PPL: Prolog Language Interface

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# 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 systems.

The system-independent features of the library are described in Section System-Independent Features. Section Compilation and Installation explains how the various incarnations of the Prolog interface are compiled and installed. Section System-Dependent Features illustrates the system-dependent features of the interface for all the supported systems.

# System-Independent Features

The Prolog interface provides access to the PPL polyhedra. A general introduction to convex polyhedra, their representation in the PPL and the operations provided by the PPL is given in Sections The Main Features, Convex Polyhedra, Representations of Convex Polyhedra and Operations on Convex Polyhedra of this manual. Here we just describe those aspects that are specific to the Prolog interface.

## Overview

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

- The Prolog interface to the PPL is initialized and finalized by the predicates `ppl_initialize/0` and `ppl_finalize/0`. Thus the only interface predicates callable after `ppl_finalize/0` are `ppl_finalize/0` itself (this further call has no effect) and `ppl_initialize/0`, 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 PPL-Prolog interface have been written so that `ppl_initialize/0` and/or `ppl_finalize/0` are called automatically. Section 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 `ppl_initialize/0` and `ppl_finalize/0` 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 polyhedron can only be accessed by means of a Prolog term called a handle. 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 by using:

         ppl_new_Polyhedron_from_dimension/3,
         ppl_new_Polyhedron_empty_from_dimension/3,
         ppl_new_Polyhedron_from_Polyhedron/4,
         ppl_new_Polyhedron_from_constraints/3,
         ppl_new_Polyhedron_from_generators/3.
         ppl_new_Polyhedron_from_bounding_box/3.

  These predicates will create or copy a PPL polyhedron and construct a valid handle for referencing it. The first argument (in the case of `ppl_new_Polyhedron_from_Polyhedron/4`, the first and third arguments) denotes the topology and can be either `c` or `nnc` indicating a C or NNC polyhedron, respectively. The third argument (in the case of `ppl_new_Polyhedron_from_Polyhedron/4`, the fourth argument) is a Prolog term that is unified with a new valid handle for accessing this polyhedron.

- As soon as a PPL polyhedron is no longer required, the memory occupied by it should be released using the PPL predicate `ppl_delete_Polyhedron/1`. 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 a variable bound to a handle for a PPL Polyhedron dies or is uninstantiated, the handle can be garbage-collected, but the polyhedra to which the handle refers will not be released. Once a handle has been used as an argument in `ppl_delete_Polyhedron/1`, it becomes invalid.

- For a PPL polyhedron with space dimension `k`, the identifiers used for the PPL variables must lie between 0 and and correspond to the indices of the associated Cartesian axes. 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 Representations of Convex Polyhedra.

- 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 Representations of Convex Polyhedra.

- Any application using the PPL should make sure that only the intended version(s) of the library are ever used. Predicates

      ppl_version_major/1,
      ppl_version_minor/1,
      ppl_version_revision/1,
      ppl_version_beta/1,
      ppl_version/1,
      ppl_banner.

  allow run-time checking of information about the version being used.

## PPL Predicate List

Here is a list of all the PPL predicates provided by the Prolog interface.

`ppl_version_major(?Integer)`

`ppl_version_minor(?Integer)`

`ppl_version_revision(?Integer)`

`ppl_version_beta(?Integer)`

`ppl_version(?Atom)`

`ppl_banner(?Atom)`

`ppl_max_space_dimension(?Integer)`

`ppl_initialize`

`ppl_finalize`

`ppl_set_timeout_exception_atom(+Atom)`

`ppl_set_timeout(+Integer)`

`ppl_reset_timeout`

`ppl_new_Polyhedron_from_dimension(+Topology, +Integer, -Handle)`

`ppl_new_Polyhedron_empty_from_dimension(+Topology, +Integer, -Handle)`

`ppl_new_Polyhedron_from_Polyhedron(+Topology_1, +Handle_1, +Topology_2, -Handle_2)`

`ppl_new_Polyhedron_from_constraints(+Topology, +Constraint_System, -Handle)`

`ppl_new_Polyhedron_from_generators(+Topology, +Generator_System, -Handle)`

`ppl_new_Polyhedron_from_bounding_box(+Topology, +Box, -Handle)`

`ppl_Polyhedron_swap(+Handle1, +Handle2)`

`ppl_delete_Polyhedron(+Handle)`

`ppl_Polyhedron_space_dimension(+Handle, -Integer)`

`ppl_Polyhedron_get_constraints(+Handle, -Constraint_System)`

`ppl_Polyhedron_get_minimized_constraints(+Handle, -Constraint_System)`

`ppl_Polyhedron_get_generators(+Handle, -Generator_System)`

`ppl_Polyhedron_get_minimized_generators(+Handle, -Generator_System)`

`ppl_Polyhedron_relation_with_constraint(+Handle, +Constraint, -Relation)`

`ppl_Polyhedron_relation_with_generator(+Handle, +Generator, -Relation)`

`ppl_Polyhedron_get_bounding_box(+Handle, +Complexity, -Box)`

`ppl_Polyhedron_is_empty(+Handle)`

`ppl_Polyhedron_is_universe(+Handle)`

`ppl_Polyhedron_is_bounded(+Handle)`

`ppl_Polyhedron_bounds_from_above(+Handle, +LinExpr)`

`ppl_Polyhedron_bounds_from_below(+Handle, +LinExpr)`

`ppl_Polyhedron_maximize(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool)`

`ppl_Polyhedron_maximize_with_point(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool, ?Point)`

`ppl_Polyhedron_minimize(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool)`

`ppl_Polyhedron_minimize_with_point(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool, ?Point)`

`ppl_Polyhedron_is_topologically_closed(+Handle)`

`ppl_Polyhedron_contains_Polyhedron(+Handle_1, +Handle_2)`

`ppl_Polyhedron_strictly_contains_Polyhedron(+Handle_1, +Handle_2)`

`ppl_Polyhedron_is_disjoint_from_Polyhedron(+Handle_1, +Handle_2)`

`ppl_Polyhedron_equals_Polyhedron(+Handle_1, +Handle_2)`

`ppl_Polyhedron_OK(+Handle)`

`ppl_Polyhedron_add_constraint(+Handle, +Constraint)`

`ppl_Polyhedron_add_constraint_and_minimize(+Handle, +Constraint)`

`ppl_Polyhedron_add_generator(+Handle, +Generator)`

`ppl_Polyhedron_add_generator_and_minimize(+Handle, +Generator)`

`ppl_Polyhedron_add_constraints(+Handle, +Constraint_System)`

`ppl_Polyhedron_add_constraints_and_minimize(+Handle, +Constraint_System)`

`ppl_Polyhedron_add_generators(+Handle, +Generator_System)`

`ppl_Polyhedron_add_generators_and_minimize(+Handle, +Generator_System)`

`ppl_Polyhedron_intersection_assign(+Handle_1, +Handle_2)`

`ppl_Polyhedron_intersection_assign_and_minimize(+Handle_1, +Handle_2)`

`ppl_Polyhedron_poly_hull_assign(+Handle_1, +Handle_2)`

`ppl_Polyhedron_poly_hull_assign_and_minimize(+Handle_1, +Handle_2)`

`ppl_Polyhedron_poly_difference_assign(+Handle_1, +Handle_2)`

`ppl_Polyhedron_affine_image(+Handle, +PPL_Var, +LinExpr, +Integer)`

`ppl_Polyhedron_affine_preimage(+Handle, +PPL_Var, +LinExpr, +Integer)`

`ppl_Polyhedron_generalized_affine_image(+Handle, +PPL_Var, +Relation_Symbol, +LinExpr, +Integer)`

`ppl_Polyhedron_generalized_affine_image_lhs_rhs(+Handle, +LinExpr1, +Relation_Symbol, +LinExpr2)`

`ppl_Polyhedron_time_elapse_assign(+Handle_1, +Handle_2)`

`ppl_Polyhedron_BHRZ03_widening_assign_with_token(+Handle_1, +Handle_2, ?Integer)`

`ppl_Polyhedron_BHRZ03_widening_assign(+Handle_1, +Handle_2)`

`ppl_Polyhedron_limited_BHRZ03_extrapolation_assign_with_token(+Handle_1, +Handle_2, +Constraint_System, ?Integer)`

`ppl_Polyhedron_limited_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

`ppl_Polyhedron_bounded_BHRZ03_extrapolation_assign_with_token(+Handle_1, +Handle_2, +Constraint_System, ?Integer)`

`ppl_Polyhedron_bounded_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

`ppl_Polyhedron_H79_widening_assign_with_token(+Handle_1, +Handle_2, ?Integer)`

`ppl_Polyhedron_H79_widening_assign(+Handle_1, +Handle_2)`

`ppl_Polyhedron_limited_H79_extrapolation_assign_with_token(+Handle_1, +Handle_2, +Constraint_System, ?Integer)`

`ppl_Polyhedron_limited_H79_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

`ppl_Polyhedron_bounded_H79_extrapolation_assign_with_token(+Handle_1, +Handle_2, +Constraint_System)`

`ppl_Polyhedron_bounded_H79_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System, ?Integer)`

`ppl_Polyhedron_topological_closure_assign(+Handle)`

`ppl_Polyhedron_add_dimensions_and_embed(+Handle, +Integer)`

`ppl_Polyhedron_add_dimensions_and_project(+Handle, +Integer)`

`ppl_Polyhedron_concatenate_assign(+Handle1, +Handle2)`

`ppl_Polyhedron_remove_dimensions(+Handle, +List_of_PPL_Vars)`

`ppl_Polyhedron_remove_higher_dimensions(+Handle, +Integer))`

`ppl_Polyhedron_expand_dimension(+Handle, +PPL_Var, +Integer))`

`ppl_Polyhedron_fold_dimensions(+Handle, +List_of_PPL_Vars, +PPL_Var))`

`ppl_Polyhedron_map_dimensions(+Handle, +P_Func))`

## PPL Predicate Specifications

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

     Handle      --> Prolog term

     Topology    --> c | nnc

     VarId       --> number | + number      variable identifier

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

     LinExpr     --> PPL_Var                PPL variable
                | number
                | + LinExpr                 unary plus
                | - LinExpr                 unary minus
                | LinExpr + LinExpr         addition
                | LinExpr - LinExpr         subtraction
                | number * LinExpr          multiplication
                | LinExpr * number          multiplication

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

     Denominator --> number
                | + number | - number       number must be non-zero

     Constraint  --> LinExpr Relation_Symbol LinExpr
                                            constraint

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

     Generator   --> point(LinExpr)         point
                | point(LinExpr, Denominator)
                                            point
                | closure_point(LinExpr)    closure point
                | closure_point(LinExpr, Denominator)
                                            closure point
                  (the point or closure point is defined by LinExpr/Denominator.)
                | ray(LinExpr)              ray
                | line(LinExpr)             line

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

     Atom        --> Prolog atom

     Relation    --> is_disjoint            between a constraint and a polyhedron
                | strictly_intersects       between a constraint and a polyhedron
                | is_included               between a constraint and a polyhedron
                | saturates                 between a constraint and a polyhedron
                | subsumes                  between a generator and a polyhedron

     Relation_List                          list of relations
                 --> []
                | [Relation | Relation_List]

     Complexity  --> polynomial | simplex | any

     Rational_Numerator
                 --> number | + number | - number

     Rational_Denominator
                 --> number                 number must be non-zero

     Rational    --> Rational_Numerator     rational number
                | Rational_Numerator/Rational_Denominator

     Bound       --> c(Rational)            closed rational limit
                | o(Rational)               open rational limit
                | o(pinf)                   unbounded in the positive direction
                | o(minf)                   unbounded in the negative direction

     Interval    --> i(Bound, Bound)        rational interval

     Box         --> []                     list of intervals
                | [Interval | Box]

     Vars_Pair   --> PPLVar - PPLVar        map relation

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

Below is a short description of each of the interface predicates. For full definitions of terminology used here, see Sections The Main Features, Convex Polyhedra, Representations of Convex Polyhedra and Operations on Convex Polyhedra of this manual.

## `ppl_version_major(?Integer)`

Unifies `Integer` with the major number of the PPL version.

## `ppl_version_minor(?Integer)`

Unifies `Integer` with the minor number of the PPL version.

## `ppl_version_revision(?Integer)`

Unifies `Integer` with the revision number of the PPL version.

## `ppl_version_beta(?Integer)`

Unifies `Integer` with the beta number of the PPL version.

## `ppl_version(?Atom)`

Unifies `Atom` with the PPL version.

## `ppl_banner(?Atom)`

Unifies `Atom` 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.

## `ppl_max_space_dimension(?Integer)`

Unifies `Integer` with the maximum space dimension this library can handle.

## `ppl_initialize`

Initializes the PPL interface. Multiple calls to `ppl_initialize` does no harm.

## `ppl_finalize`

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

## `ppl_set_timeout_exception_atom(+Atom)`

Sets the atom to be thrown by timeout exceptions to `Atom`. The default value is `time_out`.

## `ppl_timeout_exception_atom(?Atom)`

The atom to be thrown by timeout exceptions is unified with `Atom`.

## `ppl_set_timeout(+Integer)`

Computations taking exponential time will be interrupted some time after `Integer` ms after that call. If the computation is interrupted that way, the current timeout exception atom will be thrown. `Integer` must be strictly greater than zero.

## `ppl_reset_timeout`

Resets the timeout time so that the computation is not interrupted.

## `ppl_new_Polyhedron_from_dimension(+Topology, +Integer, -Handle)`

Creates a new universe C or NNC polyhedron , depending on the value of `Topology`, with `Integer` dimensions. `Handle` is unified with the handle for . Thus the query

       ?- ppl_new_Polyhedron_from_dimension(c, 3, X).

creates the C polyhedron defining the 3-dimensional vector space with `X` bound to a valid handle for accessing it.

## `ppl_new_Polyhedron_empty_from_dimension(+Topology, +Integer, -Handle)`

Creates a new empty C or NNC polyhedron , depending on the value of `Topology`, with `Integer` dimensions. `Handle` is unified with the handle for . Thus the query

       ?- ppl_new_Polyhedron_empty_from_dimension(nnc, 3, X).

creates an empty NNC polyhedron embedded in with `X` bound to a valid handle for accessing it.

## `ppl_new_Polyhedron_from_Polyhedron(+Topology_1, +Handle_1, +Topology_2, -Handle_2)`

If `Handle_1` refers to a C or NNC polyhedron (depending on the value of `Topology_1`), then this creates a copy of with topology C or NNC, depending on the value of `Topology_2`. `Handle_2` is unified with the handle for . Thus the query

       ?- ppl_new_Polyhedron_empty_from_dimension(nnc, 3, X),
          ppl_new_Polyhedron_from_Polyhedron(c, X, nnc, Y).

creates an empty C polyhedron embedded in referenced by `X` and then makes a copy, converting the topology to an NNC polyhedron. with `Y` bound to a valid handle for accessing it.

When using `ppl_new_Polyhedron_from_Polyhedron/2`, when the source polyhedron is NNC and the copy is C, care must be taken that the source polyhedron referenced by `Handle1` is topologically closed.

## `ppl_new_Polyhedron_from_constraints(+Topology, +Constraint_System, -Handle)`

Creates a polyhedron represented by `Constraint_System` with topology C or NNC, depending on the value of `Topology`. `Handle` is unified with the handle for .

## `ppl_new_Polyhedron_from_generators(+Topology, +Generator_System, -Handle)`

Creates a polyhedron represented by `Generator_System` with topology C or NNC, depending on the value of `Topology`. `Handle` is unified with the handle for .

## `ppl_new_Polyhedron_from_bounding_box(+Topology, +Box, -Handle)`

Creates a polyhedron represented by `Box` with topology C or NNC, depending on the value of `Topology`, and `Handle` is unified with the handle for . A bound of the form `o(Rational)` can be included in an interval in `Box` only if `Topology` is `nnc`.

## `ppl_Polyhedron_swap(+Handle1, +Handle2)`

Swaps the polyhedron referenced by `Handle1` with the one referenced by `Handle2`. The polyhedra and must have the same topology.

## `ppl_delete_Polyhedron(+Handle)`

Deletes the polyhedron referenced by `Handle`. After execution, `Handle` is no longer a valid handle for a PPL polyhedron.

## `ppl_Polyhedron_space_dimension(+Handle, ?Integer)`

Unifies the space dimension of the polyhedron referenced by `Handle` with `Integer`.

## `ppl_Polyhedron_get_constraints(+Handle, ?Constraint_System)`

Unifies `Constraint_System` with a list of the constraints in the constraints system representing the polyhedron referenced by `Handle`.

## `ppl_Polyhedron_get_minimized_constraints(+Handle, ?Constraint_System)`

Unifies `Constraint_System` with a minimized list of the constraints in the constraints system representing the polyhedron referenced by `Handle`.

## `ppl_Polyhedron_get_generators(+Handle, ?Generator_System)`

Unifies `Generator_System` with a list of the generators in the generators system representing the polyhedron referenced by `Handle`.

## `ppl_Polyhedron_get_minimized_generators(+Handle, ?Generator_System)`

Unifies `Generator_System` with a minimized list of the generators in the generators system representing the polyhedron referenced by `Handle`.

## `ppl_Polyhedron_relation_with_constraint(+Handle, +Constraint, ?Relation_List)`

Unifies `Relation_List` with the list of relations the polyhedron referenced by `Handle` has with `Constraint`. The possible relations are listed in the grammar rules above; their meaning is given in the paragraph specifying the relation_with operations in Section Operations on Convex Polyhedra.

## `ppl_Polyhedron_relation_with_generator(+Handle, +Generator, ?Relation_List)`

Unifies `Relation_List` with the list of relations the polyhedron referenced by `Handle` has with `Generator`. The possible relations are listed in the grammar rules above; their meaning is given in the paragraph specifying the relation_with operations in Section Operations on Convex Polyhedra.

## `ppl_Polyhedron_get_bounding_box(+Handle, +Complexity, ?Box)`

Succeeds if and only if the bounding box of the polyhedron referenced by `Handle` unifies with the box defined by `Box`. E.g.,

       ?- A = '$VAR'(0), B = '$VAR'(1),
          ppl_new_Polyhedron_from_constraints(nnc, [B > 0, 4*A =< 2], X),
          ppl_Polyhedron_get_bounding_box(X, any, Box).

       Box = [i(o(minf), c(1/2)), i(o(0), o(pinf))].

Note that the rational numbers in `Box` are in canonical form. E.g., the following will fail:

       ?- A = '$VAR'(0), B = '$VAR'(1),
          ppl_new_Polyhedron_from_constraints(nnc, [B > 0, 4*A =< 2], X),
          ppl_Polyhedron_get_bounding_box(X, any, Box),
          Box = [i(o(minf), c(2/4)), i(o(0), o(pinf))].

The complexity class `Complexity` determining the algorithm to be used has the following meaning:

- `polynomial` allows code of the worst-case polynomial complexity class;
- `simplex` allows code of the worst-case exponential but typically polynomial complexity class;
- `any` allows code of the universal complexity class.

## `ppl_Polyhedron_is_empty(+Handle)`

Succeeds if and only if the polyhedron referenced by `Handle` is empty.

## `ppl_Polyhedron_is_universe(+Handle)`

Succeeds if and only if the polyhedron referenced by `Handle` is the universe.

## `ppl_Polyhedron_is_bounded(+Handle)`

Succeeds if and only if the polyhedron referenced by `Handle` is bounded.

## `ppl_Polyhedron_bounds_from_above(+Handle, +LinExpr)`

Succeeds if and only if `LinExpr` is bounded from above in the polyhedron referenced by `Handle`.

## `ppl_Polyhedron_bounds_from_below(+Handle, +LinExpr)`

Succeeds if and only if `LinExpr` is bounded from below in the polyhedron referenced by `Handle`.

### `ppl_Polyhedron_maximize(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool)`

Succeeds if and only if the polyhedron referenced by `Handle` is not empty and `LinExpr` is bounded from above in .

`Integer1` is unified with the numerator of the supremum value and `Integer2` with the denominator of the supremum value. If the supremum is also the maximum, `Bool` is unified with the atom `true` and, otherwise, unified with the atom `false`.

## `ppl_Polyhedron_maximize_with_point(+Handle, +LinExpr, ?Integer1, ?Integer2, ?Bool, ?Point)`

Succeeds if and only if the polyhedron referenced by `Handle` is not empty and `LinExpr` is bounded from above in .

`Integer1` is unified with the numerator of the supremum value, `Integer2` with the denominator of the supremum value, and `Point` with a point or closure point where `LinExpr` reaches this value. If the supremum is also the maximum, `Bool` is unified with the atom `true` and, otherwise, unified with the atom `false`.

## `ppl_Polyhedron_minimize(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool)`

Succeeds if and only if the polyhedron referenced by `Handle` is not empty and `LinExpr` is bounded from below in .

`Integer1` is unified with the numerator of the infimum value and `Integer2` with the denominator of the infimum value. If the infimum is also the minimum, `Bool` is unified with the atom `true` and, otherwise, unified with the atom `false`.

## `ppl_Polyhedron_minimize_with_point(+Handle, +LinExpr, ?Integer, ?Integer, ?Bool, ?Point)`

Succeeds if and only if the polyhedron referenced by `Handle` is not empty and `LinExpr` is bounded from below in .

`Integer1` is unified with the numerator of the infimum value, `Integer2` with the denominator of the infimum value, and `Point` with a point or closure point where `LinExpr` reaches this value. If the infimum is also the minimum, `Bool` is unified with the atom `true` and, otherwise, unified with the atom `false`.

## `ppl_Polyhedron_is_topologically_closed(+Handle)`

Succeeds if and only if the polyhedron referenced by `Handle` is topologically closed.

## `ppl_Polyhedron_contains_Polyhedron(+Handle_1, +Handle_2)`

Succeeds if and only if the polyhedron referenced by `Handle_1` is included in or equal to the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_strictly_contains_Polyhedron(+Handle_1, +Handle_2)`

Succeeds if and only if the polyhedron referenced by `Handle_1` is included in but not equal to the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_is_disjoint_from_Polyhedron(+Handle_1, +Handle_2)`

Succeeds if and only if the polyhedron referenced by `Handle_1` is disjoint from the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_equals_Polyhedron(+Handle_1, +Handle_2)`

Succeeds if and only if the polyhedron referenced by `Handle_1` is equal to the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_OK(+Handle)`

Succeeds only if the polyhedron referenced by `Handle` is well formed, i.e., if it satisfies all its implementation invariants. Useful for debugging purposes.

## `ppl_Polyhedron_add_constraint(+Handle, +Constraint)`

## `ppl_Polyhedron_add_constraint_and_minimize(+Handle, +Constraint)`

Updates the polyhedron referenced by `Handle` to one obtained by adding `Constraint` to its constraint system. Thus, the query

       ?- ppl_new_Polyhedron_from_dimension(c, 3, X),
          A = '$VAR'(0), B = '$VAR'(1), C = '$VAR'(2),
          ppl_Polyhedron_add_constraint(X, 4*A + B - 2*C >= 5).

will update the polyhedron with handle `X` to consist of the set of points in the vector space satisfying the constraint .

Note that `ppl_Polyhedron_add_constraint_and_minimize/2` will fail if, after adding the constraint, the polyhedron is empty.

## `ppl_Polyhedron_add_generator(+Handle, +Generator)`

## `ppl_Polyhedron_add_generator_and_minimize(+Handle, +Generator)`

Updates the polyhedron referenced by `Handle` to one obtained by adding `Generator` to its generator system. Thus, after the query

       ?- ppl_new_Polyhedron_from_dimension(c, 3, X),
          A = '$VAR'(0), B = '$VAR'(1), C = '$VAR'(2),
          ppl_Polyhedron_add_generator(X, point(-100*A - 5*B, 8)).

will update the polyhedron with handle `X` to be the single point in the vector space .

## `ppl_Polyhedron_add_constraints(+Handle, +Constraint_System)`

Updates the polyhedron referenced by `Handle` to one obtained by adding to its constraint system the constraints in `Constraint_System`. E.g.,

       | ?- ppl_new_Polyhedron_from_dimension(c, 2, X),
            A = '$VAR'(0), B = '$VAR'(1),
            ppl_Polyhedron_add_constraints(X, [4*A + B >= 3, A = 1]),
            ppl_Polyhedron_get_constraints(X, CS).

       CS = [4*A+1*B>=3,1*A=1] ?

The updated polyhedron referenced by `Handle` can be empty and a query will succeed even when `Constraint_System` is unsatisfiable.

## `ppl_Polyhedron_add_constraints_and_minimize(+Handle, +Constraint_System)`

Updates the polyhedron referenced by `Handle` to one obtained by adding to its constraint system the constraints in `Constraint_System`. E.g.,

       ?- ppl_new_Polyhedron_from_dimension(c, 2, X),
          A = '$VAR'(0), B = '$VAR'(1),
          ppl_Polyhedron_add_constraints_and_minimize(X, [4*A + B >= 3, A = 1]),
          ppl_Polyhedron_get_constraints(X, CS).

       CS = [1*B>= -1,1*A=1]

This will fail if, after adding the constraints, the polyhedron is empty. E.g., the following will fail,

       ?- A = '$VAR'(0), B = '$VAR'(1),
          ppl_new_Polyhedron_from_dimension(c, 2, X),
          ppl_Polyhedron_add_constraints_and_minimize(X,
            [4*A + B >= 3, A = 0, B =< 0]),
          ppl_Polyhedron_get_constraints(X, CS).

## `ppl_Polyhedron_add_generators(+Handle, +Generator_System)`

Updates the polyhedron referenced by `Handle` to one obtained by adding to its generator system the generators in `Generator_System`.

If the system of generators representing a polyhedron is non-empty, then it must include a point (see the paragraph on generator representation in Section Representations of Convex Polyhedra). Thus care must be taken to ensure that, before calling this predicate, either the polyhedron referenced by `Handle` is non-empty or that whenever `Generator_System` is non-empty the first element defines a point. E.g.,

       ?- ppl_new_Polyhedron_empty_from_dimension(c, 3, X),
          A='$VAR'(0), B = '$VAR'(1), C = '$VAR'(2),
          ppl_Polyhedron_add_generators(X,
            [point(1*A + 1*B + 1*C, 1), ray(1*A), ray(2*A)]),
          ppl_Polyhedron_get_generators(X, GS).

       GS = [ray(2*A), point(1*A+1*B+1*C), ray(1*A)]

## `ppl_Polyhedron_add_generators_and_minimize(+Handle, +Generator_System)`

Updates the polyhedron referenced by `Handle` to one obtained by adding to its generator system the generators in `Generator_System`.

Unlike the predicate `ppl_add_generators`, the order of the generators in `Generator_System` is not important. E.g.,

       ?- ppl_new_Polyhedron_empty_from_dimension(c, 3, X),
          A='$VAR'(0), B = '$VAR'(1), C = '$VAR'(2),
          ppl_Polyhedron_add_generators_and_minimize(X,
            [ray(1*A), ray(2*A), point(1*A + 1*B + 1*C, 1)]),
          ppl_Polyhedron_get_generators(X, GS).

       GS = [point(1*A+1*B+1*C), ray(1*A)]

## `ppl_Polyhedron_intersection_assign(+Handle_1, +Handle_2)`

## `ppl_Polyhedron_intersection_assign_and_minimize(+Handle_1, +Handle_2)`

Assigns to the polyhedron referenced by `Handle_1` its intersection with the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_poly_hull_assign(+Handle_1, +Handle_2)`

## `ppl_Polyhedron_poly_hull_assign_and_minimize(+Handle_1, +Handle_2)`

Assigns to the polyhedron referenced by `Handle_1` its poly-hull with the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_poly_difference_assign(+Handle_1, +Handle_2)`

Assigns to the polyhedron referenced by `Handle_1` its poly-difference with the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_affine_image(+Handle, +PPL_Var, +LinExpr, +Integer)`

Transforms the polyhedron referenced by `Handle` assigning the affine expression `LinExpr`/`Integer` to `PPL_Var`.

## `ppl_Polyhedron_affine_preimage(+Handle, +PPL_Var, +LinExpr, +Integer)`

This is the inverse transformation to that for `ppl_affine_image`.

## `ppl_Polyhedron_generalized_affine_image(+Handle, +PPL_Var, +Relation_Symbol +LinExpr, +Integer)`

Transforms the polyhedron referenced by `Handle` assigning the generalized affine image with respect to the transfer function `PPL_Var` `Relation_Symbol` `LinExpr`/`Integer`.

## `ppl_Polyhedron_generalized_affine_image_lhs_rhs(+Handle, +LinExpr1, +Relation_Symbol +LinExpr2)`

Transforms the polyhedron referenced by `Handle` assigning the generalized affine image with respect to the transfer function `LinExpr1` `Relation_Symbol` `LinExpr2`.

## `ppl_Polyhedron_time_elapse_assign(+Handle_1, +Handle_2)`

Assigns to the polyhedron referenced by `Handle_1` the time-elapse with the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_BHRZ03_widening_assign_with_token(+Handle_1, +Handle_2, ?Integer)`

The polyhedra referenced by `Handle_1` and `Handle_2` are unaltered. The token `Integer` is 0 if a BHRZ03 widening would have changed the polyhedron referenced by `Handle_1` and is 1 otherwise.

## `ppl_Polyhedron_BHRZ03_widening_assign(+Handle_1, +Handle_2)`

Assigns to the polyhedron referenced by `Handle_1` its BHRZ03-widening with the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_limited_BHRZ03_extrapolation_assign_with_token( +Handle_1, +Handle_2, +Constraint_System, ?Integer)`

The polyhedra referenced by `Handle_1` and `Handle_2` are unaltered. The token `Integer` is 0 if a BHRZ03-widening with the polyhedron referenced by `Handle_2`, improved by enforcing those constraints in `Constraint_System` would have changed the polyhedron referenced by `Handle_1` and is 1 otherwise.

## `ppl_Polyhedron_limited_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

Assigns to the polyhedron referenced by `Handle_1` the result of its BHRZ03-widening with the polyhedron referenced by `Handle_2`, improved by enforcing those constraints in `Constraint_System`.

## `ppl_Polyhedron_bounded_BHRZ03_extrapolation_assign_with_token( +Handle_1, +Handle_2, +Constraint_System, ?Integer)`

The polyhedra and referenced by `Handle_1` and `Handle_2`, respectively are unaltered. The token `Integer` is 0 if a BHRZ03-widening with , improved by enforcing all the constraints of the form and that are satisfied by all the points of together with the constraints in `Constraint_System` would have changed the polyhedron referenced by `Handle_1` and is 1 otherwise.

## `ppl_Polyhedron_bounded_BHRZ03_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

Assigns to the polyhedron referenced by `Handle_1` the result of its BHRZ03-widening with the polyhedron referenced by `Handle_2` improved by enforcing all the constraints of the form and that are satisfied by all the points of together with the constraints in `Constraint_System`.

## `ppl_Polyhedron_H79_widening_assign_with_token(+Handle_1, +Handle_2, ?Integer)`

The polyhedra referenced by `Handle_1` and `Handle_2` are unaltered. The token `Integer` is 0 if an H79 widening would have changed the polyhedron referenced by `Handle_1` and is 1 otherwise.

## `ppl_Polyhedron_H79_widening_assign(+Handle_1, +Handle_2)`

Assigns to the polyhedron referenced by `Handle_1` its H79-widening with the polyhedron referenced by `Handle_2`.

## `ppl_Polyhedron_limited_H79_extrapolation_assign_with_token(+Handle_1, +Handle_2, +Constraint_System, ?Integer)`

The polyhedra referenced by `Handle_1` and `Handle_2` are unaltered. The token `Integer` is 0 if a H79-widening with the polyhedron referenced by `Handle_2`, improved by enforcing those constraints in `Constraint_System` would have changed the polyhedron referenced by `Handle_1` and is 1 otherwise.

## `ppl_Polyhedron_limited_H79_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

Assigns to the polyhedron referenced by `Handle_1` its H79-widening with the polyhedron referenced by `Handle_2`, improved by enforcing those constraints in `Constraint_System`.

## `ppl_Polyhedron_bounded_H79_extrapolation_assign_with_token(+Handle_1, +Handle_2, +Constraint_System, ?Integer)`

The polyhedra and referenced by `Handle_1` and `Handle_2`, respectively are unaltered. The token `Integer` is 0 if a H79-widening with , improved by enforcing all the constraints of the form and that are satisfied by all the points of together with the constraints in `Constraint_System` would have changed the polyhedron referenced by `Handle_1` and is 1 otherwise.

## `ppl_Polyhedron_bounded_H79_extrapolation_assign(+Handle_1, +Handle_2, +Constraint_System)`

Assigns to the polyhedron referenced by `Handle_1` the result of its H79-widening with the polyhedron referenced by `Handle_2` improved by enforcing all the constraints of the form and that are satisfied by all the points of together with the constraints in `Constraint_System`.

## `ppl_Polyhedron_topological_closure_assign(+Handle)`

Assigns to the polyhedron referenced by `Handle` its topological closure.

## `ppl_Polyhedron_add_dimensions_and_embed(+Handle, +Integer)`

Embeds the polyhedron referenced by `Handle` in a space that is enlarged by `Integer` dimensions, E.g.,

       ?- ppl_new_Polyhedron_empty_from_dimension(c, 0, X),
          ppl_Polyhedron_add_dimensions_and_embed(X, 2),
          ppl_Polyhedron_get_constraints(X, CS),
          ppl_Polyhedron_get_generators(X, GS).

       CS = [],
       GS = [point(0),line(1*A),line(1*B)]

## `ppl_Polyhedron_concatenate_assign(+Handle1, +Handle2)`

Updates the polyhedron referenced by `Handle1` by first embedding in a new space enlarged by the space dimensions of the polyhedron referenced by `Handle2`, and then adds to its system of constraints a renamed-apart version of the constraints of .

E.g.,

       ?- ppl_new_Polyhedron_from_dimension(nnc, 2, X),
          A = '$VAR'(0), B = '$VAR'(1), C = '$VAR'(2),
          D = '$VAR'(3), E = '$VAR'(4),
          ppl_new_Polyhedron_from_constraints(nnc, [A > 1, B >= 0, C >= 0], Y),
          ppl_Polyhedron_concatenate_assign(X, Y),
          ppl_Polyhedron_get_constraints(X, CS).

       CS = [1*C > 1, 1*D >= 0, 1*E >= 0]

## `ppl_Polyhedron_add_dimensions_and_project(+Handle, +Integer)`

Projects the polyhedron referenced by `Handle` onto a space that is enlarged by `Integer` dimensions, E.g.,

       ?- ppl_new_Polyhedron_empty_from_dimension(c, 0, X),
          ppl_Polyhedron_add_dimensions_and_project(X, 2),
          ppl_Polyhedron_get_constraints(X, CS),
          ppl_Polyhedron_get_generators(X, GS).

       CS = [1*A = 0, 1*B = 0],
       GS = [point(0)]

## `ppl_Polyhedron_remove_dimensions(+Handle, +List_of_PPL_Vars)`

Removes the space dimensions given by the identifiers of the PPL variables in list `List_of_PPL_Vars` from the polyhedron referenced by `Handle`. The identifiers for the remaining PPL variables are renumbered so that they are consecutive and the maximum index is less than the number of dimensions. E.g.,

       ?- ppl_new_Polyhedron_empty_from_dimension(c, 3, X),
          A='$VAR'(0), B = '$VAR'(1), C = '$VAR'(2),
          ppl_Polyhedron_remove_dimensions(X, [B]),
          ppl_Polyhedron_space_dimension(X, K),
          ppl_Polyhedron_get_generators(X, GS).

       K = 2,
       GS = [point(0),line(1*A),line(1*B),line(0)]

## `ppl_Polyhedron_remove_higher_dimensions(+Handle, +Integer))`

Projects the polyhedron referenced to by `Handle` onto the first `Integer` dimension. E.g.,

       ?- ppl_new_Polyhedron_empty_from_dimension(c, 5, X),
          ppl_Polyhedron_remove_higher_dimensions(X, 3),
          ppl_Polyhedron_space_dimension(X, K).

## `ppl_Polyhedron_expand_dimension(+Handle, +PPL_Var, +Integer))`

`Integer` copies of the space dimension referenced by `PPL_Var` are added to the polyhedron referenced to by `Handle`.

## `ppl_Polyhedron_fold_dimensions(+Handle, +List_of_PPL_Vars, +PPL_Var))`

The space dimensions referenced by the PPL variables in list `List_of_PPL_Vars` are folded into the dimension referenced by `PPL_Var` and removed. The result is undefined if `List_of_PPL_Vars` does not have the properties described in the paragraph specifying the fold_dimensions operator in Section Operations on Convex Polyhedra.

## `ppl_Polyhedron_map_dimensions(+Handle, +P_Func))`

Maps the dimensions of the polyhedron referenced by `Handle` using the partial function defined by `P_Func`. The result is undefined if `P_Func` does not encode a partial function with the properties described in the paragraph specifying the map_dimensions operator in Section Operations on Convex Polyhedra.

# Compilation and Installation

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.

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

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 `-DPROLOG_TRACK_ALLOCATION` to the options passed to the C++ compiler. Your configure command would then look like

      path/to/configure --with-cxxflags="-DPROLOG_TRACK_ALLOCATION" ...

# System-Dependent Features

## CIAO Prolog

Support for CIAO Prolog is under development and will be available in a future release. Only Ciao Prolog 1.9 #44 or later is supported.

## GNU Prolog

The GNU Prolog interface to the PPL library is available both as ``PPL enhanced'' GNU Prolog interpreter and as a library that can be linked to GNU Prolog programs. Only GNU Prolog version 1.2.12 or later is supported.

Notice that GNU Prolog version 1.2.12 suffers from a serious limitation as far as foreign code is concerned. In order to be safe you must configure GNU Prolog with the `--disable-ebp` option (note that this has a negative effect on performance). See http://www.cs.unipr.it/pipermail/ppl-devel/2002-June/001777.html, http://www.cs.unipr.it/pipermail/ppl-devel/2002-June/001780.html, http://www.cs.unipr.it/pipermail/ppl-devel/2002-June/001788.html and http://www.cs.unipr.it/pipermail/ppl-devel/2002-June/001789.html for more information.

We have experienced other serious problems with the GNU Prolog interface, up to and including GNU Prolog version 1.2.16: see http://www.cs.unipr.it/pipermail/ppl-devel/2002-October/002657.html for more information.

### The `ppl_gprolog` Executable

If an appropriate version of GNU Prolog is installed on the machine on which you compiled the library, the command `make install` will install the executable `ppl_gprolog` in the directory `prefix/bin`. The `ppl_gprolog` 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 `ppl_initialize/0` before any other PPL predicate and to call `ppl_finalize/0` when you are done with the library.

### Linking the Library To GNU Prolog Programs

In order to allow linking GNU Prolog programs to the PPL, the following files are installed in the directory `prefix/lib/ppl`: `ppl_gprolog.pl` contains the required foreign declarations; `libppl_gprolog.*` 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, `source1.pl` and `source2.pl` and you want to create the executable `myprog`, your compilation command may look like

    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++'

## SICStus Prolog

The SICStus Prolog interface to the PPL library is available both as a statically linked module or as a dynamically linked one. Only SICStus Prolog version 3.9.0 or later is supported.

### The Statically Linked `ppl_sicstus` Executable

If an appropriate version of SICStus Prolog is installed on the machine on which you compiled the library, the command `make install` will install the executable `ppl_sicstus` in the directory `prefix/bin`. The `ppl_sicstus` 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 `prefix/lib/ppl/ppl_sicstus.pl`.

### Loading the SICStus Interface Dynamically

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

## SWI-Prolog

The SWI-Prolog interface of the library is available both as a statically linked module or as a dynamically linked one. Only SWI-Prolog version 5.0 or later is supported.

### The `ppl_pl` Executable

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

### Loading the SWI-Prolog Interface Dynamically

In order to dynamically load the library from SWI-Prolog you should simply load `prefix/lib/ppl/ppl_swiprolog.pl`. This will invoke `ppl_initialize/0` automatically but, at least for SWI-Prolog versions up to 5.0.7, it is the programmer's responsibility to call `ppl_finalize/0`. Alternatively, you can load the library directly with

    :- load_foreign_library('prefix/lib/ppl/libppl_swiprolog').

This will call `ppl_initialize/0` automatically. Analogously,

    :- unload_foreign_library('prefix/lib/ppl/libppl_swiprolog').

will, as part of the unload process, invoke `ppl_finalize/0`.

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

## XSB

The XSB Prolog interface to the PPL library is available as a dynamically linked module. Only XSB version 2.5 and following is supported.

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

## YAP

The YAP Prolog interface to the PPL library is available as a dynamically linked module. Only YAP version 4.4 or later is supported.

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