderno 286&rdquo;.
 Available at <a href="http://www.cs.unipr.it/Publications/">http://www.cs.unipr.it/Publications/</a>,
  2002.
 See \ref BRZH02b "[BRZH02b]".

</DD>


<DT>[CC76]</DT>
<DD>
\anchor CC76
P.&nbsp;Cousot and R.&nbsp;Cousot.
 Static determination of dynamic properties of programs.
 In B.&nbsp;Robinet, editor, <em>Proceedings of the Second International
  Symposium on Programming</em>, pages 106-130, Paris, France, 1976. Dunod, Paris,
  France.


</DD>


<DT>[CC79]</DT>
<DD>
\anchor CC79
P.&nbsp;Cousot and R.&nbsp;Cousot.
 Systematic design of program analysis frameworks.
 In <em>Proceedings of the Sixth Annual ACM Symposium on Principles
  of Programming Languages</em>, pages 269-282, New York, 1979. ACM Press.


</DD>


<DT>[CC92]</DT>
<DD>
\anchor CC92
P.&nbsp;Cousot and R.&nbsp;Cousot.
 Comparing the Galois connection and widening/narrowing approaches
  to abstract interpretation.
 In M.&nbsp;Bruynooghe and M.&nbsp;Wirsing, editors, <em>Proceedings of the 4th
  International Symposium on Programming Language Implementation and Logic
  Programming</em>, volume 631 of <em>Lecture Notes in Computer Science</em>, pages
  269-295, Leuven, Belgium, 1992. Springer-Verlag, Berlin.


</DD>


<DT>[CH78]</DT>
<DD>
\anchor CH78
P.&nbsp;Cousot and N.&nbsp;Halbwachs.
 Automatic discovery of linear restraints among variables of a
  program.
 In <em>Conference Record of the Fifth Annual ACM Symposium on
  Principles of Programming Languages</em>, pages 84-96, Tucson, Arizona, 1978.
  ACM Press.


</DD>


<DT>[Che64]</DT>
<DD>
\anchor Che64
N.&nbsp;V. Chernikova.
 Algorithm for finding a general formula for the non-negative
  solutions of system of linear equations.
 <em>U.S.S.R. Computational Mathematics and Mathematical Physics</em>,
  4(4):151-158, 1964.

</DD>


<DT>[Che65]</DT>
<DD>
\anchor Che65
N.&nbsp;V. Chernikova.
 Algorithm for finding a general formula for the non-negative
  solutions of system of linear inequalities.
 <em>U.S.S.R. Computational Mathematics and Mathematical Physics</em>,
  5(2):228-233, 1965.

</DD>


<DT>[Che68]</DT>
<DD>
\anchor Che68
N.&nbsp;V. Chernikova.
 Algorithm for discovering the set of all solutions of a linear
  programming problem.
 <em>U.S.S.R. Computational Mathematics and Mathematical Physics</em>,
  8(6):282-293, 1968.

</DD>


<DT>[Dan63]</DT>
<DD>
\anchor Dan63
G.&nbsp;B. Dantzig.
 <em>Linear Programming and Extensions</em>.
 Princeton University Press, Princeton, NJ, 1963.

</DD>


<DT>[FCB07]</DT>
<DD>
\anchor FCB07
P.&nbsp;Feautrier, J.-F. Collard, and C.&nbsp;Bastoul.
 <em>PIP/PipLib: A Solver for Parametric Integer Programming
  Problems</em>, 5.0 edition, July 2007.
 Distributed with PIP/PipLib 1.4.0.

</DD>


<DT>[Fea88]</DT>
<DD>
\anchor Fea88
P.&nbsp;Feautrier.
 Parametric integer programming.
 <em>RAIRO Recherche Op&eacute;rationnelle</em>, 22(3):243-268, 1988.

</DD>


<DT>[FP96]</DT>
<DD>
\anchor FP96
K.&nbsp;Fukuda and A.&nbsp;Prodon.
 Double description method revisited.
 In M.&nbsp;Deza, R.&nbsp;Euler, and Y.&nbsp;Manoussakis, editors, <em>Combinatorics
  and Computer Science, 8th Franco-Japanese and 4th Franco-Chinese Conference,
  Brest, France, July 3-5, 1995, Selected Papers</em>, volume 1120 of <em>Lecture
  Notes in Computer Science</em>, pages 91-111. Springer-Verlag, Berlin, 1996.


</DD>


<DT>[Fuk98]</DT>
<DD>
\anchor Fuk98
K.&nbsp;Fukuda.
 Polyhedral computation FAQ.
 Swiss Federal Institute of Technology, Lausanne and Zurich,
  Switzerland, available at
  <a href="http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html">http://www.ifor.math.ethz.ch/~fukuda/polyfaq/polyfaq.html</a>, 1998.

</DD>


<DT>[GDD<sup>+</sup>04]</DT>
<DD>
\anchor GDDetal04
D.&nbsp;Gopan, F.&nbsp;DiMaio, N.&nbsp;Dor, T.&nbsp;W. Reps, and M.&nbsp;Sagiv.
 Numeric domains with summarized dimensions.
 In K.&nbsp;Jensen and A.&nbsp;Podelski, editors, <em>Tools and Algorithms for
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<DD>
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D.&nbsp;K. Wilde.
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</DD>
</DL>


\if Include_Implementation_Details

\section prelims Further Notation and Terminology

\subsection Linear_Independence Linear Independence

A finite set of points
\f$\{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n\f$
is <EM>linearly independent</EM> if,
for all \f$\lambda_1, \ldots, \lambda_k \in \Rset\f$,
the set of equations
\f[
  \sum_{i = 1}^k \lambda_i \vect{x}_i = \vect{0}
\f]
implies that, for each \f$i = 1\f$, \f$\ldots\f$, \f$k\f$,
\f$\lambda_i = 0\f$.

The maximum number of linearly independent points in \f$\Rset^n\f$ is \f$n\f$.
Note that linear independence implies affine independence,
but the converse is not true.

<B><EM>Proposition</EM></B>

If \f$A\f$ is an \f$m \times n\f$ matrix, the maximum number of
linearly independent rows of \f$A\f$, viewed as vectors of \f$\Rset^n\f$,
equals the maximum number of linearly independent columns of \f$A\f$,
viewed as vectors of \f$\Rset^m\f$.

\subsection Rank Rank

The maximum number of linearly independent rows (columns) of a
matrix \f$A\f$ is the <EM>rank</EM> of \f$A\f$ and is denoted by
\f$\prank(A)\f$.

<B><EM>Proposition</EM></B>

A polyhedron is a convex set.

\subsection Minkowskis_Theorem Minkowski's Theorem

Let
\f$\cP = \{\, \vect{x} \in \Rset^n \mid A\vect{x} \geq \vect{b} \,\}\f$
be a non-empty polyhedron where \f$\prank(A) = n\f$.
Let \f$V\f$ be the set of vertices and
\f$R\f$ the set of extreme rays of \f$\cP\f$.
Let also \f$\mathcal{V}\f$ be the set of convex combinations of \f$V\f$
and \f$\mathcal{R}\f$ the set of positive combinations of \f$R\f$.
Then
\f[
  \cP = \mathcal{V} + \mathcal{R}.
\f]

Informally, this theorem states that,
whenever a polyhedron \f$\cP\f$ has a vertex,
there exists a decomposition such that
- \f$V\f$ is the set of all <EM>vertices</EM> of \f$\cP\f$;
- \f$R\f$ is the set of all <EM>extreme</EM> rays of \f$\cP\f$; and
- \f$L = \emptyset\f$.

The conditions that \f$\cP\f$ is not empty and \f$\prank(A) = n\f$
are equivalent to the condition that \f$\cP\f$ has a vertex.
(See also Nemhauser and Wolsey - Integer and Combinatorial Optimization -
propositions 4.1 and 4.2 on pages 92 and 93).

<B><EM>Proposition</EM></B>

Under the same hypotheses of Minkowski's theorem,
if \f$\cP\f$ is a rational polyhedron then
all the vertices in \f$V\f$ have rational coefficients and
we can consider a set \f$R\f$ of extreme rays having
rational coefficients only.

The second theorem, called Weyl's theorem, states that any system of
generators having rational coefficients defines a rational polyhedron:

\subsection Weyls_Theorem Weyl's Theorem

If \f$A\f$ is a rational \f$m \times n\f$ matrix,
\f$B\f$ is a rational \f$m' \times n\f$ matrix and
\f[
  \cQ = \sset{
          \vect{x} \in \Rset^n
        }{
          \vect{x}^\transpose = \vect{y}^\transpose A
                                  + \vect{z}^\transpose B, \\
          \vect{y} = (y_0, \ldots, y_{m-1})^\transpose \in \nonnegRset^{m},
          \sum_{k=0}^{m-1} y_k = 1, \\
	  \vect{z} \in \nonnegRset^{m'}
        },
\f]
then \f$\cQ\f$ is a rational polyhedron.

In fact, since \f$\cQ\f$ consists of the sum of convex combinations of
the rows of \f$A\f$ with positive combinations of the rows of \f$B\f$,
we can think of \f$A\f$ as the matrix of vertices and \f$B\f$ as
the matrix of rays.

\subsection Cone Cone

A set \f$C \sseq \Rset^n\f$ is a <EM>cone</EM> if
\f[
  \vect{x} \in C \Rightarrow \lambda \vect{x} \in C
  \text{ for all } \lambda \in \nonnegRset.
\f]

\subsection Polyhedral_Cone Polyhedral Cone

The polyhedron
\f$\cP = \{\,\vect{x} \in \Rset^n \mid A\vect{x} \geq \vect{0}\,\}\f$
is a convex cone and is called <EM>polyhedral cone</EM>.

A polyhedral cone is either <EM>pointed</EM>,
having the origin as its only vertex, or has no vertices at all.

\subsection Lineality_Space Lineality Space

Given a polyhedron
\f$\cP = \{\,\vect{x} \in \Rset^n \mid A\vect{x} \geq \vect{b}\,\}\f$,
the <EM>lineality space</EM> of \f$\cP\f$ is the set
\f[
  \{\, \vect{x} \in \cP \mid A\vect{x} = \vect{0} \,\}
\f]
and it is denoted by \f$\linspace(\cP)\f$.


\section homogeneous Homogeneous Systems

To simplify the operations on polyhedra,
each polyhedron is first transformed to a homogeneous cone
in which the original polyhedron is embedded.

\subsection Corresponding_Polyhedral_Cone Corresponding Polyhedral Cone

The transformation changes the inhomogeneous system of constraints
in \f$n\f$ variables, representing a polyhedron \f$\cP \in \Rset^n\f$,
into a homogeneous system in \f$n + 1\f$ variables, representing a
polyhedral cone \f$C \in \Rset^{n + 1}\f$,
so that each point \f$\vect{x} \in \cP\f$
corresponds to a point
\f$\vect{x}' = (\xi \vect{x}^\transpose, \xi)^\transpose \in C\f$
where \f$\xi \geq 0\f$. That is,
\f[
  \cP = \{\,\vect{x} \mid A\vect{x} \geq \vect{b}\,\}
      = \{\,\vect{x} \mid A\vect{x} - \vect{b} \geq \vect{0}\,\}
\f]
\f[
  C = \{\,
           (\xi \vect{x}^\transpose, \xi)^\transpose
      \mid
           \xi A\vect{x} - \xi \vect{b}
	   \geq \vect{0}, \xi \geq 0
      \,\}
    = \{\,\vect{x}' \mid A'\vect{x}' \geq \vect{0}\,\}
\f]
where:
\f$\vect{x}' = (\xi \vect{x}^\transpose, \xi)^\transpose \in \Rset^{n + 1}\f$;
\f$A'\f$ is the \f$(m+1) \times (n+1)\f$ matrix having, for
its first \f$m\f$ rows, the submatrix
\f$(A, -\vect{b}) \in \Rset^m \times \Rset^{n + 1}\f$;
and, for the (\f$m + 1\f$)'st row,
\f$(\vect{0}^\transpose, 1)\f$ where \f$\vect{0} \in \Rset^n\f$.
We call  \f$C\f$ the
<EM>corresponding polyhedral cone for \f$\cP\f$</EM>.

The (\f$m+1\f$)'st row
\f$(\vect{0}^\transpose, 1)\f$
represents the <EM>positivity constraint</EM> \f$1 \geq 0\f$.

Note that \f$\cP\f$ is contained in \f$C\f$
since the intersection of \f$C\f$ with
the hyperplane defined by the equality \f$\xi = 1\f$ is \f$\cP\f$.
Therefore, it is always possible to
transform  a polyhedron \f$\cP\f$ to its corresponding polyhedral cone
\f$C\f$ and then recover \f$\cP\f$ by means of this intersection.

As  \f$C\f$ always includes the origin and, hence, is non-empty,
by Minkowski's theorem,
it can also be represented by a system of generators.

The systems of generators for \f$\cP\f$ and \f$C\f$ are such that:
- Each vertex \f$\vect{v}\f$ in \f$\cP\f$
  corresponds to a ray \f$(\vect{v}^\transpose, d)^\transpose\f$
  with \f$d \neq 0\f$, in \f$C\f$.
- Each ray \f$\vect{r}\f$ in \f$\cP\f$
  corresponds to the ray \f$(\vect{r}^\transpose, 0)^\transpose\f$
  in \f$C\f$.
- Every ray in \f$C\f$ corresponds to a vertex or ray in \f$\cP\f$.
- The origin in \f$\Rset^{n+1}\f$ is a point in \f$C\f$.

Thus, in the cone \f$C\f$, a ray derived from a vertex
in \f$\cP\f$ differs from a ray derived from a ray in \f$\cP\f$
only in that, for a vertex,
the (\f$n+1\f$)'st term is different from zero and,
for a ray, it is zero.

\subsection Devref_Double_Description Double Description

Let \f$\cP \in \Rset^n \f$ be a polyhedron and  \f$C \in \Rset^{n+1} \f$
the corresponding polyhedral cone.
Then the dual representations, the systems of constraints
and generators representing \f$C\f$, form the
<EM>double description</EM> for \f$\cP\f$.

Note that, in a double description for a non-empty polyhedron,
the system of constraints subsumes the positivity constraint \f$1 \geq 0\f$
while the system of generators (which has only rays and lines
corresponding to the vertices, rays and lines for \f$\cP\f$)
implicitly assumes the origin in \f$\Rset^{n+1}\f$
as a point so that the cone \f$C\f$ represented by the generators is non-empty.

\subsection PPL_Polyhedron_Representation PPL Polyhedron Representation

In the PPL,
a polyhedron is represented by one or both of the representations in
its double description.
Thus, in the sequel, by <EM>PPL representation of a polyhedra</EM>,
we are referring to the corresponding representation of
its corresponding polyhedral cone.

\subsection Valid_Linear_Inequalities Valid Linear Inequalities

Let \f$\cP\f$ be a convex polyhedron (or polytope) in
\f$\Rset^n\f$. For a real \f$n\f$-vector \f$\vect{c}\f$ and a real number
\f$b\f$, a linear inequality \f$\langle \vect{c}, \vect{x} \rangle \geq b\f$
(briefly denoted by \f$(\vect{c},b)\f$) is called <EM>valid</EM>
for \f$\cP\f$ if it is satisfied by all points \f$\vect{x} \in \cP\f$.

\subsection Redundancy Redundancy

-# In a system of equalities, if an equality is a linear combination
   of the others, it is said to be <EM>dependent</EM> upon them;
   the dependent equality is called <EM>redundant</EM>.
   A system containing no redundant equality is called <EM>independent</EM>.
-# In a system of inequalities, an inequality is said to be <EM>redundant</EM>
   if it can be eliminated from the system obtaining a system
   equivalent to the previous one, i.e., having the same solutions.

Given a polyhedron \f$\cP\f$ generated by \f$V\f$ vertices,
\f$R\f$ rays and \f$L\f$ lines, we say that:

-# \f$L\f$ is <EM>irredundant</EM> if
  \f$L\f$ is a set of linearly independent lines; and
-# a ray \f$\vect{r}_1\in R\f$ is <EM>redundant</EM>
  if there exists another ray \f$\vect{r}_2 \in R\f$ and
  there exists \f$\lambda \in \Rset, \lambda > 0\f$
  such that \f$\vect{r}_1 = \lambda \vect{r}_2\f$.

Note that, in the PPL representation of a polyhedron \f$\cP\f$,
vertices are represented as rays so that
this concept of a redundant ray also applies to the vertices of \f$\cP\f$.


\subsection Face Face

If \f$(\vect{c},b)\f$ is a valid inequality for \f$\cP\f$, and
\f$F = \{\,\vect{x} \in \cP \mid \langle \vect{c}, \vect{x} \rangle = b\,\}\f$,
\f$F\f$ is called a <EM>face</EM> of \f$\cP\f$ and we say that the inequality
represents \f$F\f$.
A face \f$F\f$ is said to be <EM>proper</EM> if \f$F \neq \emptyset\f$ and
\f$F \neq \cP\f$.

When \f$F\f$ is non-empty, we say that \f$(\vect{c},b)\f$ <EM>supports</EM>
\f$\cP\f$.

The empty polyhedron and the universe polyhedron both have no proper faces,
because the only face of an empty polyhedron is itself, while the faces of
the universe polyhedron are itself and the emptyset.

Let \f$\cP\f$ be a non-empty polyhedron. The set
\f[
  F = \{ \vect{p} \} + \linspace(\cP),
\f]
where \f$\vect{p}\f$ is a point of \f$\cP\f$ and the symbol
'\f$+\f$' denotes the Minkowski's sum, is a <EM>minimal proper face</EM> of
the polyhedron if \f$F\f$ is a proper face of \f$\cP\f$.

\subsection Facet Facet

A proper face \f$F\f$ of \f$\cP\f$ is a <EM>facet</EM> (or <EM>maximal proper
face</EM>) of \f$\cP\f$ if it is not strictly included into any other proper
face of \f$\cP\f$.
The affine dimension of a facet is equal to \f$\pdim(\cP) - 1\f$.

<B><EM>Proposition</EM></B>

Let \f$\cP\f$ a polyhedron in \f$\Rset^n\f$. The set of all faces is a lattice
under inclusion: the minimal face is the emptyset, while the maximal face
is the polyhedron.

<B><EM>Proposition</EM></B>

Let \f$\cP \neq \emptyset\f$ be a polyhedron in \f$\Rset^n\f$ and
\f$C\f$ be the polyhedral cone in \f$\Rset^{n+1}\f$ obtained from \f$\cP\f$
by homogenization, then:
-# the only <EM>minimal proper face</EM> of \f$C\f$ is
   \f$\linspace(C)\f$;
-# let \f$\vect{y} \in C\f$ be different from \f$\vect{0}\f$ and
   \f$\cone\{\vect{y}\}\f$ be defined as
   \f$\{\, \lambda \vect{y} \mid \lambda \geq 0 \,\}\f$. If the set
   \f$F = \cone\{\vect{y}\} + \linspace(C)\f$ is
   a proper face of \f$C\f$, then \f$\vect{y}\f$ is an <EM>extremal ray</EM>
   of \f$C\f$.

\subsection Ray_Space Ray Space

Given the decomposition \f$\mathcal{V} + \mathcal{R} + \mathcal{L}\f$
of a polyhedron \f$\cP\f$ the set \f$\mathcal{V} + \mathcal{R}\f$
is called the <EM>ray space</EM> of \f$\cP\f$
and denoted by \f$\mathop{\mathrm{ray space}}(\cP)\f$.

Thus a polyhedron \f$\cP\f$ can be always decomposed in its
\f$\linspace\f$ and its \f$\mathop{\mathrm{ray space}}\f$.

Note that, since \f$\linspace(\cP)\f$ and
\f$\mathop{\mathrm{ray space}}(\cP)\f$ are polyhedra, their
affine dimensions can be computed using the definition of affine
dimension given for polyhedra.

The spaces defined are connected by some consistency rules shown below.

\subsection Dimensionality_Rules Dimensionality Rules

In \f$\Rset^n\f$

- The dimension of the \f$\linspace\f$
  is the rank of any set of lines that span the space.
- The dimension of the polyhedron is the dimension of the
  \f$\mathop{\mathrm{ray space}}\f$ plus the dimension of the
  \f$\linspace\f$.
- The dimension of the \f$\mathop{\mathrm{ray space}}\f$
  is \f$n\f$ minus the number of irredundant
  lines minus the number of irredundant equalities.

The proofs of these properties can be obtained considering the definitions of
affine dimension and the decomposition of a polyhedron.

\subsection Saturation Saturation

Let us consider a ray \f$\vect{r} \in \Rset^n\f$ and an inequality
\f$(\vect{a}, 0)\f$ where \f$\vect{a} \in \Rset^n\f$.
Then we say that:
- \f$r\f$ <EM>saturates</EM> the inequality
  if \f$\langle \vect{a}, \vect{r} \rangle = 0\f$;
- \f$r\f$ <EM>verifies</EM>  the inequality
  if \f$\langle \vect{a}, \vect{r} \rangle > 0\f$;
- \f$r\f$ <EM>violates</EM>  the inequality
  if \f$\langle \vect{a}, \vect{r} \rangle < 0\f$.

Similarly, considering an equality
\f$\langle \vect{a}, \vect{x} \rangle = 0\f$:
- \f$\vect{r}\f$ <EM>saturates</EM> the equality
if \f$\langle \vect{a}, \vect{r} \rangle = 0\f$;
- \f$\vect{r}\f$ <EM>does not verify</EM> the equality
if \f$\langle \vect{a}, \vect{r} \rangle \neq 0\f$.

A constraint (i.e., an equality or an inequality) is
<EM>satisfied</EM> by a ray if the ray saturates or verifies the constraint.

<B><EM>Proposition</EM></B>

Let \f$C \sseq \Rset^n\f$ be a polyhedral cone and
\f$\vect{y}_1, \vect{y}_2 \in C\f$. If the sets
\f$F_i = \cone\{\vect{y}_i\} + \linspace(C)\f$ with
\f$i = 1, 2\f$ are proper faces of \f$C\f$, \f$F_1\f$ is equal to \f$F_2\f$
if and only if the set of constraints that are saturated by \f$\vect{y}_1\f$
is equal to the set of constraints that are saturated by \f$\vect{y}_2\f$.


\subsection Saturation_Matrix Saturation Matrix

A <EM>saturation matrix</EM> is a bit matrix that represents
the connection between constraints and generators of a polyhedron.
There are two kinds of saturation matrices, one having rows indexed
by constraints and columns indexed by generators (<EM>sat_g</EM>),
and one (that is the transposed version of the previous one) having
rows indexed by generators and columns indexed by constraints (<EM>sat_c</EM>).

For instance, in the saturation matrix sat_g, the elements are defined
as follows:
\f[
s_{ij} =
\begin{cases}
0, \text{if the constraint indexed by } i \text{ is saturated by
   the generator indexed by } j;\\
1, \text{if the constraint indexed by } i \text{ is only
  verified by the generator indexed by } j.
\end{cases}
\f]
For efficiency reasons, the PPL uses both the sat_g and sat_c matrices.

\subsection Saturation_Rule Saturation Rule

In an \f$n\f$-dimensional \f$\mathop{\mathrm{ray space}}\f$,
-# Every inequality must be saturated by at least \f$n\f$ vertices/rays.
-# Every vertex must saturate at least \f$n\f$ inequalities and a ray
   must saturate at least \f$n - 1\f$ inequalities plus the
   positivity constraint.
-# Every equality must be saturated by all lines and vertices/rays.
-# Every line must saturate all equalities and inequalities.

These rules are a consequence of the saturation concept.

<B><EM>Proposition</EM></B>

Let \f$C = \{\vect{x} \mid A\vect{x} \geq \vect{0}\}\f$ be a polyhedral cone.
Then the minimal proper face of \f$C\f$ in an \f$n\f$-dimensional space can
also be represented as
\f$
  F = \{\,\vect{x} \mid A\vect{x} = \vect{0}\,\}.
\f$

To see this, note that the minimal proper face of a polyhedral cone is equal
to its lineality space. This for definition is composed by all \f$\vect{y}\f$
of \f$C\f$ that satisfies \f$A \vect{x} = \vect{0}.\f$

\subsection Adjacent_Rays Adjacent Rays

Let \f$A\f$ be representing matrix of constraints of a cone \f$C\f$
and \f$Q\f$ the set of rays that generate \f$C\f$. Then two rays
\f$\vect{r}_1\f$ and \f$\vect{r}_2\f$ are <EM>adjacent rays</EM> if
-# there exists at least a row of \f$A\f$ (i.e., a constraint) that
   is saturated by both \f$\vect{r}_1\f$ and \f$\vect{r}_2\f$
-# and none of the rays of \f$Q\f$, except \f$\vect{r}_1\f$ and
   \f$\vect{r}_2\f$, saturates all the constraints saturated by both
   \f$\vect{r}_1\f$ and \f$\vect{r}_2\f$.

\subsection Independence_Rule Independence Rule

-# No inequality is a positive combination of any other two inequalities
   or equalities.
-# No ray is a linear combination of any other two rays or lines.
-# The set of equalities must be linearly independent.
-# The set of lines must be linearly independent.

To remove redundant constraints/generators we will use the following
characterization:

\subsection Redundancy_Rules Redundancy Rules

- An inequality is not redundant if it
  satisfies both point (1) of the saturation rule and point (1)
  of the independence rule.
- A vertex/ray is irredundant if it satisfies
  both point (2) of the saturation rule and point (2) of the independence rule.

It is useful to note that:
- All rays saturate the positivity constraint and no vertex saturates
  the positivity constraint; in fact in the homogeneous form
  the positivity constraint is represented by the vector
  \f$\vect{a}^\transpose = (0, \ldots, 0, 1)\f$, rays are of the form
  \f$\vect{r} = (r_0, \ldots, r_{n-1}, 0)^\transpose\f$ and vertices
  \f$\vect{v} = (v_0, \ldots, v_{n-1}, d)^\transpose\f$ with \f$d \neq 0\f$,
  thus \f$\langle \vect{a}, \vect{r} \rangle = 0\f$ for each ray
  \f$\vect{r}\f$ and \f$\langle \vect{a}, \vect{v} \rangle \neq 0\f$
  for each vertex \f$\vect{v}\f$.
- The positivity constraint will be irredundant if and only if
  the size of the set
  of rays is \f$\geq n\f$, where \f$n\f$ is the dimension of the ray space,
  and the rank of the ray set is \f$n\f$; in fact a constraint is
  irredundant if it is saturated by at least \f$n\f$ vertices/rays (see
  above), but since only rays saturate the positivity constraint, then in
  a system with \f$n\f$ vertices/rays the positivity constraint
  is irredundant.

\section integer_floats Integers Represented by Floating Point Numbers

Floating point numbers can be used to represent finite families
of integer numbers.  In this section we collect some closure properties
of these families that are exploited in the PPL.

In order not to depend on the particular family of floating point
numbers considered, we consider an abstraction that is parametric
in the number \f$b\f$ of bits in the mantissa and gives no limit
to the magnitude of the exponent \f$e\f$.
For \f$b \in \Nset \setminus \{ 0 \}\f$ let
\f[
\begin{aligned}
  F_b^+
    &=
      \bigl\{\,
        x \in \Nset
      \bigm|
        x = (1 + m / 2^b) \cdot 2^e,
        e \in \Nset,
        m \in \Nset \cap [0, 2^b - 1]
      \,\bigr\}, \\
  F_b
    &=
      F_b^+ \cup \{ 0 \} \cup \{\, -x \mid x \in F_b^+ \,\}.
\end{aligned}
\f]

Let \f$\phi \colon \Rset \to \Zset\f$ denote the function defined by
\f[
  \phi(t)
    =
      \begin{cases}
        \lfloor t \rfloor, & \text{if $t \ge 0$;} \\
        \lceil  t \rceil,  & \text{if $t  <  0$.}
      \end{cases}
\f]
Notice that \f$\phi\f$ is an <EM>odd</EM> function, that is, it satisfies
\f$\phi(-t) = -\phi(t)\f$ for all \f$t \in \Rset\f$.
For \f$x\f$, \f$y \in \Zset\f$ with \f$y \ne 0\f$, we also write
\f[
\begin{aligned}
  x \bdiv y &= \phi(x / y), \\
  x \brem y &= x - (x \bdiv y) y.
\end{aligned}
\f]
These are the integer division and remainder function as defined by
the C99 standard
[ISO/IEC 9899:1999(E), Programming Languages - C (ISO and ANSI C99 Standard)].

<B><EM>Proposition A</EM></B>
If \f$x\f$, \f$y \in F_b\f$ and \f$y \ne 0\f$, then \f$x \brem y \in F_b\f$.

The proof is given in the next three lemmas.

<B><EM>Lemma 1</EM></B>
Let \f$G_b = \{\, n \in F_b^+ \mid \text{$n$ is odd} \,\}\f$.
Then \f$G_b = \{ 1, 3, 5, \ldots, 2^{b + 1} - 1 \}\f$.
Furthermore, if \f$x \in F_b^+\f$ then there exist \f$n \in G_b\f$ and
\f$f \in \Nset\f$ such that \f$x = n \cdot 2^f\f$.

<B><EM>Proof</EM></B>
Let \f$n \in \{ 1, 3, 5, \ldots, 2^{b + 1} - 1 \}\f$.
There is a non negative integer \f$\beta \le b\f$ such that
\f$2^\beta \le n < 2^{\beta + 1}\f$.
Then \f$n = (1 + m / 2^b) \cdot 2^e\f$ with
\f$m = (n - 2^\beta) \cdot 2^{b - \beta}\f$ and \f$e = \beta\f$.
Here \f$m < (2^{\beta + 1} - 2^\beta) \cdot 2^{b - \beta} = 2^b\f$
so that \f$n \in G_b\f$.
The same argument shows that odd integers larger than \f$2^{b+1}\f$ do not
in fact belong to \f$G_b\f$, since the corresponding value of \f$m\f$ would
exceed the bound \f$2^b - 1\f$ in the definition.

For the second part, let \f$x = (1 + m / 2^b) \cdot 2^e \in F_b^+\f$.
Let \f$m = 2^d \cdot m_1\f$ with \f$m_1\f$ odd and \f$d < b\f$.
Then \f$n = 2^{b - d} + m_1\f$ is an odd integer that belongs to \f$G_b\f$
since \f$2^{b - d} + m_1 \le 2^{b - d} + (2^b - 1) / 2^d < 2^{b - d + 1}
\le 2^{b + 1}\f$, using the first part.
Hence we may take \f$f = e + d - b\f$ which is non negative since
otherwise \f$m \cdot 2^{e - b} = m_1 \cdot 2^{e + d - b}\f$ would not be
an integer as assumed.

<B><EM>Lemma 2</EM></B>
If \f$x\f$, \f$y \in F_b^+\f$ and \f$y\f$ does not divide \f$x\f$, then
\f$x \bmod y \in F_b^+\f$.

<B><EM>Proof</EM></B>
By Lemma 1 above we may assume that \f$x = n \cdot 2^e\f$ and
\f$y = m \cdot 2^f\f$ with \f$n\f$, \f$m \in G_b\f$ odd integers, and \f$e\f$,
\f$f \in \Nset\f$.
Let \f$k = \lfloor x / y \rfloor\f$.
The goal is to prove that \f$x - k y \in F_b^+\f$: we may assume that
\f$k > 0\f$, that is, that \f$x > y\f$ for otherwise \f$x \bmod y = x\f$
and there is nothing to prove.
- If \f$e < f\f$ then \f$x - k y = 2^e (n - k m \cdot 2^{f - e})\f$.
  The integer \f$n - k m \cdot 2^{f - e}\f$ is positive, odd and smaller
  than \f$n\f$, and therefore belongs to \f$G_b\f$.
- If \f$e = f\f$ then \f$x - k y = 2^e (n - k m)\f$.
  The integer \f$n - k m\f$ is positive and smaller than \f$n\f$, and therefore
  belongs to \f$F_b^+\f$.
- If \f$e > f\f$ then \f$x - k y = 2^f (n \cdot 2^{e - f} - k m)\f$.
  The integer \f$n \cdot 2^{e - f} - k m\f$ is positive and smaller
  than \f$m\f$, and therefore belongs to \f$F_b^+\f$: in fact
  \f[
    n \cdot 2^{e - f} - k m
    =
    n \cdot 2^{e - f} - \Bigl\lfloor \frac xy \Bigr\rfloor m
    =
    n \cdot 2^{e - f} - \Bigl\lfloor \frac{n \cdot 2^{e - f}}m \Bigr\rfloor m.
  \f]
  In other words, this integer is \f$n \cdot 2^{e - f} \bmod m\f$ and
  therefore it is smaller than \f$m\f$.

In all cases, we wrote \f$x - k y\f$ as the product of a power of 2 and an
element of \f$F_b^+\f$, and this product is another element of \f$F_b^+\f$.


<B><EM>Lemma 3</EM></B>
For \f$x\f$, \f$y \in \Zset\f$ with \f$y \ne 0\f$, we have
\f[
  x \brem y
  =
  \begin{cases}
       x  \brem \abs{y},  & \text{if $x \ge 0$;} \\
    -(\abs{x} \brem \abs{y}), & \text{if $x  <  0$.}
  \end{cases}
\f]

<B><EM>Proof</EM></B>
Throughout the proof we write \f$x_0 = \abs{x}\f$ and \f$y_0 = \abs{y}\f$.
First, assume that \f$x \ge 0\f$ and that \f$y < 0\f$.
Let \f$k = \phi(x / y_0) = - \phi(x / y)\f$, by the property above.
We have
\f[
  x \brem y
  =
  x - (x \bdiv y) y
  =
  x - (-k) y
  =
  x - k (-y)
  =
  x - k y_0.
\f]
Next, assume that \f$x < 0\f$ and that \f$y < 0\f$.
Let \f$k = \phi(x_0 / y_0) = \phi(x / y)\f$.
We have
\f[
  x \brem y
  =
  x - (x \bdiv y) y
  =
  x - k y
  =
  -( -x - k (-y))
  =
  -(x_0 - k y_0).
\f]
Finally, assume that \f$x < 0\f$ and that \f$y > 0\f$.
Let \f$k = \phi(x_0 / y) = -\phi(x / y)\f$, again by the property above.
We have
\f[
  x \brem y
  =
  x - (x \bdiv y) y
  =
  x - (-k) y
  =
  -( -x - k y)
  =
  -(x_0 - k y).
\f]
This completes the proof.

<B><EM>Lemma 4</EM></B>
If \f$x\f$, \f$y \in F_b^+\f$ then \f$\gcd(x, y) \in F_b^+\f$.

<B><EM>Proof</EM></B>
Let \f$x = n \cdot 2^e\f$ and \f$y = m \cdot 2^f\f$ with \f$n\f$, \f$m \in G_b\f$
odd integers, and \f$e\f$, \f$f \in \Nset\f$.
Then \f$\gcd(x, y) = \gcd(n, m) \cdot 2^{\min(e, f)}\f$, and therefore it
belongs to \f$F_b^+\f$, since \f$\gcd(n, m) \le \min(n, m)\f$ so that it belongs
to \f$G_b\f$.

<B><EM>Lemma 5</EM></B>
If \f$x\f$, \f$y \in F_b^+\f$, then \f$x / \gcd(x, y) \in F_b^+\f$.

<B><EM>Proof</EM></B>
With the same notation as in the previous Lemma, both \f$n\f$ and
\f$\gcd(n, m) \in G_b\f$: but all positive odd integers up to and
including \f$n\f$ belong to \f$G_b\f$, so that \f$n / \gcd(n, m)\f$
does as well.
By Lemma 1
\f$x / \gcd(x, y) = n \cdot 2^e / (\gcd(n, m) \cdot 2^{\min(e, f)})
                = (n / \gcd(n, m)) \cdot 2^{e - \min(e, f)} \in F_b^+\f$.

\endif

*/ /* \mainpage */
