/* Configuration Independent Documentation for Prolog.
   Copyright (C) 2001-2010 Roberto Bagnara <bagnara@cs.unipr.it>
   Copyright (C) 2010-2011 BUGSENG srl (http://bugseng.com)

This file is part of the Parma Polyhedra Library (PPL).

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.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software Foundation,
Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02111-1307, USA.

For the most up-to-date information see the Parma Polyhedra Library
site: http://www.cs.unipr.it/ppl/ . */

/*!
  \defgroup PPL_Prolog_interface Prolog Language Interface

  The Parma Polyhedra Library comes equipped with an interface
  for the Prolog language.
*/

/*! \mainpage Prolog Language Interface

The Parma Polyhedra Library comes equipped with a Prolog interface.
Despite the lack of standardization of Prolog's foreign language interfaces,
the PPL Prolog interface supports several Prolog systems and, to the
extent this is possible, provides a uniform view of the library from
each such system.

The system-independent features of the library are described in
Section \ref PI_SI_Features "System-Independent Features".
Section \ref PI_Compilation "Compilation and Installation"
explains how the Prolog interface is compiled and installed.
Section \ref PI_SD_Features "System-Dependent Features"
illustrates the system-dependent features of the interface
for all the supported systems.

The structure of this section is as follows:
- \ref PI_SI_Features "System-Independent Features"
    - \ref Prolog_Interface_Overview "Overview"
    - \ref Predicate_Specifications "Predicate Specifications"
    - \ref predicate_descriptions "Predicate Descriptions"
        - \ref di_predicates "Domain Independent Predicates"
        - \ref mip_predicates "MIP Predicates"
        - \ref pip_predicates "PIP Predicates"
        - \ref Polyhedron_predicates "Predicates for C Polyhedra"
        - \ref other_domains "Ad hoc Predicates for Other Domains"
- \ref PI_Compilation "Compilation and Installation"
- \ref PI_SD_Features "System-Dependent Features"
    - \ref GNU_Prolog "GNU Prolog"
    - \ref CIAO_Prolog "CIAO Prolog"
    - \ref SICStus_Prolog "SICStus Prolog"
    - \ref SWI_Prolog "SWI Prolog"
    - \ref XSB_Prolog "XSB"
    - \ref YAP_Prolog "YAP"

In the sequel, <CODE>prefix</CODE> is the prefix under which
you have installed the library (typically <CODE>/usr</CODE>
or <CODE>/usr/local</CODE>).

\anchor PI_SI_Features
<H1>System-Independent Features</H1>

The Prolog interface provides access to the numerical abstractions
(convex polyhedra, BD shapes, octagonal shapes, etc.) implemented
by the PPL library.
A general introduction to the numerical abstractions,
their representation in the PPL and the operations provided
by the PPL is given in the main \extref{preamble, PPL user manual}.
Here we just describe those aspects that are specific to the Prolog interface.

\anchor Prolog_Interface_Overview
<H2>Overview</H2>

First, here is a list of notes with general information and advice
on the use of the interface.

- The numerical abstract domains available to the Prolog user consist
  of the <EM>simple</EM> domains, <EM>powersets</EM> of a simple domain and
  <EM>products</EM> of simple domains.
  - The simple domains are:
     - convex polyhedra, which consist of C_Polyhedron and
       NNC_Polyhedron;<BR>
     - weakly relational, which consist of BD_Shape_N and
       Octagonal_Shape_N
       where N is one of the numeric types
       int8, int16, int32, int64, mpz_class, mpq_class,
       float, double, long_double;<BR>
     - boxes which consist of
       Int8_Box, Int16_Box,
       Int32_Box, Int64_Box,
       Uint8_Box, Uint16_Box,
       Uint32_Box, Uint64_Box,
       Double_Box, Long_Double_Box,
       Z_Box, Rational_Box, Float_Box; and<BR>
     - the Grid domain.
  - The powerset domains are Pointset_Powerset_S where S is
    a simple domain.
  - The product domains consist of
    Direct_Product_S_T,
    Smash_Product_S_T,
    Constraints_Product_S_T and
    Shape_Preserving_Product_S_T where S
    and T are simple domains.
- In the following, any of the above numerical
  abstract domains  is called a PPL <EM>domain</EM>
  and any element of a PPL domain is called a <EM>PPL object</EM>.
- The Prolog interface to the PPL is initialized and finalized by the
  predicates <CODE>ppl_initialize/0</CODE> and <CODE>ppl_finalize/0</CODE>.
  Thus the only interface predicates callable after
  <CODE>ppl_finalize/0</CODE> are <CODE>ppl_finalize/0</CODE> itself
  (this further call has no effect) and <CODE>ppl_initialize/0</CODE>,
  after which the interface's services are usable again.
  Some Prolog systems allow the specification of initialization
  and deinitialization functions in their foreign language interfaces.
  The corresponding incarnations of the Prolog interface
  have been written so that <CODE>ppl_initialize/0</CODE> and/or
  <CODE>ppl_finalize/0</CODE> are called automatically.
  Section \ref PI_SD_Features "System-Dependent Features" will detail
  in which cases initialization and finalization is automatically
  performed or is left to the Prolog programmer's responsibility.
  However, for portable applications, it is best
  to invoke <CODE>ppl_initialize/0</CODE> and <CODE>ppl_finalize/0</CODE>
  explicitly: since they can be called multiple times without problems,
  this will result in enhanced portability at a cost that is, by all means,
  negligible.
- A PPL object such as a polyhedron can only be accessed
  by means of a Prolog term called a <EM>handle</EM>.
  Note, however, that the data structure of a handle,
  is implementation-dependent, system-dependent and
  version-dependent, and, for this reason, deliberately left unspecified.
  What we do guarantee is that the handle requires very little memory.
- A Prolog term can be bound to a valid handle for a PPL object by using
  predicates such as
\code
  ppl_new_C_Polyhedron_from_space_dimension/3,
  ppl_new_C_Polyhedron_from_C_Polyhedron/2,
  ppl_new_C_Polyhedron_from_constraints/2,
  ppl_new_C_Polyhedron_from_generators/2,
\endcode
  These predicates will create or copy a PPL polyhedron
  and construct a valid handle for referencing it.
  The last argument is a Prolog term that is
  unified with a new valid handle for accessing this polyhedron.

- As soon as a PPL object is no longer required,
  the memory occupied by it should be released
  using the PPL predicate such as <CODE>ppl_delete_Polyhedron/1</CODE>.
  To understand why this is important,
  consider a Prolog program and a variable that is bound to
  a Herbrand term.
  When the variable dies (goes out of scope) or is uninstantiated
  (on backtracking), the term it is bound to is amenable to garbage collection.
  But this only applies for the standard domain of the language:
  Herbrand terms.
  In Prolog+PPL, when, for example,
  a variable bound to a handle for a Polyhedron dies
  or is uninstantiated,
  the handle can be garbage-collected, but the polyhedron to which
  the handle refers will not be released.
  Once a handle has been used as an argument in
  <CODE>ppl_delete_Polyhedron/1</CODE>,
  it becomes invalid.
- For a PPL object with space dimension \p k,
  the identifiers used for the PPL variables
  must lie between 0 and \f$k-1\f$ and correspond to the indices of the
  associated Cartesian axes.
  For example, when using the predicates that combine PPL polyhedra
  or add constraints or generators to a representation of
  a PPL polyhedron,
  the polyhedra referenced and any constraints or generators in the call
  should follow all the (space) dimension-compatibility rules stated in
  Section \extref{representation, Representations of Convex Polyhedra}
  of the main \extref{preamble, PPL user manual}.
- As explained above, a polyhedron has a fixed topology C or NNC,
  that is determined at the time of its initialization.
  All subsequent operations on the polyhedron must respect all the
  topological compatibility rules stated in Section
  \extref{representation, Representations of Convex Polyhedra}
  of the main \extref{preamble, PPL user manual}.
- Any application using the PPL should make sure that only the
  intended version(s) of the library are ever used.
  Predicates
\code
  ppl_version_major/1,
  ppl_version_minor/1,
  ppl_version_revision/1,
  ppl_version_beta/1,
  ppl_version/1,
  ppl_banner.
\endcode
  allow run-time checking of information about the version being used.

\anchor Predicate_Specifications
<H2>Predicate Specifications</H2>

The PPL predicates provided by the Prolog interface are specified below.
The specification uses the following grammar rules:
\code

 Number      --> unsigned integer	ranging from 0 to an upper bound
					depending on the actual Prolog system.

 C_int       --> Number | - Number	C integer

 C_unsigned  --> Number			C unsigned integer

 Coeff       --> Number			used in linear expressions;
					the upper bound will depend on how
					the PPL has been configured

 Dimension_Type
             --> Number			used for the number of affine and
					space dimensions and the names of
 					the dimensions;
					the upper bound will depend on
					the maximum number of dimensions
					allowed by the PPL
					(see ppl_max_space_dimensions/1)

 Boolean     --> true | false

 Handle      --> Prolog term		used to identify a Polyhedron

 Topology    --> c | nnc		Polyhedral kind;
					c is closed and nnc is NNC

 VarId       --> Dimension_Type 	variable identifier

 PPL_Var     --> '$VAR'(VarId)		PPL variable

 Lin_Expr    --> PPL_Var		PPL variable
            | Coeff
            | Lin_Expr			unary plus
            | - Lin_Expr		unary minus
            | Lin_Expr + Lin_Expr	addition
            | Lin_Expr - Lin_Expr	subtraction
            | Coeff * Lin_Expr	        multiplication
            | Lin_Expr * Coeff	        multiplication

 Relation_Symbol --> =			equals
            | =< 			less than or equal
            | >=			greater than or equal
            | < 			strictly less than
            | > 			strictly greater than

 Constraint  --> Lin_Expr Relation_Symbol Lin_Expr
					constraint

 Constraint_System			list of constraints
             --> []
            | [Constraint | Constraint_System]

 Modulus     --> Coeff | - Coeff

 Congruence  --> Lin_Expr =:= Lin_Expr  congruence with modulo 1
            | (Lin_Expr =:= Lin_Expr) / Modulus
                                        congruence with modulo Modulus

 Congruence_System			list of congruences
             --> []
            | [Congruence | Congruence_System]

 Generator_Denominator --> Coeff	must be non-zero
	    | - Coeff

 Generator   --> point(Lin_Expr)	point
            | point(Lin_Expr, Generator_Denominator)
					point
            | closure_point(Lin_Expr)	closure point
            | closure_point(Lin_Expr, Generator_Denominator)
					closure point
            | ray(Lin_Expr)		ray
            | line(Lin_Expr)		line

 Generator_System			list of generators
             --> []
	    | [Generator | Generator_System]

 Grid_Generator
             --> grid_point(Lin_Expr)	grid point
            | grid_point(Lin_Expr, Generator_Denominator)
					grid point
            | parameter(Lin_Expr)	parameter
            | parameter(Lin_Expr, Generator_Denominator)
                                        parameter
            | grid_line(Lin_Expr)       grid line

 Grid_Generator_System			list of grid generators
             --> []
	    | [Grid_Generator | Grid_Generator_System]

 Artificial_Parameter  --> Lin_Expr / Coeff

 Artificial_Parameter_List --> []
	    | [Artificial_Parameter | Artificial_Parameter_List]

 Atom        --> Prolog atom

 Universe_or_Empty			PPL object
             --> universe | empty

 Poly_Relation --> is_disjoint          with a constraint or congruence
	    | strictly_intersects	with a constraint or congruence
	    | is_included		with a constraint or congruence
	    | saturates			with a constraint or congruence
            | subsumes			with a (grid) generator

 Relation_List --> []
	    | [Poly_Relation | Relation_List]

 Complexity  --> polynomial | simplex | any

 Vars_Pair   --> PPLVar - PPLVar        map relation

 P_Func      --> []    			list of map relations
            | [Vars_Pair | P_Func].

 Width  --> bits_8 | bits_16 | bits_32 | bits_64 | bits_128

 Representation  --> unsigned | signed_2_complement

 Overflow  --> overflow_wraps | overflow_undefined | overflow_impossible

 Optimization_Mode --> max | min

 Problem_Status --> unfeasible
            | unbounded
            | optimized

 Control_Parameter_Name --> pricing            for MIP problems
                        | control_strategy     for PIP problems
                        | pivot_row_strategy   for PIP problems

 Control_Parameter_Value
             --> pricing_steepest_edge_float
            | pricing_steepest_edge_exact
            | pricing_textbook
            | control_strategy_first
            | control_strategy_deepest
            | control_strategy_all
            | pivot_row_strategy_first
            | pivot_row_strategy_max_column

 Vars_List   --> []    			list of PPL variables
            | [PPL_Var | Vars_List].
\endcode

\anchor predicate_descriptions
<H2> Predicate Descriptions </H2>

Below is a short description of many of the interface predicates.
For full definitions of terminology used here,
see the main \extref{preamble, PPL user manual}.

\anchor di_predicates
<H3> Domain Independent Predicates </H3>

First we describe the domain independent predicates
that are included with all instantiations of the Prolog interfaces.

<P><CODE> ppl_version_major(?C_int) </CODE><BR>
  <EM>Unifies <CODE>C_int</CODE> with the major number of the PPL version.</EM>

<P><CODE> ppl_version_minor(?C_int) </CODE><BR>
  <EM>Unifies <CODE>C_int</CODE> with the minor number of the PPL version.</EM>

<P><CODE> ppl_version_revision(?C_int) </CODE><BR>
  <EM>Unifies <CODE>C_int</CODE> with the revision number
  of the PPL version.</EM>

<P><CODE> ppl_version_beta(?C_int) </CODE><BR>
  <EM>Unifies <CODE>C_int</CODE> with the beta number of the PPL version.</EM>

<P><CODE> ppl_version(?Atom) </CODE><BR>
  <EM>Unifies <CODE>Atom</CODE> with the PPL version.</EM>

<P><CODE> ppl_banner(?Atom) </CODE><BR>
  <EM>Unifies <CODE>Atom</CODE> with
  information about the PPL version, the licensing, the lack of any
  warranty whatsoever, the C++ compiler used to build the library,
  where to report bugs and where to look for further information.</EM>

<P><CODE> ppl_Coefficient_bits(?Bits) </CODE><BR>

  <EM>Unifies <CODE>Bits</CODE> with the number of bits used to encode a
  Coefficient in the C++ interface; 0 if unbounded.</EM>

<P><CODE> ppl_Coefficient_is_bounded </CODE><BR>

  <EM>Succeeds if and only if the Coefficients in the C++ interface are bounded.</EM>

<P><CODE> ppl_Coefficient_max(Max) </CODE><BR>

  <EM>If the Coefficients in the C++ interface are bounded,
  then the maximum coefficient the C++ interface can handle is
  unified with <CODE>Max</Code>.
  If the Prolog system cannot handle this coefficient, then
  an exception is thrown.
  It fails if the Coefficients in the C++ interface are unbounded.</EM>

<P><CODE> ppl_Coefficient_min(Min) </CODE><BR>

  <EM>If the Coefficients in the C++ interface are bounded,
  then the minimum coefficient the C++ interface can handle is
  unified with <CODE>Min</Code>.
  If the Prolog system cannot handle this coefficient, then
  an exception is thrown.
  It fails if the Coefficients in the C++ interface are unbounded.</EM>

<P><CODE> ppl_max_space_dimension(?Dimension_Type) </CODE><BR>

  <EM>Unifies <CODE>Dimension_Type</CODE> with the maximum space dimension
  this library can handle.</EM>

<P><CODE>
  ppl_initialize
 </CODE><BR>

  <EM>Initializes the PPL interface.
  Multiple calls to <CODE>ppl_initialize</CODE> does no harm.</EM>

<P><CODE>
  ppl_finalize
 </CODE><BR>

  <EM>Finalizes the PPL interface.
  Once this is executed, the next call to an interface predicate must
  either be to <CODE>ppl_initialize</CODE> or to <CODE>ppl_finalize</CODE>.
  Multiple calls to <CODE>ppl_finalize</CODE> does no harm.</EM>

<P><CODE> ppl_set_timeout_exception_atom(+Atom) </CODE><BR>

   <EM>Sets the atom to be thrown by timeout exceptions
   to <CODE>Atom</CODE>.
   The default value is <CODE>time_out</CODE>.</EM>

<P><CODE> ppl_timeout_exception_atom(?Atom) </CODE><BR>

   <EM>The atom to be thrown by timeout exceptions
   is unified with <CODE>Atom</CODE>.</EM>

<P><CODE> ppl_set_timeout(+Hsecs) </CODE><BR>

   <EM>Computations taking exponential time will be interrupted
   some time after <CODE>Hsecs</CODE> hundreths of seconds after that call.
   If the computation is interrupted that way, the current timeout
   exception atom will be thrown.
   <CODE>Hsecs</CODE> must be strictly greater than zero.</EM>

<P><CODE> ppl_reset_timeout </CODE><BR>

   <EM>Resets the timeout time so that the computation is not interrupted.</EM>

<P><CODE> ppl_set_deterministic_timeout(+Weight) </CODE><BR>

   <EM>Computations taking exponential time will be interrupted
   some time after reaching the <CODE>Weight</CODE> complexity threshold.
   If the computation is interrupted that way, the current timeout
   exception atom will be thrown.
   <CODE>Weight</CODE> must be strictly greater than zero.</EM>
   <P>
   <EM>NOTE:</EM> This "timeout" checking functionality is said to be
   <EM>deterministic</EM> because it is not based on actual elapsed time.
   Its behavior will only depend on (some of the) computations performed
   in the PPL library and it will be otherwise independent from the
   computation environment (CPU, operating system, compiler, etc.).
   The weight mechanism is under alpha testing: client applications
   should be ready to reconsider the tuning of these weight thresholds
   when upgrading to newer version of the PPL.

<P><CODE> ppl_reset_deterministic_timeout </CODE><BR>

   <EM>Resets the deterministic timeout so that the computation is not
   interrupted.</EM>

<P><CODE> ppl_set_rounding_for_PPL </CODE><BR>

   <EM>Sets the FPU rounding mode so that the PPL abstractions based on
  floating point numbers work correctly.
  This is performed automatically at initialization-time.  Calling
  this function is needed only if restore_pre_PPL_rounding() has
  previously been called.</EM>


<P><CODE> ppl_restore_pre_PPL_rounding </CODE><BR>

   <EM>Sets the FPU rounding mode as it was before initialization of the PPL.
  After calling this function it is absolutely necessary to call
  set_rounding_for_PPL() before using any PPL abstractions based on
  floating point numbers.
  This is performed automatically at finalization-time.</EM>

<P><CODE> ppl_irrational_precision(?Precision) </CODE><BR>

  <EM>Unifies <CODE>Precision</CODE> with the precision parameter for
  irrational calculations.</EM>

<P><CODE> ppl_set_irrational_precision(+Precision) </CODE><BR>

  <EM>Sets the precision parameter for irrational calculations to
  <CODE>Precision</CODE>. In the following irrational calculations
  returning an unbounded rational (e.g., when computing a square root),
  the lesser between numerator and denominator will be limited to
  <CODE>2**Precision</CODE>.</EM>


\anchor mip_predicates
<H3> MIP Predicates </H3>
Here we describe the predicates available for PPL objects
defining mixed integer (linear) programming problems.

<P><CODE>
  ppl_new_MIP_Problem_from_space_dimension(+Dimension_Type, -Handle)
</CODE><BR>

   <EM>Creates an MIP Problem \f$\mathrm{MIP}\f$ with the feasible region
   the vector space of dimension <CODE>Dimension_Type</CODE>,
   objective function \f$0\f$ and optimization mode <CODE>max</CODE>.
   <CODE>Handle</CODE> is unified with the handle for \f$\mathrm{MIP}\f$.</EM>

<P><CODE>
  ppl_new_MIP_Problem(+Constraint_System, +Lin_Expr, +Optimization_Mode,
                      -Handle)
</CODE><BR>

   <EM>Creates an MIP Problem \f$\mathrm{MIP}\f$ with
   the feasible region represented by <CODE>Constraint_System</CODE>,
   objective function <CODE>Lin_Expr</CODE> and optimization mode
   <CODE>Optimization_Mode</CODE>.
   <CODE>Handle</CODE> is unified with the handle for \f$\mathrm{MIP}\f$.</EM>

<P><CODE>
  ppl_new_MIP_Problem_from_MIP_Problem(+Handle_1, -Handle_2)
</CODE><BR>

   <EM>Creates an MIP Problem \f$\mathrm{MIP}\f$ from the MIP Problem
   referenced by \c Handle_1.
   \c Handle_2 is unified with the handle for \f$\mathrm{MIP}\f$.</EM>

<P><CODE>
  ppl_MIP_Problem_swap(+Handle_1, +Handle_2)
</CODE><BR>

   <EM>Swaps the MIP Problem referenced by <CODE>Handle_1</CODE>
   with the one referenced by <CODE>Handle_2</CODE>.</EM>

<P><CODE>
  ppl_delete_MIP_Problem(+Handle)
</CODE><BR>

   <EM>Deletes the MIP Problem referenced by <CODE>Handle</CODE>.
   After execution,
   <CODE>Handle</CODE> is no longer a valid handle for a PPL MIP Problem.</EM>

<P><CODE>
  ppl_MIP_Problem_space_dimension(+Handle, ?Dimension_Type)
</CODE><BR>

   <EM>Unifies the dimension of the vector space in which the
   MIP Problem referenced by <CODE>Handle</CODE> is embedded
   with <CODE>Dimension_Type</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_integer_space_dimensions(+Handle, ?Vars_List)
</CODE><BR>

   <EM>Unifies <CODE>Vars_List</CODE> with a list of variables representing
   the integer space dimensions of the MIP Problem
   referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_constraints(+Handle, -Constraint_System)
</CODE><BR>

   <EM>Unifies <CODE>Constraint_System</CODE> with a list of
   the constraints in the constraints system
   representing the feasible region for the MIP Problem
   referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_objective_function(+Handle, ?Lin_Expr)
</CODE><BR>

   <EM>Unifies <CODE>Lin_Expr</CODE> with the objective function
   for the MIP Problem referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_optimization_mode(+Handle, ?Optimization_Mode)
</CODE><BR>

   <EM>Unifies <CODE>Optimization_Mode</CODE> with the optimization mode
   for the MIP Problem referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_clear(+Handle)
</CODE><BR>

  <EM>Resets the MIP problem referenced by <CODE>Handle</CODE>
  to be the trivial problem with
  the feasible region the \f$0\f$-dimensional universe,
  objective function \f$0\f$ and optimization mode <CODE>max</CODE>.</EM>

<P><CODE> ppl_MIP_Problem_add_space_dimensions_and_embed(
             +Handle, +Dimension_Type)
 </CODE><BR>

   <EM>Embeds the MIP problem referenced by <CODE>Handle</CODE>
   in a space that is enlarged by <CODE>Dimension_Type</CODE> dimensions.</EM>

<P><CODE>
  ppl_MIP_Problem_add_to_integer_space_dimensions(+Handle, +Vars_List)
</CODE><BR>

   <EM>Updates the MIP Problem referenced by <CODE>Handle</CODE>
   so that the variables in <CODE>Vars_List</CODE> are added to
   the set of integer space dimensions.</EM>

<P><CODE>
  ppl_MIP_Problem_add_constraint(+Handle, +Constraint)
</CODE><BR>

   <EM>Updates the MIP Problem referenced by <CODE>Handle</CODE>
   so that the feasible region is represented by the original constraint
   system together with the constraint <CODE>Constraint</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_add_constraints(+Handle, +Constraint_System)
</CODE><BR>

   <EM>Updates the MIP Problem referenced by <CODE>Handle</CODE>
   so that the feasible region is represented by the original constraint
   system together with all the constraints in
   <CODE>Constraint_System</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_set_objective_function(+Handle, +Lin_Expr)
</CODE><BR>

   <EM>Updates the MIP Problem referenced by <CODE>Handle</CODE>
   so that the objective function is changed to <CODE>Lin_Expr</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_set_control_parameter(+Handle, +Control_Parameter_Value)
</CODE><BR>

   <EM>Updates the MIP Problem referenced by <CODE>Handle</CODE>
   so that the value for the relevant control parameter name is
   changed to <CODE>Control_Parameter_Value</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_get_control_parameter(+Handle, +Control_Parameter_Name,
                                        ?Control_Parameter_Value)
</CODE><BR>

   <EM>Unifies \c Control_Parameter_Value with the value of the
   control parameter \c Control_Parameter_Name.</EM>

<P><CODE>
  ppl_MIP_Problem_set_optimization_mode(+Handle, +Optimization_Mode)
</CODE><BR>

   <EM>Updates the MIP Problem referenced by <CODE>Handle</CODE>
   so that the optimization mode is changed to
   <CODE>Optimization_Mode</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_is_satisfiable(+Handle)
</CODE><BR>

  <EM>Succeeds if and only if the MIP Problem referenced by
  <CODE>Handle</CODE> is satisfiable.</EM>

<P><CODE>
  ppl_MIP_Problem_solve(+Handle, ?MIP_Problem_Status)
</CODE><BR>

  <EM>Solves the MIP problem  referenced by
  <CODE>Handle</CODE> and unifies <CODE>MIP_Problem_Status</CODE>
  with:
  <CODE>unfeasible</CODE>, if the MIP problem is not satisfiable;
  <CODE>unbounded</CODE>, if the MIP problem is satisfiable but
  there is no finite bound to the value of the objective function;
  <CODE>optimized</CODE>, if the MIP problem admits an optimal solution.</EM>

<P><CODE>
  ppl_MIP_Problem_feasible_point(+Handle, ?Generator)
</CODE><BR>

  <EM>Unifies  <CODE>Generator</CODE> with a feasible point
  for the MIP problem referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_optimizing_point(+Handle, ?Generator)
</CODE><BR>

  <EM>Unifies  <CODE>Generator</CODE> with an optimizing point for
  the MIP problem referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_optimal_value(+Handle, ?Coeff_1, ?Coeff_2)
</CODE><BR>

  <EM>Unifies <CODE>Coeff_1</CODE> and <CODE>Coeff_2</CODE>
  with the numerator and denominator, respectively, for the optimal value
  for the MIP problem  referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_evaluate_objective_function(+Handle, +Generator,
                                             ?Coeff_1, ?Coeff_2)
</CODE><BR>
  <EM>Evaluates the objective function of the MIP problem  referenced by
  <CODE>Handle</CODE> at point <CODE>Generator</CODE>.
  <CODE>Coefficient_1</CODE> is unified with the numerator and
  <CODE>Coefficient_2</CODE> is unified with the denominator of the
  objective function value at <CODE>Generator</CODE>.</EM>

<P><CODE>
  ppl_MIP_Problem_OK(+Handle)
</CODE><BR>

   <EM>Succeeds only if the MIP Problem referenced by
   <CODE>Handle</CODE> is well formed, i.e., if it
   satisfies all its implementation invariants.
   Useful for debugging purposes.</EM>

<BR>

\anchor pip_predicates
<H3> PIP Predicates </H3>
Here we describe some functions available for PPL objects
defining parametric integer programming problems.

<P><CODE>
  ppl_new_PIP_Problem_from_space_dimension(+Dimension_Type, -Handle)
</CODE><BR>

   <EM>Creates a PIP Problem \f$\mathrm{PIP}\f$ with the
   feasible region the vector space of dimension <CODE>dimension</CODE>,
   empty constraint_system and empty set of parametric variables.
   <CODE>Handle</CODE> is unified with the handle for \f$\mathrm{PIP}\f$.</EM>

<P><CODE>
  ppl_new_PIP_Problem(+Constraint_System, +Lin_Expr, +Vars_List,
                      -Handle)
</CODE><BR>

   <EM>Creates a PIP Problem \f$\mathrm{PIP}\f$ having space
   dimension <CODE>dimension</CODE>, a feasible region represented by
   <CODE>constraint_system</CODE> and parametric variables represented
   by <CODE>Vars_List</CODE>.
   <CODE>Handle</CODE> is unified with the handle for \f$\mathrm{PIP}\f$.</EM>

<P><CODE>
  ppl_new_PIP_Problem_from_PIP_Problem(+Handle_1, -Handle_2)
</CODE><BR>
   <EM>Creates a PIP Problem \f$\mathrm{PIP}\f$ from the PIP Problem
   referenced by \c Handle_1.
   \c Handle_2 is unified with the handle for \f$\mathrm{PIP}\f$.</EM>

<P><CODE>
  ppl_PIP_Problem_swap(+Handle_1, +Handle_2)
</CODE><BR>
   <EM>Swaps the PIP Problem referenced by <CODE>Handle_1</CODE>
   with the one referenced by <CODE>Handle_2</CODE>.</EM>

<P><CODE>
  ppl_delete_PIP_Problem(+Handle)
</CODE><BR>
   <EM>Deletes the PIP Problem referenced by <CODE>Handle</CODE>.
   After execution,
   <CODE>Handle</CODE> is no longer a valid handle for a PPL PIP Problem.</EM>

<P><CODE>
  ppl_PIP_Problem_space_dimension(+Handle, ?Dimension_Type)
</CODE><BR>
   <EM>Unifies the dimension of the vector space in which the
   PIP Problem referenced by <CODE>Handle</CODE> is embedded
   with <CODE>Dimension_Type</CODE>.</EM>

<P><CODE>
  ppl_PIP_Problem_parameter_space_dimensions(+Handle, ?Vars_List)
</CODE><BR>
   <EM>Unifies <CODE>Vars_List</CODE> with a list of variables representing
   the parameter space dimensions of the PIP Problem
   referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_PIP_Problem_constraints(+Handle, ?Constraint_System)
</CODE><BR>
   <EM>Unifies <CODE>Constraint_System</CODE> with a list of
   the constraints in the constraints system
   representing the feasible region for the PIP Problem
   referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_PIP_Problem_clear(+Handle)
</CODE><BR>
  <EM>Resets the PIP problem referenced by <CODE>Handle</CODE>
  to be the trivial problem with
  the feasible region the \f$0\f$-dimensional universe.</EM>

<P><CODE> ppl_PIP_Problem_add_space_dimensions_and_embed(
             +Handle, +Dimension_Type1, +Dimension_Type2)
 </CODE><BR>
   <EM>Embeds the PIP problem referenced by <CODE>handle</CODE>
   in a space that is enlarged by <CODE>dimension1</CODE> non-parameter
   dimensions and <CODE>dimension2</CODE> parameter dimensions.</EM>

<P><CODE>
  ppl_PIP_Problem_add_to_parameter_space_dimensions(+Handle, +Vars_List)
</CODE><BR>
   <EM>Updates the PIP Problem referenced by <CODE>Handle</CODE>
   so that the variables in <CODE>Vars_List</CODE> are added to
   the set of parameter space dimensions.</EM>

<P><CODE>
  ppl_PIP_Problem_add_constraint(+Handle, +Constraint)
</CODE><BR>
   <EM>Updates the PIP Problem referenced by <CODE>Handle</CODE>
   so that the feasible region is represented by the original constraint
   system together with the constraint <CODE>Constraint</CODE>.</EM>

<P><CODE>
  ppl_PIP_Problem_add_constraints(+Handle, +Constraint_System)
</CODE><BR>
   <EM>Updates the PIP Problem referenced by <CODE>Handle</CODE>
   so that the feasible region is represented by the original constraint
   system together with all the constraints in
   <CODE>Constraint_System</CODE>.</EM>

<P><CODE>
  ppl_PIP_Problem_set_big_parameter_dimension(+Handle, +Dimension_Type)
</CODE><BR>

   <EM>Updates the PIP Problem referenced by <CODE>Handle</CODE>
       so that the dimension for the big parameter is \p Dimension_Type.</EM>

<P><CODE>
  ppl_PIP_Problem_has_big_parameter_dimension(+Handle, +Dimension_Type)
</CODE><BR>
   <EM>Succeeds if and only if the PIP Problem referenced by
       <CODE>Handle</CODE> has a dimension \p dim for the big parameter
       and \p Dimension_Type unifies with \p dim.</EM>

<P><CODE>
  ppl_PIP_Problem_is_satisfiable(+Handle)
</CODE><BR>
  <EM>Succeeds if and only if the PIP Problem referenced by
  <CODE>Handle</CODE> is satisfiable.</EM>

<P><CODE>
  ppl_PIP_Problem_solve(+Handle, ?PIP_Problem_Status)
</CODE><BR>
  <EM>Solves the PIP problem  referenced by
  <CODE>Handle</CODE> and unifies <CODE>PIP_Problem_Status</CODE>
  with:
  <CODE>unfeasible</CODE>, if the PIP problem is not satisfiable;
  <CODE>optimized</CODE>, if the PIP problem admits an optimal solution.</EM>

<P><CODE>
  ppl_PIP_Problem_solution(+Handle1, ?Handle2)
</CODE><BR>
  <EM>Solves the PIP problem referenced by <CODE>Handle1</CODE> and
  creates a PIP tree node \f$\mathrm{Node}\f$ representing this a solution
  if it exists and bottom otherwise
  \c Handle_2 is unified with the handle for \f$\mathrm{Sol}\f$.</EM>

<P><CODE>
  ppl_PIP_Problem_optimizing_solution(+Handle, ?PIP_Tree_Node)
</CODE><BR>
  <EM>Solves the PIP problem referenced by <CODE>Handle1</CODE> and
  creates a PIP tree node \f$\mathrm{Node}\f$ representing this an optimizing
  solution if a solution exists and bottom otherwise
  \c Handle_2 is unified with the handle for \f$\mathrm{Sol}\f$.</EM>

<P><CODE>ppl_PIP_Problem_ascii_dump(+Handle)</CODE><BR>
  <EM>Dumps an ascii representation of the PPL internal state for
  the PIP problem referenced by \c Handle on the standard output.</EM>

<P><CODE>
  ppl_PIP_Problem_OK(+Handle)
</CODE><BR>
   <EM>Succeeds only if the PIP Problem referenced by
   <CODE>Handle</CODE> is well formed, i.e., if it
   satisfies all its implementation invariants.
   Useful for debugging purposes.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_swap(+Handle_1, +Handle_2)
</CODE><BR>
   <EM>Swaps the PIP tree node referenced by <CODE>Handle_1</CODE>
   with the one referenced by <CODE>Handle_2</CODE>.</EM>

<P><CODE>ppl_PIP_Tree_Node_ascii_dump(+Handle)</CODE><BR>
  <EM>Dumps an ascii representation of the PPL internal state for
  the PIP tree node referenced by \c Handle on the standard output.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_OK(+Handle)
</CODE><BR>
   <EM>Succeeds only if the PIP tree node referenced by
   <CODE>Handle</CODE> is well formed, i.e., if it
   satisfies all its implementation invariants.
   Useful for debugging purposes.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_constraints(+Handle, ?Constraint_System)
</CODE><BR>
   <EM>Unifies <CODE>Constraint_System</CODE> with a list of
   the parameter constraints in the PIP tree node
   referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_artificials(+Handle, ?Artificial_Parameter_List)
</CODE><BR>
   <EM>Unifies <CODE>Artificial_Parameter_List</CODE> with a list of
   the artificial parameters in the PIP tree node
   referenced by <CODE>Handle</CODE>.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_is_bottom(+Handle)
</CODE><BR>
   <EM>Succeeds if and only if <CODE>handle</CODE> represents bottom.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_is_decision(+Handle)
</CODE><BR>
   <EM>Succeeds if and only if <CODE>handle</CODE> represents
   a decision node.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_is_solution(+Handle)
</CODE><BR>
   <EM>Succeeds if and only if <CODE>handle</CODE> represents
   a solution node.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_parametric_values(+Handle, +Var, ?Lin_Expr)
</CODE><BR>
   <EM>Unifies \p Lin_Expr with a linear expression representing
   the values of problem variable \p Var in the solution node
   represented by <CODE>Handle</CODE>.
   The  linear expression may involve problem parameters
   as well as artificial parameters.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_true_child(+Handle1, ?Handle2)
</CODE><BR>
  <EM>If the PIP_Tree_Node represented by \p Handle1 is a decision node
   unifies the PIP tree node referenced by <CODE>Handle2</CODE>
   with the child on the true branch of the
   PIP tree node represented by <CODE>Handle1</CODE>.
   An exception is thrown if this is not a decision node.</EM>

<P><CODE>
  ppl_PIP_Tree_Node_false_child(+Handle1, ?Handle2)
</CODE><BR>
  <EM>If the PIP_Tree_Node represented by \p Handle1 is a decision node
   unifies the PIP tree node referenced by <CODE>Handle2</CODE>
   with the child on the false branch of the
   PIP tree node represented by <CODE>Handle1</CODE>.
   An exception is thrown if this is not a decision node.</EM>

<BR>

\anchor Polyhedron_predicates
<H1>Predicates for the C Polyhedron Domain</H1>
  Here we provide a short description for each of the predicates
  available for the domain of C polyhedra.
Note that predicates for other domains will follow a similar pattern.

<H2>Constructor, copy, conversion and destructor predicates</H2>
<H3>Constructor predicates for C polyhedra</H3>
The constructor predicates build a
C polyhedron from a specification
and binds the given variable to a handle for future referencing.
The specification can be:
- the number of space dimensions and an atom indicating if it
  is to be the universe or empty element.
- a representation for the particular class of semantic geometric
  descriptors to which the element being built belongs.
For example,
  a C Polyhedron can be built from a list of
  non-strict inequality or equality constraints or a list of equality
  congruences or a list of generators that contains no closure points.

<P><CODE>ppl_new_C_Polyhedron_from_space_dimension(+Dimension_Type, +Universe_or_Empty, -Handle)</CODE><BR>
   <EM>Builds a new C polyhedron \f$\cP\f$
   with \c Dimension_Type dimensions; it is empty
   or the universe depending on whether \c Atom
   is \c empty or \c universe, respectively.
   \c Handle is unified with the handle for \f$\cP\f$.
   Thus the query
\code
   ?- ppl_new_C_Polyhedron_from_space_dimension(3, universe, X).
\endcode
   creates the C polyhedron defining the
   3-dimensional vector space
   \f$\Rset^3\f$ with \c X bound to a valid handle for accessing it.</EM>

<P><CODE>ppl_new_C_Polyhedron_from_constraints(+Constraint_System, -Handle)</CODE><BR>
  <EM>Builds a new C polyhedron \p P from
  \c Constraint_System.
  \c Handle is unified with the handle for \p P.</EM>

<P><CODE>ppl_new_C_Polyhedron_from_congruences(+Congruence_System, -Handle)</CODE><BR>
  <EM>Builds a new C polyhedron \p P from
  \c Congruence_System.
  \c Handle is unified with the handle for \p P.</EM>

<P><CODE>ppl_new_C_Polyhedron_from_generators(+Generator_System, -Handle)</CODE><BR>
  <EM>Builds a new C polyhedron \p P from
  \c Generator_System.
  \c Handle is unified with the handle for \p P.</EM>


<H3>Predicates that build new C polyhedra
  by copying or converting from
  other semantic geometric descriptions</H3>
  Besides the constructors listed above, the library also
  provides:
- copy constructors that will copy an element belonging
  to the same class of semantic geometric descriptions
- conversion operators that build a new semantic geometric
  description starting from a \b friend;
  that is, a semantic geometric description in different class
  (e.g., ppl_new_Grid_from_C_Polyhedron,
  ppl_new_C_Polyhedron_from_BD_Shape_mpq_class, etc.).<BR>

  The copy and conversion predicates have two versions,
  one with arity 2 for the source and target handles and one
  with an extra argument denoting the maximum complexity to be used
  in the conversion;
  this complexity argument is ignored when the
  the friend and the element being built are in the same class.

<P><CODE>ppl_new_C_Polyhedron_from_C_Polyhedron(+Handle_1, -Handle_2)</CODE><BR>
  <EM>Builds a new C polyhedron \p P_1 from
   the c polyhedron referenced by handle \c Handle_1.
  \c Handle_2 is unified with the handle for \p P_1.</EM>

<P><CODE>ppl_new_C_Polyhedron_from_NNC_Polyhedron(+Handle_1, -Handle_2)</CODE><BR>
  <EM>Builds a new C polyhedron \p P_1 from
   the nnc polyhedron referenced by handle \c Handle_1.
  \c Handle_2 is unified with the handle for \p P_1.</EM>

<P><CODE>ppl_new_C_Polyhedron_from_C_Polyhedron_with_complexity(+Handle, +Complexity, -Handle)</CODE><BR>
  <EM>Builds a new C polyhedron \p P_1 from
   the c polyhedron referenced by handle \c Handle_1
  using an algorithm whose complexity does not exceed \c Complexity;
  \c Handle_2 is unified with the handle for \p P_1.</EM>

<P><CODE>ppl_new_C_Polyhedron_from_NNC_Polyhedron_with_complexity(+Handle, +Complexity, -Handle)</CODE><BR>
  <EM>Builds a new C polyhedron \p P_1 from
   the nnc polyhedron referenced by handle \c Handle_1
  using an algorithm whose complexity does not exceed \c Complexity;
  \c Handle_2 is unified with the handle for \p P_1.</EM>


<H3>Destructor predicate</H3>
Below is the destructor predicate for the Polyhedron domain.

<P><CODE>ppl_delete_Polyhedron(+Handle)</CODE><BR>
  <EM>Invalidates the handle  referenced by \c Handle:
  this makes sure the corresponding resources will eventually be released.</EM>


<H2>Predicates that do not change the polyhedron</H2>
<H3>Test Predicates</H3>
These predicates test the polyhedron for different properties
and succeed or fail depending on the outcome.

<P><CODE>ppl_Polyhedron_is_empty(+Handle)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle is empty.</EM>

<P><CODE>ppl_Polyhedron_is_universe(+Handle)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle is the universe.</EM>

<P><CODE>ppl_Polyhedron_is_bounded(+Handle)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle is bounded.</EM>

<P><CODE>ppl_Polyhedron_contains_integer_point(+Handle)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle contains an integer point.</EM>

<P><CODE>ppl_Polyhedron_is_topologically_closed(+Handle)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle is topologically closed.</EM>

<P><CODE>ppl_Polyhedron_is_discrete(+Handle)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle is discrete.</EM>

<P><CODE>ppl_Polyhedron_bounds_from_above(+Handle, +Lin_Expr)</CODE><BR>
   <EM>Succeeds if and only if <CODE>Lin_Expr</CODE> is bounded from above
   in the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_bounds_from_below(+Handle, +Lin_Expr)</CODE><BR>
   <EM>Succeeds if and only if <CODE>Lin_Expr</CODE> is bounded from below
   in the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_contains_Polyhedron(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   <CODE>Handle_2</CODE> is included in or equal to
   the polyhedron referenced by <CODE>Handle_1</CODE>.</EM>

<P><CODE>ppl_Polyhedron_strictly_contains_Polyhedron(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   <CODE>Handle_2</CODE> is included in but not equal to
   the polyhedron referenced by <CODE>Handle_1</CODE>.</EM>

<P><CODE>ppl_Polyhedron_is_disjoint_from_Polyhedron(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   <CODE>Handle_2</CODE> is disjoint from
   the polyhedron referenced by <CODE>Handle_1</CODE>.</EM>

<P><CODE>ppl_Polyhedron_equals_Polyhedron(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Succeeds if and only if the polyhedron referenced by
   <CODE>Handle_1</CODE> is
   equal to the polyhedron referenced by <CODE>Handle_2</CODE>.</EM>

<P><CODE>ppl_Polyhedron_OK(+Handle)</CODE><BR>
   <EM>Succeeds only if the polyhedron referenced by
   \c Handle is well formed, i.e., if it
   satisfies all its implementation invariants.
   Useful for debugging purposes.</EM>

<P><CODE>ppl_Polyhedron_constrains(+Handle, +PPL_Var)</CODE><BR>
  <EM>Succeeds if and only if the polyhedron referenced by
   \c Handle constrains the dimension \c PPL_Var.</EM>


<H3>Predicates that return information about the polyhedron</H3>
These predicates will obtain more detailed information about the
polyhedron
unifying some of their arguments with the results.

<P><CODE>ppl_Polyhedron_space_dimension(+Handle, ?Dimension_Type)</CODE><BR>
   <EM>Unifies \c Dimension_Type with the dimension of the vector space enclosing
   the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_affine_dimension(+Handle, ?Dimension_Type)</CODE><BR>
   <EM>Unifies \c Dimension_Type with the affine dimension of
   the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_relation_with_constraint(+Handle, +Constraint, ?Relation_List)</CODE><BR>
  <EM>Unifies \c Relation_List with the list of relations the
  polyhedron referenced by \c Handle has with \c Constraint.
  The possible relations are listed in the
  grammar rules above.</EM>

<P><CODE>ppl_Polyhedron_relation_with_generator(+Handle, +Generator, ?Relation_List)</CODE><BR>
  <EM>Unifies \c Relation_List with the list of relations the
  polyhedron referenced by \c Handle has with \c Generator.
  The possible relations are listed in the
  grammar rules above.</EM>

<P><CODE>ppl_Polyhedron_relation_with_congruence(+Handle, +Congruence, ?Relation_List)</CODE><BR>
  <EM>Unifies \c Relation_List with the list of relations the
  polyhedron referenced by \c Handle has with \c Congruence.
  The possible relations are listed in the
  grammar rules above.</EM>

<P><CODE>ppl_Polyhedron_get_constraints(+Handle, ?Constraint_System)</CODE><BR>
   <EM>Unifies \c Constraint_System with
   the constraints (in the form of a list) in the constraint system
   satisfied by the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_get_congruences(+Handle, ?Congruence_System)</CODE><BR>
   <EM>Unifies \c Congruence_System with
   the congruences (in the form of a list) in the congruence system
   satisfied by the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_get_generators(+Handle, ?Generator_System)</CODE><BR>
   <EM>Unifies \c Generator_System with
   the generators (in the form of a list) in the generator system
   for the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_get_minimized_constraints(+Handle, ?Constraint_System)</CODE><BR>
   <EM>Unifies \c Constraint_System with
   the constraints (in the form of a list)
   in the minimized constraint system
   satisfied by the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_get_minimized_congruences(+Handle, ?Congruence_System)</CODE><BR>
   <EM>Unifies \c Congruence_System with
   the congruences (in the form of a list)
   in the minimized congruence system
   for the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_get_minimized_generators(+Handle, ?Generator_System)</CODE><BR>
   <EM>Unifies \c Generator_System with
   the generators (in the form of a list)
   in the minimized generator system
   satisfied by the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_maximize(+Handle, +Lin_Expr, ?Coeff_1, ?Coeff_2, ?Boolean)</CODE><BR>
  <EM>Succeeds if and only if polyhedron \p P referenced by \c Handle
  is not empty and \c Lin_Expr is bounded from above in \p P.

  \c Coeff_1 is unified with the numerator of the supremum value
  and \c Coeff_2 with the denominator of the supremum value.
  If the supremum is also the maximum, \c Boolean is unified
  with the atom \c true and, otherwise, unified with
  the atom \c false.</EM>

<P><CODE>ppl_Polyhedron_minimize(+Handle, +Lin_Expr, ?Coeff_1, ?Coeff_2, ?Boolean)</CODE><BR>
  <EM>Succeeds if and only if polyhedron \p P referenced by \c Handle
  is not empty and \c Lin_Expr is bounded from below in \p P.

  \c Coeff_1 is unified with the numerator of the infinum value
  and \c Coeff_2 with the denominator of the infinum value.
  If the infinum is also the minimum, \c Boolean is unified
  with the atom \c true and, otherwise, unified with
  the atom \c false.</EM>

<P><CODE>ppl_Polyhedron_maximize_with_point(+Handle, +Lin_Expr, ?Coeff_1, ?Coeff_2, ?Boolean, ?Point)</CODE><BR>
  <EM>Succeeds if and only if polyhedron \p P referenced by \c Handle
  is not empty and \c Lin_Expr is bounded from above in \p P.

  \c Coeff_1 is unified with the numerator of the supremum value
  and \c Coeff_2 with the denominator of the supremum value
  and \c Point with a point or closure point where \c Lin_Expr reaches
  this value.
  If the supremum is also the maximum, \c Boolean is unified
  with the atom \c true and, otherwise, unified with the
  atom \c false.</EM>

<P><CODE>ppl_Polyhedron_minimize_with_point(+Handle, +Lin_Expr, ?Coeff_1, ?Coeff_2, ?Boolean, ?Point)</CODE><BR>
  <EM>Succeeds if and only if polyhedron \p P referenced by \c Handle
  is not empty and \c Lin_Expr is bounded from below in \p P.

  \c Coeff_1 is unified with the numerator of the infinum value
  and \c Coeff_2 with the denominator of the infinum value
  and \c Point with a point or closure point where \c Lin_Expr reaches
  this value.
  If the infinum is also the minimum, \c Boolean is unified
  with the atom \c true and, otherwise, unified with the
  atom \c false.</EM>

<P><CODE>ppl_Polyhedron_external_memory_in_bytes(+Handle, ?Number)</CODE><BR>
<EM>Unifies \c Number with the size of the total memory in
bytes occupied by the polyhedron referenced by \c Handle.</EM>

<P><CODE>ppl_Polyhedron_total_memory_in_bytes(+Handle, ?Number)</CODE><BR>
<EM>Unifies \c Number with the size of the external memory in
bytes occupied by the polyhedron referenced by \c Handle.</EM>


<H3>Ascii dump predicate</H3>
This output predicate is useful for debugging.

<P><CODE>ppl_Polyhedron_ascii_dump(+Handle)</CODE><BR>
  <EM>Dumps an ascii representation of the PPL internal state for
  the polyhedron referenced by \c Handle on the standard output.</EM>


<H2>Space-dimension preserving predicates that may change
    the polyhedron</H2>
These predicates may modify the polyhedron referred to
by the handle in first argument;
the (dimension of the) vector space in which it is embedded is unchanged.
<H3>Predicates that may change the polyhedron by adding
to its constraint or generator descriptions</H3>
Note that there are two forms of these predicates
differentiated in the names by the
words "add" or "refine with";
see Section \extref{Generic_Operations_on_Semantic_Geometric_Descriptors, Generic Operations on Semantic Geometric Descriptors} in the
main \extref{preamble, PPL User Manual}
for the differences in the semantics and therefore, the expected behavior,
between these forms.

<P><CODE>ppl_Polyhedron_add_constraint(+Handle, +Constraint)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by adding
   \c Constraint to its constraint system.
   For a C polyhedron, \c Constraint must be
   an equality or a non-strict inequality.</EM>

<P><CODE>ppl_Polyhedron_add_congruence(+Handle, +Congruence)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by adding
   \c Congruence to its congruence system.
   For a C polyhedron, \c Congruence must be
   an equality.</EM>

<P><CODE>ppl_Polyhedron_add_generator(+Handle, +Generator)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by adding
   \c Generator to its generator system.
   For a C polyhedron, \c Generator must be
   a line, ray or point.</EM>

<P><CODE>ppl_Polyhedron_add_constraints(
                  +Handle,
                  +Constraint_System)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by adding to its constraint system the constraints in
   \c Constraint_System.
   For a C polyhedron, \c Constraints must be a list of
   equalities and non-strict inequalities.</EM>

<P><CODE>ppl_Polyhedron_add_congruences(
                  +Handle,
                  +Congruence_System)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by adding to its congruence system the congruences in
   \c Congruence_System.
   For a C polyhedron, \c Congruences must be a list of
   equalities.</EM>

<P><CODE>ppl_Polyhedron_add_generators(
                  +Handle,
                  +Generator_System)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by adding to its generator system the generators in
   \c Generator_System.
   For a C polyhedron, \c Generators must be a list of
   lines, rays and points.</EM>

<P><CODE>ppl_Polyhedron_refine_with_constraint(
                  +Handle,
                  +Constraint)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by refining its constraint system with
   \c Constraint.</EM>

<P><CODE>ppl_Polyhedron_refine_with_congruence(
                  +Handle,
                  +Congruence)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by refining its congruence system with
   \c Congruence.</EM>

<P><CODE>ppl_Polyhedron_refine_with_constraints(
                  +Handle,
                  +Constraint_System)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by refining its constraint system with
   the constraints in \c Constraint_System.</EM>

<P><CODE>ppl_Polyhedron_refine_with_congruences(
                  +Handle,
                  +Congruence_System)</CODE><BR>
   <EM>Updates the polyhedron referenced by \c Handle to
   one obtained by refining its congruence system with
   the congruences in \c Congruence_System.</EM>


<H3>Predicates that transform the polyhedron</H3>
These predicates enable transformations
such as taking the topological closure (which for the domain of
C polyhedron is the identity transformation),
unconstraining a specified dimension
as explained in the
main \extref{preamble, PPL User Manual} in Section
\extref{Cylindrification, Cylindrification Operator}
and several different image and preimage affine transfer relations;
for details of the latter see Sections
\extref{Images_and_Preimages_of_Affine_Transfer_Relations, Images and Preimages of Affine Transfer Relations}
and
\extref{Generalized_Affine_Relations, Generalized Affine Relations}
<P><CODE>ppl_Polyhedron_topological_closure_assign(+Handle)</CODE><BR>
  <EM>Assigns to the polyhedron referenced by \c Handle
  its topological closure.</EM>

<P><CODE>ppl_Polyhedron_unconstrain_space_dimension(+Handle, +PPL_Var)</CODE><BR>
  <EM>Modifies the polyhedron \p P referenced by \c Handle by unconstraining
  the space dimension \c PPL_Var.</EM>

<P><CODE>ppl_Polyhedron_unconstrain_space_dimensions(+Handle, +List_of_PPL_Var)</CODE><BR>
  <EM>Modifies the polyhedron \p P referenced by \c Handle by unconstraining
  the space dimensions that are specified in \c List_of_PPL_Var.
  The presence of duplicates in \c List_of_PPL_Var is a waste
  but an innocuous one.</EM>

<P><CODE>ppl_Polyhedron_affine_image(+Handle, +PPL_Var, +Lin_Expr, +Coeff)</CODE><BR>
   <EM>Transforms the polyhedron referenced by \c Handle
   assigning the affine expression for \c Lin_Expr/\c Coeff to \c PPL_Var.</EM>

<P><CODE>ppl_Polyhedron_affine_preimage(+Handle, +PPL_Var, +Lin_Expr, +Coeff)</CODE><BR>
   <EM>Transforms the polyhedron referenced by \c Handle
   substituting the affine expression for \c Lin_Expr/\c Coeff to \c PPL_Var.</EM>

<P><CODE>ppl_Polyhedron_bounded_affine_image(+Handle, +PPL_Var, +Lin_Expr_1, +Lin_Expr_2, +Coeff)</CODE><BR>
  <EM>Assigns to polyhedron \p P referenced by \c Handle
  the generalized image with respect to the
  generalized affine transfer relation
  \c Lin_Expr_1/Coeff
    \f$\leq\f$ \c PPL_Var
      \f$\leq\f$ \c Lin_Expr_2/Coeff.</EM>

<P><CODE>ppl_Polyhedron_bounded_affine_preimage(+Handle, +PPL_Var, +Lin_Expr_1, +Lin_Expr_2, +Coeff)</CODE><BR>
  <EM>Assigns to polyhedron \p P referenced by \c Handle
  the generalized preimage with respect to the
  generalized affine transfer relation
  \c Lin_Expr_1/Coeff
    \f$\leq\f$ \c PPL_Var
      \f$\leq\f$ \c Lin_Expr_2/Coeff.</EM>

<P><CODE>ppl_Polyhedron_generalized_affine_image(+Handle, +PPL_Var, +Relation_Symbol, +Lin_Expr, +Coeff)</CODE><BR>
  <EM>Assigns to polyhedron \p P referenced by \c Handle
  the generalized image with respect to the
  generalized affine transfer relation
   \c PPL_Var \f$\bowtie\f$ \c Lin_Expr/\c Coeff,
   where \f$\bowtie\f$ is the symbol
   represented by \c Relation_Symbol.</EM>

<P><CODE>ppl_Polyhedron_generalized_affine_preimage(+Handle, +PPL_Var, +Relation_Symbol, +Lin_Expr, +Coeff)</CODE><BR>
  <EM>Assigns to polyhedron \p P referenced by \c Handle
  the generalized preimage with respect to the
  generalized affine transfer relation
   \c PPL_Var \f$\bowtie\f$ \c Lin_Expr/\c Coeff,
   where \f$\bowtie\f$ is the symbol
   represented by \c Relation_Symbol.</EM>

<P><CODE>ppl_Polyhedron_generalized_affine_image_lhs_rhs(+Handle, +Lin_Expr_1, +Relation_Symbol, +Lin_Expr_2)</CODE><BR>
  <EM>Assigns to polyhedron \p P referenced by \c Handle
  the generalized image with respect to the
  generalized affine transfer relation
   \c Lin_Expr_1 \f$\bowtie\f$ \c Lin_Expr_2,
   where \f$\bowtie\f$ is the symbol
   represented by \c Relation_Symbol.</EM>

<P><CODE>ppl_Polyhedron_generalized_affine_preimage_lhs_rhs(+Handle, +Lin_Expr_1, +Relation_Symbol, +Lin_Expr_2)</CODE><BR>
  <EM>Assigns to polyhedron \p P referenced by \c Handle
  the generalized preimage with respect to the
  generalized affine transfer relation
   \c Lin_Expr_1 \f$\bowtie\f$ \c Lin_Expr_2,
   where \f$\bowtie\f$ is the symbol
   represented by \c Relation_Symbol.</EM>


<H3>Predicates whose results depend on more than one polyhedron</H3>
These predicates include the binary operators which will assign to the
polyhedron referred to by the first argument
its combination with the polyhedron referred to by the second argument
as described in the
main \extref{preamble, PPL User Manual} in Sections
\extref{Intersection_and_Convex_Polyhedral_Hull, Intersection and Convex Polyhedral Hull}
and
\extref{Convex_Polyhedral_Difference, Convex Polyhedral Difference};
and a linear partitioning operator described below.

<P><CODE>ppl_Polyhedron_intersection_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the intersection of \p P and the polyhedron referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_upper_bound_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the upper bound of \p P and the polyhedron referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_difference_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the difference of \p P and the polyhedron referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_time_elapse_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the time elapse of \p P and the polyhedron referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_poly_hull(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the poly-hull of \p P and the polyhedron referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_poly_difference(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the poly-difference of \p P and the polyhedron referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_upper_bound_assign_if_exact(+Handle_1, +Handle_2)</CODE><BR>
    <EM>Succeeds if the least upper bound of the
    polyhedron \p P_1 referenced by \c Handle_1
    with the polyhedron referenced by \c Handle_2 is exact;
    in which case the least upper bound is assigned
    to \p P_1;
    fails otherwise.</EM>

<P><CODE>ppl_Polyhedron_poly_hull_assign_if_exact(+Handle_1, +Handle_2)</CODE><BR>
    <EM>Succeeds if the least upper bound of the
    polyhedron \p P_1 referenced by \c Handle_1
    with the polyhedron referenced by \c Handle_2 is exact;
    in which case the least upper bound is assigned
    to \p P_1;
    fails otherwise.</EM>

<P><CODE>ppl_Polyhedron_simplify_using_context_assign(+Handle_1, +Handle_2, ?Boolean)</CODE><BR>
    <EM>Succeeds if and only if the intersection of polyhedron \p P_1
    referenced by \c Handle_1 and the polyhedron \p P_2 referenced by \c Handle_2
    is non-empty.
    Assigns to \p P_1 its meet-preserving simplification with
    respect to \p P_2.</EM>

<P><CODE>ppl_Polyhedron_linear_partition(+Handle_1, +Handle_2, -Handle_3,
                                        -Handle_4)</CODE><BR>
  <EM>\c Handle_1 and \c Handle_2 are handles for elements \p P_1 and \p P_2
  in the Polyhedron domain. The predicate unifies handle
  \c Handle_3 to a reference to the intersection of
  \p P_1 and \p P_2 and \c Handle_4 to a reference to
  a pointset powerset of nnc polyhedra \p P_4;
  where \p P_4 is the linear partition of \p P_1 with respect to \p P_2.
  This predicate is only provided if the
  class \c Pointset_Powerset_NNC_Polyhedron
  has been enabled when configuring the library.</EM>


<H3>Predicates for widening and extrapolation</H3>
In addition to the above binary operators, there are also a number
of widening, extrapolation and narrowing operators
as described in the
main \extref{preamble, PPL User Manual} in Sections
\extref{Widening_Operators, Widening Operators},
\extref{Widening_with_Tokens, Widening with Tokens} and
\extref{Extrapolation_Operators, Extrapolation Operators}.
Note that for all these widening and extrapolation predicates
to behave as specified
the polyhedron referred to by the second argument has to be
contained in (or equal to) the polyhedron referred to by the first argument.

<P><CODE>ppl_Polyhedron_BHRZ03_widening_assign_with_tokens(+Handle_1, +Handle_2, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the BHRZ03-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2.
   The widening with tokens delay
   technique is applied with \c C_unsigned_1 tokens;
   \c C_unsigned_2 is unified with the number of tokens
   remaining at the end of the operation.</EM>

<P><CODE>ppl_Polyhedron_H79_widening_assign_with_tokens(+Handle_1, +Handle_2, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the H79-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2.
   The widening with tokens delay
   technique is applied with \c C_unsigned_1 tokens;
   \c C_unsigned_2 is unified with the number of tokens
   remaining at the end of the operation.</EM>

<P><CODE>ppl_Polyhedron_BHRZ03_widening_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the BHRZ03-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_H79_widening_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the H79-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2.</EM>

<P><CODE>ppl_Polyhedron_widening_assign_with_tokens(+Handle_1, +Handle_2, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
<EM>Same as predicate <CODE>ppl_Polyhedron_H79_widening_assign_with_tokens</CODE>/4</EM>

<P><CODE>ppl_Polyhedron_widening_assign(+Handle_1, +Handle_2)</CODE><BR>
<EM>Same as predicate <CODE>ppl_Polyhedron_H79_widening_assign</CODE>/2</EM>

<P><CODE>ppl_Polyhedron_limited_BHRZ03_extrapolation_assign_with_tokens(+Handle_1, +Handle_2, +Constraint_System, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the BHRZ03-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1.
   The widening with tokens delay
   technique is applied with \c C_unsigned_1 tokens;
   \c C_unsigned_2 is unified with the number of tokens
   remaining at the end of the operation.</EM>

<P><CODE>ppl_Polyhedron_bounded_BHRZ03_extrapolation_assign_with_tokens(+Handle_1, +Handle_2, +Constraint_System, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the BHRZ03-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1, further intersected with the smallest box containing \p P_1.
   The widening with tokens delay
   technique is applied with \c C_unsigned_1 tokens;
   \c C_unsigned_2 is unified with the number of tokens
   remaining at the end of the operation.</EM>

<P><CODE>ppl_Polyhedron_limited_H79_extrapolation_assign_with_tokens(+Handle_1, +Handle_2, +Constraint_System, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the H79-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1.
   The widening with tokens delay
   technique is applied with \c C_unsigned_1 tokens;
   \c C_unsigned_2 is unified with the number of tokens
   remaining at the end of the operation.</EM>

<P><CODE>ppl_Polyhedron_bounded_H79_extrapolation_assign_with_tokens(+Handle_1, +Handle_2, +Constraint_System, +C_unsigned_1, ?C_unsigned_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the H79-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1, further intersected with the smallest box containing \p P_1.
   The widening with tokens delay
   technique is applied with \c C_unsigned_1 tokens;
   \c C_unsigned_2 is unified with the number of tokens
   remaining at the end of the operation.</EM>

<P><CODE>ppl_Polyhedron_limited_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the BHRZ03-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1.</EM>

<P><CODE>ppl_Polyhedron_bounded_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the BHRZ03-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1, further intersected with the smallest box containing \p P_1.</EM>

<P><CODE>ppl_Polyhedron_limited_H79_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the H79-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1.</EM>

<P><CODE>ppl_Polyhedron_bounded_H79_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)</CODE><BR>
   <EM>Assigns to the polyhedron \p P_1 referenced by \c Handle_1
   the H79-widening of \p P_1 with the polyhedron
   referenced by \c Handle_2 intersected with the constraints in
   \c Constraint_System that are
   satisfied by all the points of \p P_1, further intersected with the smallest box containing \p P_1.</EM>


<H2>Predicates that may modify the vector space</H2>
These predicates enable the modification of the vector space
of the polyhedron referred to in the first argument.
<H3>Predicate for concatenation</H3>
For more information on this operation, see Section
\extref{Concatenating_Polyhedra, Concatenating Polyhedra},
of the main \extref{preamble, PPL User Manual}.

<P><CODE>ppl_Polyhedron_concatenate_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the polyhedron \p P referenced by \c Handle_1
   the concatenation of \p P and the polyhedron referenced by \c Handle_2.</EM>


<H3>Predicates for mapping dimensions or changing the vector space</H3>
These predicates enable the modification of the vector space
of the polyhedron referred to in the first argument.
These predicates enable the modification of the vector space
of the polyhedron referred to in the first argument.
Detailed descriptions of these can be found in the
main \extref{preamble, PPL User Manual} in Sections
\extref{Adding_New_Dimensions_to_the_Vector_Space, Adding New Dimensions to the Vector Space},
\extref{Removing_Dimensions_from_the_Vector_Space, Removing Dimensions from the Vector Space},
\extref{Mapping_the_Dimensions_of_the_Vector_Space, Mapping the Dimensions of the Vector Space},
\extref{expand_space_dimension, Expanding One Dimension of the Vector Space to Multiple Dimensions}
and
\extref{Folding_Multiple_Dimensions_of_the_Vector_Space_into_One_Dimension, Folding Multiple Dimensions of the Vector Space into One Dimension}.

<P><CODE>ppl_Polyhedron_add_space_dimensions_and_embed(+Handle, +Dimension_Type)</CODE><BR>
  <EM>Adds \c Dimension_Type new dimensions to the space enclosing
  the polyhedron \p P referenced by \c Handle
  and and_embeds \p P in this space.</EM>

<P><CODE>ppl_Polyhedron_add_space_dimensions_and_project(+Handle, +Dimension_Type)</CODE><BR>
  <EM>Adds \c Dimension_Type new dimensions to the space enclosing
  the polyhedron \p P referenced by \c Handle
  and and_projects \p P in this space.</EM>

<P><CODE>ppl_Polyhedron_remove_space_dimensions(+Handle, +List_of_PPL_Vars)</CODE><BR>
  <EM>Removes from the vector space enclosing
  the polyhedron \p P referenced by \c Handle the space dimensions that
  are specified in \c List_of_PPL_Var.  The presence
  of duplicates in  \c List_of_PPL_Var is a waste but an innocuous one.</EM>

<P><CODE>ppl_Polyhedron_remove_higher_space_dimensions(+Handle, +Dimension_Type)</CODE><BR>
  <EM>Removes the higher dimensions from the vector space enclosing
  the polyhedron \p P referenced by \c Handle
  so that, upon successful return, the new space dimension is
  \c Dimension_Type.</EM>

<P><CODE>ppl_Polyhedron_expand_space_dimension(+Handle, +PPL_Var, +Dimension_Type)</CODE><BR>
  <EM>Expands the \c  PPL_Var-th
  dimension of the vector space enclosing
  the polyhedron referenced by \c Handle to
  \c Dimension_Type new space dimensions.</EM>

<P><CODE>ppl_Polyhedron_fold_space_dimensions(+Handle, +List_of_PPL_Vars, +PPL_Var)</CODE><BR>
  <EM>Modifies the polyhedron referenced by \c Handle
  by folding the
  space dimensions contained in \c List_of_PPL_Vars
  into dimension \c PPL_Var.
  The presence of duplicates in \c List_of_PPL_Vars is a waste
  but an innocuous one.</EM>

<P><CODE>ppl_Polyhedron_map_space_dimensions(+Handle, +P_Func)</CODE><BR>
  <EM>Remaps the dimensions of the vector space according to a
  partial function.
  This function is specified by means of the \c P_Func,
  which has \p n entries.
  The result is undefined if \c P_Func does not encode a partial
  function.</EM>


\anchor other_domains
<H1>Ad hoc Predicates for Other Domains</H1>
<H2>Extra Predicates Specifically for the Pointset Powerset Domains</H2>
  The powerset domains can be instantiated by taking as a base domain
  any fixed semantic geometric description
  (C and NNC polyhedra, BD and octagonal shapes, boxes and grids).
  An element of the powerset domain represents a disjunctive collection
  of base objects (its disjuncts), all having the same space dimension.
  For more information on this construct,
  see Section \extref{The_Powerset_Domain, The Powerset Domain}
  in the main \extref{preamble, PPL User Manual}.

  Besides the predicates that are available in all semantic geometric
  descriptions (whose documentation is not repeated here),
  the powerset domain also provides several ad hoc predicates.
  These are specified below, instantiated for the
  PPL domain Pointset_Powerset_C_Polyhedron.
  Note that predicates for other pointset powerset domains
  will follow similar patterns.

  <H3>Predicates for pointset powerset iterators and disjuncts.</H3>
Iterators allow the user to examine and change individual elements
(called here disjuncts) of a pointset powerset.
Detailed descriptions for adding and removing disjuncts can be found in the
main \extref{preamble, PPL User Manual} in Section
\extref{Adding_a_Disjunct, Adding a Disjunct}.
The following predicates support useful operations
on these iterators and disjuncts via the usual handles.

<P><CODE>ppl_new_Pointset_Powerset_C_Polyhedron_iterator_from_iterator(+Iterator_1, -Iterator_2)</CODE><BR>
  <EM>Builds a new iterator \p it from the iterator referenced by
  \c Iterator_1. <CODE>Iterator_2</CODE> is unified with the handle
  for \p it.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_begin_iterator(+Handle, -Iterator)</CODE><BR>
  <EM>Unifies \c Iterator with a handle to an iterator "pointing"
  to the beginning of
  the sequence of disjuncts of the powerset referred to by \c Handle.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_end_iterator(+Handle, -Iterator)</CODE><BR>
  <EM>Unifies \c Iterator with a handle to an iterator "pointing"
  to the end of
  the sequence of disjuncts of the powerset referred to by \c Handle.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_iterator_equals_iterator(+Iterator_1, +Iterator_2)</CODE><BR>
   <EM>Succeeds if and only if the iterator referenced by
   \c Iterator_1 is equal to the iterator referenced by
   \c Iterator_2.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_iterator_increment(+Iterator)</CODE><BR>
  <EM>Increments the iterator referenced by \c Iterator
  so that it "points" to the next disjunct.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_iterator_decrement(+Iterator)</CODE><BR>
  <EM>Decrements the iterator referenced by \c Iterator
  so that it "points" to the previous disjunct.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_iterator_get_disjunct(+Iterator, -Handle)</CODE><BR>
  <EM>Unifies with \c Handle a reference to the disjunct referred
  to by \c Iterator_1.</EM>

<P><CODE>ppl_delete_Pointset_Powerset_C_Polyhedron_iterator(+Iterator)</CODE><BR>
  <EM>Invalidates the handle referenced by <CODE>Iterator</CODE>:
  this makes sure the corresponding resources
  will eventually be released.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_add_disjunct(+Handle_1, +Handle_2)</CODE><BR>
  <EM>Adds to the pointset powerset referenced by \c Handle_1 a disjunct
  referred to by \c Handle_2.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_drop_disjunct(+Handle, +Iterator)</CODE><BR>
  <EM>If \p it is the iterator referred to by \c Iterator,
  drops from the pointset powerset referenced by \c Handle
  the disjunct pointed to by \p it and
  assigns to \p it an iterator to the next disjunct.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_drop_disjuncts(+Handle, +Iterator_1, +Iterator_2)</CODE><BR>
  <EM>If \p it_1 and \p it_2 are the iterators referred to by \c Iterator_1
  and \c Iterator_2, respectively,
  drops from the pointset powerset referenced by \c Handle
   all the disjuncts from \p it_1 to \p it_2 (excluded).</EM>

  <H3>Other Ad Hoc Predicates for the pointset powerset domains.</H3>
Collected here are some other predicates that are specific to
pointset powersets of C polyhedra;
these provide operations for
simplifying the powerset,
geometric comparisons
and widening and extrapolation.
Detailed descriptions of these can be found in the
main \extref{preamble, PPL User Manual} in Sections
\extref{Geometric_Comparisons, Geometric Comparisons},
\extref{Certificate_Based_Widenings, Certificate-Based Widenings},
\extref{Powerset_Extrapolation_Operators, Powerset Extrapolation Operators}.

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_pairwise_reduce(+Handle)</CODE><BR>
   <EM>Assigns the result of pairwise reduction on
   the pointset powerset referenced by \c Handle.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_omega_reduce(+Handle)</CODE><BR>
   <EM>Assigns the result of omega reduction on
   the pointset powerset referenced by \c Handle.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_geometrically_covers_Pointset_Powerset_C_Polyhedron(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Succeeds if and only if the pointset powerset referenced by
   \c Handle_2 geometrically covers
   the pointset powerset referenced by \c Handle_1;
   see Section \extref{Geometric_Comparisons, Geometric Comparisons}
  in the main \extref{preamble, PPL User Manual}.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_geometrically_equals_Pointset_Powerset_C_Polyhedron(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Succeeds if and only if the pointset powerset referenced by
   \c Handle_2 geometrically equals
   the pointset powerset referenced by \c Handle_1;
   see Section \extref{Geometric_Comparisons, Geometric Comparisons}
  in the main \extref{preamble, PPL User Manual}.</EM>


<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_BHZ03_BHRZ03_BHRZ03_widening_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the pointset powerset \p P_1 referenced by
   \c Handle_1 the \extref{pps_certificate_widening, BHZ03-widening}
    between \p P_1 and the pointset powerset referenced by
   \c Handle_2, using the BHRZ03-widening
    certified by the convergence certificate for BHRZ03.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_BHZ03_H79_H79_widening_assign(+Handle_1, +Handle_2)</CODE><BR>
   <EM>Assigns to the pointset powerset \p P_1 referenced by
   \c Handle_1 the \extref{pps_certificate_widening, BHZ03-widening}
    between \p P_1 and the pointset powerset referenced by
   \c Handle_2, using the H79-widening
    certified by the convergence certificate for H79.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_BGP99_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, C_unsigned)</CODE><BR>
    <EM>Assigns to the pointset powerset \p P_1 referenced by
   \c Handle_1 the result of applying the
    \extref{pps_bgp99_extrapolation, BGP99 extrapolation operator}
    between \p P_1 and the pointset powerset referenced by
   \c Handle_2, using the BHRZ03-widening
    and the cardinality threshold \c C_unsigned.</EM>

<P><CODE>ppl_Pointset_Powerset_C_Polyhedron_BGP99_H79_extrapolation_assign(+Handle_1, +Handle_2, C_unsigned)</CODE><BR>
    <EM>Assigns to the pointset powerset \p P_1 referenced by
   \c Handle_1 the result of applying the
    \extref{pps_bgp99_extrapolation, BGP99 extrapolation operator}
    between \p P_1 and the pointset powerset referenced by
   \c Handle_2, using the H79-widening
    and the cardinality threshold \c C_unsigned.</EM>


\anchor PI_Compilation
<H1>Compilation and Installation</H1>

When the Parma Polyhedra Library is configured, it tests for the existence
of each supported Prolog system.  If a supported Prolog system is
correctly installed in a standard location, things are arranged
so that the corresponding interface is built and installed.

The Prolog interface files are all installed in the directory
<CODE>prefix/lib/ppl</CODE>.  Since this includes shared and
dynamically loaded libraries, you must make your dynamic
linker/loader aware of this fact.  If you use a GNU/Linux system,
try the commands <CODE>man ld.so</CODE> and <CODE>man ldconfig</CODE>
for more information.

As an option, the Prolog interface can track the creation and disposal
of polyhedra.  In fact, differently from native Prolog data, PPL polyhedra
must be explicitly disposed and forgetting to do so is a very common mistake.
To enable this option, configure the library adding
<CODE>-DPROLOG_TRACK_ALLOCATION</CODE> to the options passed to the
C++ compiler.
Your configure command would then look like
\code
  path/to/configure --with-cxxflags="-DPROLOG_TRACK_ALLOCATION" ...
\endcode

\anchor PI_SD_Features
<H1>System-Dependent Features</H1>

\anchor CIAO_Prolog
<H2>CIAO Prolog</H2>

The Ciao Prolog interface to the PPL is available
both as `PPL enhanced' Ciao Prolog interpreter
and as a library that can be linked to Ciao Prolog programs.
Only Ciao Prolog versions 1.10 #5 and later are supported.

So that it can be used with the Ciao Prolog PPL interface, the
Ciao Prolog installation must be configured with the
<CODE>--disable-regs</CODE> option.

<H3>The <CODE>ppl_ciao</CODE> Executable</H3>

If an appropriate version of Ciao Prolog is installed on the machine
on which you compiled the library, the command <CODE>make install</CODE>
will install the executable <CODE>ppl_ciao</CODE> in the directory
<CODE>prefix/bin</CODE>.
The <CODE>ppl_ciao</CODE> executable is simply the Ciao Prolog
interpreter with the Parma Polyhedra Library linked in.
The only thing you should do to use the library is to
call <CODE>ppl_initialize/0</CODE> before any other PPL predicate
and to call <CODE>ppl_finalize/0</CODE> when you are done with the
library.

<H3>Linking the Library To Ciao Prolog Programs</H3>

In order to allow linking Ciao Prolog programs to the PPL,
the following files are installed in the directory
<CODE>prefix/lib/ppl</CODE>:
<CODE>ppl_ciao.pl</CODE> contains the required foreign declarations;
<CODE>libppl_ciao.*</CODE> contain the executable code for the
Ciao Prolog interface in various formats (static library, shared library,
libtool library).
If your Ciao Prolog program is constituted by, say, <CODE>source1.pl</CODE>
and <CODE>source2.pl</CODE> and you want to create the executable
<CODE>myprog</CODE>, your compilation command may look like
\code
ciaoc -o myprog prefix/lib/ppl/ppl_ciao.pl ciao_pl_check.pl \
  -L '-Lprefix/lib/ppl -lppl_ciao -Lprefix/lib -lppl -lgmpxx -lgmp -lstdc++'
\endcode

\anchor GNU_Prolog
<H2>GNU Prolog</H2>

The GNU Prolog interface to the PPL is available both as a
`PPL enhanced' GNU Prolog interpreter and as a library that can be
linked to GNU Prolog programs.
The only GNU Prolog version that is known to work is a patched
version of the `unstable version' tagged
<A HREF="ftp://ftp.inria.fr/INRIA/Projects/contraintes/gprolog/unstable/gprolog-20040608.tgz">20040608</A>
(which unpacks to a directory called <CODE>gprolog-1.2.18</CODE>).
The patch is contained in the
<CODE>interfaces/Prolog/GNU/README</CODE> file of the PPL's distribution.

So that it can be used with the GNU Prolog PPL interface
(and, for that matter, with any foreign code),
the GNU Prolog installation must be configured with the
<CODE>--disable-regs</CODE> option.

<H3>The <CODE>ppl_gprolog</CODE> Executable</H3>

If an appropriate version of GNU Prolog is installed on the machine
on which you compiled the library, the command <CODE>make install</CODE>
will install the executable <CODE>ppl_gprolog</CODE> in the directory
<CODE>prefix/bin</CODE>.
The <CODE>ppl_gprolog</CODE> executable is simply the GNU Prolog
interpreter with the Parma Polyhedra Library linked in.
The only thing you should do to use the library is to
call <CODE>ppl_initialize/0</CODE> before any other PPL predicate
and to call <CODE>ppl_finalize/0</CODE> when you are done with the
library.

<H3>Linking the Library To GNU Prolog Programs</H3>

In order to allow linking GNU Prolog programs to the PPL,
the following files are installed in the directory
<CODE>prefix/lib/ppl</CODE>:
<CODE>ppl_gprolog.pl</CODE> contains the required foreign declarations;
<CODE>libppl_gprolog.*</CODE> contain the executable code for the
GNU Prolog interface in various formats (static library, shared library,
libtool library).
If your GNU Prolog program is constituted by, say, <CODE>source1.pl</CODE>
and <CODE>source2.pl</CODE> and you want to create the executable
<CODE>myprog</CODE>, your compilation command may look like
\code
gplc -o myprog prefix/lib/ppl/ppl_gprolog.pl source1.pl source2.pl \
  -L '-Lprefix/lib/ppl -lppl_gprolog -Lprefix/lib -lppl -lgmpxx -lgmp -lstdc++'
\endcode

GNU Prolog uses several stacks to execute a Prolog program each with a
pre-defined default size.  If the size of a stack is too small for the
application an overflow will occur.  To change the default size of a
stack, the user has to set the value of the relevant environment
variable; in particular, to execute some of the tests, we found it
necessary to increase the size of GLOBALSZ.
Thus, for the above example, the compilation command would be
\code
GLOBALSZ=32768 gplc -o myprog prefix/lib/ppl/ppl_gprolog.pl source1.pl source2.pl \
  -L '-Lprefix/lib/ppl -lppl_gprolog -Lprefix/lib -lppl -lgmpxx -lgmp -lstdc++'
\endcode
More information on
adjusting the size of the stacks can be found in Section 3.3 in the
<A HREF="http://www.gprolog.org/manual/gprolog.html">GNU Prolog
Manual</A>

\anchor SICStus_Prolog
<H2>SICStus Prolog</H2>

The SICStus Prolog interface to the PPL is available
both as a statically linked module or as a dynamically linked one.
Only SICStus Prolog versions 3.9.1 and later are supported.

<H3>The Statically Linked <CODE>ppl_sicstus</CODE> Executable</H3>

If an appropriate version of SICStus Prolog is installed on the machine
on which you compiled the library, the command <CODE>make install</CODE>
will install the executable <CODE>ppl_sicstus</CODE> in the directory
<CODE>prefix/bin</CODE>.
The <CODE>ppl_sicstus</CODE> executable is simply the SICStus Prolog
system with the Parma Polyhedra Library statically linked.
The only thing you should do to use the library is to
load <CODE>prefix/lib/ppl/ppl_sicstus.pl</CODE>.

<H3>Loading the SICStus Interface Dynamically</H3>

In order to dynamically load the library from SICStus Prolog you should
simply load <CODE>prefix/lib/ppl/ppl_sicstus.pl</CODE>.
Notice that, for dynamic linking to work, you should have configured
the library with the <CODE>--enable-shared</CODE> option.


\anchor SWI_Prolog
<H2>SWI-Prolog</H2>

The SWI-Prolog interface to the PPL is available
both as a statically linked module or as a dynamically linked one.
Only SWI-Prolog version 5.6.0 and later versions are supported.

<H3>The <CODE>ppl_pl</CODE> Executable</H3>

If an appropriate version of SWI-Prolog is installed on the machine
on which you compiled the library, the command <CODE>make install</CODE>
will install the executable <CODE>ppl_pl</CODE> in the directory
<CODE>prefix/bin</CODE>.
The <CODE>ppl_pl</CODE> executable is simply the SWI-Prolog
shell with the Parma Polyhedra Library statically linked:
from within <CODE>ppl_pl</CODE> all the services of the library
are available without further action.

<H3>Loading the SWI-Prolog Interface Dynamically</H3>

In order to dynamically load the library from SWI-Prolog you should
simply load <CODE>prefix/lib/ppl/ppl_swiprolog.pl</CODE>.
This will invoke <CODE>ppl_initialize/0</CODE> and
<CODE>ppl_finalize/0</CODE> automatically.
Alternatively, you can load the library directly with
\code
:- load_foreign_library('prefix/lib/ppl/libppl_swiprolog').
\endcode
This will call <CODE>ppl_initialize/0</CODE> automatically.
Analogously,
\code
:- unload_foreign_library('prefix/lib/ppl/libppl_swiprolog').
\endcode
will, as part of the unload process, invoke <CODE>ppl_finalize/0</CODE>.

Notice that, for dynamic linking to work, you should have configured
the library with the <CODE>--enable-shared</CODE> option.


\anchor XSB_Prolog
<H2>XSB</H2>

The XSB Prolog interface to the PPL is available
as a dynamically linked module.
Only some CVS versions of XSB starting from 2 July 2005 are known to work.
CVS versions starting from 11 November 2005 are known not to work.

In order to dynamically load the library from XSB you should
load the <CODE>ppl_xsb</CODE> module and import the predicates
you need.
For things to work, you may have to copy the files
<CODE>prefix/lib/ppl/ppl_xsb.xwam</CODE>
and
<CODE>prefix/lib/ppl/ppl_xsb.so</CODE>
in your current directory or in one of the XSB library directories.


\anchor YAP_Prolog
<H2>YAP</H2>

The YAP Prolog interface to the PPL is available
as a dynamically linked module.
Only YAP versions following 5.1.0 and CVS HEAD versions starting from
4 January 2006 are supported.  Notice that support for unbounded
integers in YAP is young and may have errors that could affect
programs using the PPL (see, e.g.,
<A HREF="http://www.cs.unipr.it/pipermail/ppl-devel/2006-January/007780.html">http://www.cs.unipr.it/pipermail/ppl-devel/2006-January/007780.html</A>).

In order to dynamically load the library from YAP you should
simply load <CODE>prefix/lib/ppl/ppl_yap.pl</CODE>.
This will invoke <CODE>ppl_initialize/0</CODE> automatically;
it is the programmer's responsibility to call <CODE>ppl_finalize/0</CODE>
when the PPL library is no longer needed.
Notice that, for dynamic linking to work, you should have configured
the library with the <CODE>--enable-shared</CODE> option.

*/ /* \mainpage */


