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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.

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 $\cP$, depending on the value of Topology, with Integer dimensions. Handle is unified with the handle for $\cP$. Thus the query

   ?- ppl_new_Polyhedron_from_dimension(c, 3, X).
creates the C polyhedron defining the 3-dimensional vector space $\Rset^3$ 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 $\cP$, depending on the value of Topology, with Integer dimensions. Handle is unified with the handle for $\cP$. Thus the query

   ?- ppl_new_Polyhedron_empty_from_dimension(nnc, 3, X).
creates an empty NNC polyhedron embedded in $\Rset^3$ 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 $\cP_1$ (depending on the value of Topology_1), then this creates a copy $\cP_2$ of $\cP_1$ with topology C or NNC, depending on the value of Topology_2. Handle_2 is unified with the handle for $\cP_2$. 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 $\Rset^3$ 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 $\cP$ represented by Constraint_System with topology C or NNC, depending on the value of Topology. Handle is unified with the handle for $\cP$.

ppl_new_Polyhedron_from_generators(+Topology, +Generator_System, -Handle)

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

ppl_new_Polyhedron_from_bounding_box(+Topology, +Box, -Handle)

Creates a polyhedron $\cP$ represented by Box with topology C or NNC, depending on the value of Topology, and Handle is unified with the handle for $\cP$. 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 $\cP$ and $\cQ$ 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:

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 $P$ referenced by Handle is not empty and LinExpr is bounded from above in $P$.

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 $P$ referenced by Handle is not empty and LinExpr is bounded from above in $P$.

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 $P$ referenced by Handle is not empty and LinExpr is bounded from below in $P$.

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 $P$ referenced by Handle is not empty and LinExpr is bounded from below in $P$.

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 $\Rset^3$ satisfying the constraint $4x + y - 2z >= 5$.

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 $(-12.5, -0.625, 0)^\transpose$ in the vector space $\Rset^3$.

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 $\cP$ referenced by Handle_1 the time-elapse $ (\cP \nearrow \cQ)$ with the polyhedron $\cQ$ 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 $\cP$ 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 $\cP_1$ and $\cP_2$ referenced by Handle_1 and Handle_2, respectively are unaltered. The token Integer is 0 if a BHRZ03-widening with $\cP_2$ , improved by enforcing all the constraints of the form $\pm x \leq r$ and $\pm x < r$ that are satisfied by all the points of $\cP_1$ 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 $\cP$ 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 $\pm x \leq r$ and $\pm x < r$ that are satisfied by all the points of $\cP$ 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 $\cP$ 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 $\cP_1$ and $\cP_2$ referenced by Handle_1 and Handle_2, respectively are unaltered. The token Integer is 0 if a H79-widening with $\cP_2$ , improved by enforcing all the constraints of the form $\pm x \leq r$ and $\pm x < r$ that are satisfied by all the points of $\cP_1$ 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 $\cP$ 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 $\pm x \leq r$ and $\pm x < r$ that are satisfied by all the points of $\cP$ 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 $\cP_1$ referenced by Handle1 by first embedding $\cP_1$ in a new space enlarged by the space dimensions of the polyhedron $\cP_2$ referenced by Handle2, and then adds to its system of constraints a renamed-apart version of the constraints of $\cP_2$.

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.


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