General Information on the PPL

0.6.1

The Main Features

The Parma Polyhedra Library (PPL) is a modern C++ library for the manipulation of numerical information that can be represented by points in some $n$

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

The main features of the library are the following:

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

In addition to the basic domains, we also provide generic support for constructing new domains from pre-existing domains. The following domains and domain constructors are provided by the PPL:

the domain of topologically closed, rational convex polyhedra;
the domain of rational convex polyhedra that are not necessarily closed;
the powerset construction;
the powerset construction, instantiated for rational convex polyhedra.

In the following sections we describe these domains and domain constructors together with their representations and operations that are available to the PPL user.

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

Convex Polyhedra

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

Vectors, Matrices and Scalar Products

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

The scalar product of $\vect{v},\vect{w} \in \Rset^n$ , denoted $\langle \vect{v}, \vect{w} \rangle$ , is the real number

$\vect{v}^\transpose \vect{w} = \sum_{i=0}^{n-1} v_i w_i.$

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

Affine Hyperplanes and Half-spaces

For each vector $\vect{a} \in \Rset^n$ and scalar $b \in \Rset$ , where $\vect{a} \neq \vect{0}$ , and for each relation symbol $\mathord{\relsym} \in \{ =, \geq, > \}$ , the linear constraint $\langle \vect{a}, \vect{x} \rangle \relsym b$ defines:

an affine hyperplane if it is an equality constraint, i.e., if $\mathord{\relsym} \in \{ = \}$ ;
a topologically closed affine half-space if it is a non-strict inequality constraint, i.e., if $\mathord{\relsym} \in \{ \geq \}$ ;
a topologically open affine half-space if it is a strict inequality constraint, i.e., if $\mathord{\relsym} \in \{ > \}$ .

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

Convex Polyhedra

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

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

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

In theoretical terms, $\Pset_n$ is a lattice under set inclusion and $\CPset_n$ is a sub-lattice of $\Pset_n$ .

Bounded Polyhedra

An NNC polyhedron $\cP \in \Pset_n$ is bounded if there exists a $\lambda \in \nonnegRset$ such that

$\cP \sseq \bigl\{\, \vect{x} \in \Rset^n \bigm| - \lambda \leq x_j \leq \lambda \text{ for } j = 0, \ldots, n-1 \,\bigr\}.$

A bounded polyhedron is also called a polytope.

Representations of Convex Polyhedra

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

Constraints representation

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

By definition, each polyhedron $\cP \in \Pset_n$ is the set of solutions to a constraint system, i.e., a finite number of constraints. By using matrix notation, we have

$\cP \defeq \{\, \vect{x} \in \Rset^n \mid A_1 \vect{x} = \vect{b}_1, A_2 \vect{x} \geq \vect{b}_2, A_3 \vect{x} > \vect{b}_3 \,\},$

where, for all $i \in \{1, 2, 3\}$ , $A_i \in \Rset^{m_i} \times \Rset^n$ and $\vect{b}_i \in \Rset^{m_i}$ , and $m_1, m_2, m_3 \in \Nset$ are the number of equalities, the number of non-strict inequalities, and the number of strict inequalities, respectively.

Combinations and Hulls

Let $S = \{ \vect{x}_1, \ldots, \vect{x}_k \} \sseq \Rset^n$ be a finite set of vectors. For all scalars $\lambda_1, \ldots, \lambda_k \in \Rset$ , the vector $\vect{v} = \sum_{j=1}^k \lambda_j \vect{x}_j$ is said to be a linear combination of the vectors in $S$ . Such a combination is said to be

a positive (or conic) combination, if $\forall j \in \{ 1, \ldots, k \} \itc \lambda_j \in \nonnegRset$ ;
an affine combination, if $\sum_{j = 1}^k \lambda_j = 1$ ;
a convex combination, if it is both positive and affine.

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

Let $P, C \sseq \Rset^n$ , where $P \union C = S$ . We denote by $\NNChull(P, C)$ the set of all convex combinations of the vectors in $S$ such that $\lambda_j > 0$ for some $\vect{x}_j \in P$ (informally, we say that there exists a vector of $P$ that plays an active role in the convex combination). Note that $\NNChull(P, C) = \NNChull(P, P \union C)$ so that, if $C \sseq P$ ,

$\convexhull(P) = \NNChull(P, \emptyset) = \NNChull(P, P) = \NNChull(P, C).$

It can be observed that $\linearhull(S)$ is an affine space, $\conichull(S)$ is a topologically closed convex cone, $\convexhull(S)$ is a topologically closed polytope, and $\NNChull(P, C)$ is an NNC polytope.

Points, Closure Points, Rays and Lines

Let $\cP \in \Pset_n$ be an NNC polyhedron. Then

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

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

Generators Representation

Each NNC polyhedron $\cP \in \Pset_n$ can be represented by finite sets of lines $L$ , rays $R$ , points $P$ and closure points $C$ of $\cP$ . The 4-tuple $\cG = (L, R, P, C)$ is said to be a generator system for $\cP$ , in the sense that

$\cP = \linearhull(L) + \conichull(R) + \NNChull(P, C),$

where the symbol ' $+$ ' denotes the Minkowski's sum.

When $\cP \in \CPset_n$ is a closed polyhedron, then it can be represented by finite sets of lines $L$ , rays $R$ and points $P$ of $\cP$ . In this case, the 3-tuple $\cG = (L, R, P)$ is said to be a generator system for $\cP$ since we have

$\cP = \linearhull(L) + \conichull(R) + \convexhull(P).$

Thus, in this case, every closure point of $\cP$ is a point of $\cP$ .

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

Minimized Representations

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

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

Double Description

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

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

Topologies and Topological-compatibility

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

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

Unless it is otherwise stated, all the polyhedra, constraints and/or generators in any library operation must obey the following topological-compatibility rules:

polyhedra are topologically-compatible if and only if they have the same topology;
all constraints except for strict inequality constraints and all generators except for closure points are topologically-compatible with both C and NNC polyhedra;
strict inequality constraints and closure points are topologically-compatible with a polyhedron if and only if it is NNC.

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

Space Dimensions and Dimension-compatibility

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

Unless it is otherwise stated, all the polyhedra, constraints and/or generators in any library operation must obey the following space dimension-compatibility rules:

polyhedra are dimension-compatible if and only if they have the same space dimension;
the constraint $\langle \vect{a}, \vect{x} \rangle \relsym b$ where $\mathord{\relsym} \in \{ =, \geq, > \}$ and $\vect{a}, \vect{x} \in \Rset^m$ , is dimension-compatible with a polyhedron having space dimension if and only if $m \leq n$ ;
the generator $\vect{x} \in \Rset^m$ is dimension-compatible with a polyhedron having space dimension if and only if $m \leq n$ ;
a system of constraints (resp., generators) is dimension-compatible with a polyhedron if and only if all the constraints (resp., generators) in the system are dimension-compatible with the polyhedron.

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

Rational Polyhedra

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

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

Operations on Convex Polyhedra

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

Intersection and Convex Polyhedral Hull

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

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

Convex Polyhedral Difference

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

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

Concatenating Polyhedra

Viewing a polyhedron as a set of tuples (its points), it is sometimes useful to consider the set of tuples obtained by concatenating an ordered pair of polyhedra. Formally, the concatenation of the polyhedra $\cP \in \Pset_n$ and $\cQ \in \Pset_m$ (taken in this order) is the polyhedron $\cR \in \Pset_{n+m}$ such that

$\cR \defeq \Bigl\{\, (x_0, \ldots, x_{n-1}, y_0, \ldots, y_{m-1})^\transpose \in \Rset^{n+m} \Bigm| (x_0, \ldots, x_{n-1})^\transpose \in \cP, (y_0, \ldots, y_{m-1})^\transpose \in \cQ \,\Bigl\}.$

Another way of seeing it is as follows: first embed polyhedron $\cP$ into a vector space of dimension $n+m$ and then add a suitably renamed-apart version of the constraints defining $\cQ$ .

Adding New Dimensions to the Vector Space

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

The operator add_dimensions_and_embed embeds the polyhedron $\cP$ into the new vector space of dimension $i+n$ and returns the polyhedron $\cQ$ defined by all and only the constraints defining $\cP$ (the variables corresponding to the added dimensions are unconstrained). For instance, when starting from a polyhedron $\cP \sseq \Rset^2$ and adding a third dimension, the result will be the polyhedron

$\cQ = \bigl\{\, (x_0, x_1, x_2)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}.$

In contrast, the operator add_dimensions_and_project projects the polyhedron $\cP$ into the new vector space of dimension $i+n$ and returns the polyhedron $\cQ$ whose constraint system, besides the constraints defining $\cP$ , will include additional constraints on the added dimensions. Namely, the corresponding variables are all constrained to be equal to 0. For instance, when starting from a polyhedron $\cP \sseq \Rset^2$ and adding a third dimension, the result will be the polyhedron

$\cQ = \bigl\{\, (x_0, x_1, 0)^\transpose \in \Rset^3 \bigm| (x_0, x_1)^\transpose \in \cP \,\bigr\}.$

Removing Dimensions from the Vector Space

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

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

$\cQ = \bigl\{ (3, 2)^\transpose \bigr\} \sseq \Rset^2.$

Given a space dimension $m$ less than or equal to that of the polyhedron, the operator remove_higher_dimensions removes the dimensions having indices greater than or equal to $m$ . For instance, letting $\cP \in \Pset_4$ defined as before, by invoking this operator with $m = 2$ the resulting polyhedron will be

$\cQ = \bigl\{ (3, 1)^\transpose \bigr\} \sseq \Rset^2.$

Mapping the Dimensions of the Vector Space

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

If $m = 0$ , i.e., if the function $\rho$ is undefined everywhere, then the operator projects the argument polyhedron $\cP \in \Pset_n$ onto the zero-dimension space $\Rset^0$ ; otherwise the result is $\cQ \in \Pset_m$ given by

$\cQ \defeq \Bigl\{\, \bigl(v_{\rho^{-1}(0)}, \ldots, v_{\rho^{-1}(m-1)}\bigr)^\transpose \Bigm| (v_0, \ldots, v_{n-1})^\transpose \in \cP \,\Bigr\}.$

Expanding One Dimension of the Vector Space to Multiple Dimensions

The operator expand_dimension provided by the library adds $m$ new dimensions to a polyhedron $\cP \in \Pset_n$ , with $n > 0$ , so that dimensions $n$ , $n+1$ , $\ldots$ , $n+m-1$ of the result $\cQ$ are exact copies of the $i$ -th dimension of $\cP$ . More formally,

$\cQ \defeq \sset{ \vect{u} \in \Rset^{n+m} }{ \exists \vect{v}, \vect{w} \in \cP \st u_i = v_i \\ \qquad \mathord{} \land \forall j = n, n+1, \ldots, n+m-1 \itc u_j = w_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_k = v_k = w_k }.$

This operation has been proposed in [GDMDRS04].

Folding Multiple Dimensions of the Vector Space into One Dimension

The operator fold_dimensions provided by the library, given a polyhedron $\cP \in \Pset_n$ , with $n > 0$ , folds a set of dimensions $J = \{ j_0, \ldots, j_{m-1} \}$ , with $m < n$ and $j < n$ for each $j \in J$ , into dimension $i < n$ , where $i \notin J$ . The result is given by

$\cQ \defeq \biguplus_{d = 0}^m \cQ_d$

where

$\cQ_m \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_i \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k }$

and, for $ d = 0 $ , $\ldots$ , $ m-1 $ ,

$\cQ_d \defeq \sset{ \vect{u} \in \Rset^{n-m} }{ \exists \vect{v} \in \cP \st u_{i'} = v_{j_d} \\ \qquad \mathord{} \land \forall k = 0, \ldots, n-1 \itc k \neq i \implies u_{k'} = v_k },$

and, finally, for $ k = 0 $ , $\ldots$ , $ n-1 $ ,

$k' \defeq k - \card \{\, j \in J \mid k > j \,\},$

( $\card S$ denotes the cardinality of the finite set $S$ ).

This operation has been proposed in [GDMDRS04].

Affine Images and Preimages

For each function mapping $\fund{\phi}{\Rset^n}{\Rset^m}$ , we denote by $\phi(S) \sseq \Rset^m$ the image under $\phi$ of the set $S \sseq \Rset^n$ ; formally,

$\phi(S) \defeq \bigl\{\, \phi(\vect{v}) \in \Rset^m \bigm| \vect{v} \in S \,\bigr\}.$

Similarly, we denote by $\phi^{-1}(S') \sseq \Rset^n$ the preimage under $\phi$ of $S' \sseq \Rset^m$ , that is the largest set $S \sseq \Rset^n$ such that $\phi(S) \sseq S'$ ; formally,

$\phi^{-1}(S') \defeq \bigl\{\, \vect{v} \in \Rset^n \bigm| \phi(\vect{v}) \in S' \,\bigr\}.$

The function mapping $\fund{\phi}{\Rset^n}{\Rset^m}$ is an affine transformation if there exist a matrix $A \in \Rset^m \times \Rset^n$ and a vector $\vect{b} \in \Rset^m$ such that, for all $\vect{x} \in \Rset^n$ , we have $\phi(\vect{x}) = A\vect{x} + \vect{b}$ . If $n = m$ , then the function $\phi$ is said to be space-dimension preserving.

Both $\Pset_n$ and $\CPset_n$ are closed under the application of any space-dimension preserving affine image and preimage operators.

The library provides two operators, one computes an affine image and the other an affine preimage of a polyhedron $\cP \in \Pset_n$ for a given variable $x_k$ and linear expression $\mathrm{expr} = \sum_{i=0}^{n-1} a_i x_i + b$ . This variable and expression determine the affine transformation $\phi$ that is to be used by the operator. That is, $\phi$ is the transformation defined by the matrix and vector

$A = \begin{pmatrix} 1 & & 0 & 0 & \cdots & \cdots & 0 \\ & \ddots & & \vdots & & & \vdots \\ 0 & & 1 & 0 & \cdots & \cdots & 0 \\ a_0 & \cdots & a_{k-1} & a_k & a_{k+1} & \cdots & a_{n-1} \\ 0 & \cdots & \cdots & 0 & 1 & & 0 \\ \vdots & & & \vdots & & \ddots & \\ 0 & \cdots & \cdots & 0 & 0 & & 1 \end{pmatrix}, \qquad \vect{b} = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ b \\ 0 \\ \vdots \\ 0 \end{pmatrix}$

where the $a_i$ (resp., $b$ ) occurs in the $(k+1)$ st row in $A$ (resp., position in $\vect{b}$ ). Thus $\phi$ transforms any point $(x_0, \ldots, x_{n-1})^\transpose$ in the polyhedron $\cP$ to

$\Bigl(x_0, \ldots, \bigl(\textstyle{\sum_{i=0}^{n-1}} a_i x_i + b\bigr), \ldots, x_{n-1}\Bigr)^\transpose.$

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

The affine preimage operator computes the affine preimage of $\cP$ under $\phi$ . For instance, suppose now that we apply the affine preimage operator as given in the first example using variable $x_0$ and linear expression $x_0 + 2 x_1 + 4$ to the parallelogram $\cP_1$ ; then we get the original square $\cP$ back. If, on the other hand, we apply the affine preimage operator as given in the second example using variable $x_0$ and linear expression $x_1$ to $\cP_2$ , then the resulting polyhedron is a line that corresponds to the $x_1$ axes.

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

Generalized Affine Images

The library provides another operator which is a generalization of the affine image operator. Given a polyhedron $\cP \in \Pset_n$ , an affine expression $\mathrm{lhs} = \sum_{i=0}^{n-1} a'_i x_i + b'$ , a relation symbol $\mathord{\relsym} \in \{ <, \leq, =, \geq, > \}$ , and an affine expression $\mathrm{rhs} = \sum_{i=0}^{n-1} a_i x_i + b$ , the image of $\cP$ with respect to the transfer function $\mathrm{lhs} \relsym \mathrm{rhs}$ is defined as

$\sset{ (w_0, \ldots, w_{n-1})^\transpose \in \Rset^n }{ (v_0, \ldots, v_{n-1})^\transpose \in \cP, \\ \bigl( i \in \{0, \ldots, n-1\} \land a'_i = 0 \implies w_i = v_i \bigr), \\ \sum_{i=0}^{n-1} a'_i w_i + b' \relsym \sum_{i=0}^{n-1} a_i v_i + b }.$

Note that, when $\mathrm{lhs} = x_k$ and $\mathord{\relsym} \in \{ = \}$ , then the above operator is equivalent to the application of the standard affine image of $\cP$ with respect to the variable $x_k$ and the affine expression $\mathrm{rhs}$ (hence the name given to this operator).

Time-Elapse Operator

The time-elapse operator has been defined in [HPR97]. Actually, the time-elapse operator provided by the library is a slight generalization of that one, since it also works on NNC polyhedra. For any two NNC polyhedra $\cP, \cQ \in \Pset_n$ , the time-elapse between $\cP$ and $\cQ$ , denoted $\cP \nearrow \cQ$ , is the smallest NNC polyhedron containing the set

$\bigl\{\, \vect{p} + \lambda \vect{q} \in \Rset^n \bigm| \vect{p} \in \cP, \vect{q} \in \cQ, \lambda \in \nonnegRset \,\bigr\}.$

Note that, if $\cP,\cQ \in \CPset_n$ are closed polyhedra, the above set is also a closed polyhedron. In contrast, when $\cQ$ is not topologically closed, the above set might not be an NNC polyhedron.

Relation-with Operators

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

Suppose $\cP$ is an NNC polyhedron and $\cC$ an arbitrary constraint system representing $\cP$ . Suppose also that $c = \bigl( \langle \vect{a}, \vect{x} \rangle \relsym b \bigr)$ is a constraint with $\mathord{\relsym} \in \{ =, \geq, > \}$ and $\cQ$ the set of points that satisfy $c$ . The possible relations between $\cP$ and $c$ are as follows.

$\cP$ is disjoint from if $\cP \inters \cQ = \emptyset$ ; that is, adding to $\cC$ gives us the empty polyhedron.
$\cP$ strictly intersects if $\cP \inters \cQ \neq \emptyset$ and $\cP \inters \cQ \subset \cP$ ; that is, adding to $\cC$ gives us a non-empty polyhedron strictly smaller than $\cP$ .
$\cP$ is included in if $\cP \sseq \cQ$ ; that is, adding to $\cC$ leaves $\cP$ unchanged.
$\cP$ saturates if $\cP \sseq \cH$ , where $\cH$ is the hyperplane induced by constraint , i.e., the set of points satisfying the equality constraint $\langle \vect{a}, \vect{x} \rangle = b$ ; that is, adding the constraint $\langle \vect{a}, \vect{x} \rangle = b$ to $\cC$ leaves $\cP$ unchanged.

The polyhedron $\cP$ subsumes the generator $g$ if adding $g$ to any generator system representing $\cP$ does not change $\cP$ .

Intervals, boxes and bounding boxes

An interval in $\Rset$ is a pair of bounds, called lower and upper. Each bound can be either (1) closed and bounded, (2) open and bounded, or (3) open and unbounded. If the bound is bounded, then it has a value in $\Rset$ . An $n$ -dimensional box $\cB$ in $\Rset^n$ is a sequence of $n$ intervals in $\Rset$ .

The polyhedron $\cP$ represents a box $\cB$ in $\Rset^n$ if $\cP$ is described by a constraint system in $\Rset^n$ that consists of one constraint for each bounded bound (lower and upper) in an interval in $\cB$ : Letting $\vect{e}_i = (0, \ldots, 1, \ldots, 0)^\transpose$ be the vector in $\Rset^n$ with 1 in the $i$ 'th position and zeroes in every other position; if the lower bound of the $i$ 'th interval in $\cB$ is bounded, the corresponding constraint is defined as $\langle \vect{e}_i, \vect{x} \rangle \relsym b$ , where $b$ is the value of the bound and $\mathord{\relsym}$ is $\mathord{\geq}$ if it is a closed bound and $\mathord{>}$ if it is an open bound. Similarly, if the upper bound of the $i$ 'th interval in $\cB$ is bounded, the corresponding constraint is defined as $\langle\vect{e}_i,\vect{x}\rangle \relsym b$ , where $b$ is the value of the bound and $\mathord{\relsym}$ is $\mathord{\leq}$ if it is a closed bound and $\mathord{<}$ if it is an open bound.

If every bound in the intervals defining a box $\cB$ is either closed and bounded or open and unbounded, then $\cB$ represents a closed polyhedron.

The bounding box of an NNC polyhedron $\cP$ is the smallest $n$ -dimensional box containing $\cP$ .

The library provides operations for computing the bounding box of an NNC polyhedron and conversely, for obtaining the NNC polyhedron representing a given bounding box.

Widening Operators

The library provides two widening operators for the domain of NNC polyhedra. The first one, that we call H79-widening, mainly follows the specification provided in the PhD thesis of N. Halbwachs [Hal79], also described in [HPR97]. There are a few differences between the H79-widening and the widening described in the cited paper. In particular, the H79-widening of an NNC polyhedron $\cP \in \Pset_n$ using the NNC polyhedron $\cQ \in \Pset_n$ :

allows for equalities in $\cP$ and $\cQ$ (the original definition is restricted to inequalities);
requires as a precondition that $\cQ \sseq \cP$ .

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

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

Widening with Tokens

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

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

Extrapolation Operators

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

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

The bounded extrapolation operators further enhance each one of the limited extrapolation operators described above, by ensuring that their results cannot be worse than the smallest bounding box enclosing the two argument polyhedra.

The Powerset Construction

The PPL provides the finite powerset construction; this takes a pre-existing domain and upgrades it to one that can represent disjunctive information (by using a finite number of disjucts). The construction follows the approach described in [Bag98], also summarised in [BHZ04] where there is an account of generic widenings for the powerset domain (some of which are supported in the instantiation of this construction by the domain of convex polyhedra and described in Section The Polyhedra Powerset Domain).

The Powerset Domain

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

As the intended semantics of an element of the powerset of the base-level domain is that of disjunction, elements of the powerset are always reduced to semantically-equivalent non-redundant elements.

A set $\cS \in \wp(D)$ is called non-redundant with respect to ` $\mathord{\entails}$ ' if and only if $\false \notin \cS$ and $\forall d_1, d_2 \in \cS \itc d_1 \entails d_2 \implies d_1 = d_2$ . The set of finite non-redundant subsets of $D$ (with respect to ` $\mathord{\entails}$ ') is denoted by $\wpfn{D}{\entails}$ . The reduction function $\fund{\nonredmap}{\wpf(D)}{\wpfn{D}{\entails}}$ mapping a finite set into its non-redundant counterpart is defined, for each $\cS \in \wpf(D)$ , by

$\nonredmap(\cS) \defeq \cS \setdiff \{\, d \in \cS \mid d = \false \text{ or } \exists d' \in \cS \st d \sentails d' \,\}.$

The restriction to the finite subsets reflects the fact that here disjunctions are implemented by explicit collections of elements of the base-level abstract domain. As a consequence of this restriction, for any $\cS \in \wpf(D)$ such that $\cS \neq \{ \false \}$ , $\nonredmap(\cS)$ is the (finite) set of the maximal elements of $\cS$ .

The finite powerset domain over a domain $D$ is the set of all finite reduced sets of $D$ and denoted by $D_{\smallP}$ . The domain includes an approximation ordering ` $\mathord{\entailsP}$ ' defined so that $\cS_1 \entailsP \cS_2$ if and only if

$\forall d_1 \in \cS_1 \itc \exists d_2 \in \cS_2 \st d_1 \entails d_2.$

Therefore the top element is $\{\true\}$ and the bottom element is the emptyset.

Operations on the Powerset Construction

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

Meet and Upper Bound

Given the sets $\cS_1$ and $\cS_2 \in D_{\smallP}$ , the meet and upper bound operators provided by the library returns the set $\nonredmap\bigl(\{d_1 \meet d_2 \,|\, d_1 \in \cS_1, d_2 \in \cS_2\}\bigr)$ and reduced set union $\nonredmap(\cS_1 \union \cS_2)$ respectively.

Adding a Disjunct

Given the powerset element $\cS \in D_{\smallP}$ and the base-level element $d \in D$ , the add_disjunct operator provided by the library returns the powerset element $\nonredmap\bigl(\cS \union \{d\}\bigr)$ .

Collapsing a Powerset Element

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

The Polyhedra Powerset Domain

The Polyhedra powerset domain $(\Pset_n)_{\smallP}$ provided by the PPL is the finite powerset domain (defined in Section The Powerset Construction) over the domain of NNC polyhedra $\Pset_n$ .

In addition to the operations described for the generic powerset domain in Section Operations on the Powerset Construction, we provide some operations that are specific to this instantiation. Of these, most correspond to the application of the equivalent operation on each of the NNC polyhedron that are in the given set. Here we just describe those operations that are particular to the polyhedra powerset domain.

Geometric Comparisons

Given the sets $\cS_1, \cS_2 \in (\Pset_n)_{\smallP}$ , then we say that $\cS_1$ geometrically_covers $\cS_2$ if every point (in some element) in a polyhedron in $\cS_2$ is also a point in a polyhedron in $\cS_1$ . If $\cS_1$ geometrically_covers $\cS_2$ and $\cS_2$ geometrically_covers $\cS_1$ , then we say that they are geometrically_equal.

Pairwise Merge

Given the powerset $\cS \in (\Pset_n)_{\smallP}$ , then the pairwise_merge operator takes pairs of distinct elements in $\cS$ whose poly-hull is the same as their set-theoretical union and replaces them by their union. This replacement is done recursively so that, for each pair $\cP, \cQ$ of distinct polyhedra in the result set, we have $\cP \uplus \cQ \neq \cP \union \cQ$ .

Extrapolation Operators

The library implements a generalization of the extrapolation operator for powerset domains proposed in [BGP99]. The operator BGP99_extrapolation_assign is made parametric by allowing for the specification of a base-level extrapolation operator different from the H79 widening (e.g., the BHRZ03 widening can be used). Note that, in the general case, this operator cannot guarantee the convergence of the iteration sequence in a finite number of steps (for a counter-example, see [BHZ04]).

Certificate-Based Widenings

The PPL library provides support for the specification of proper widening operators on the powerset domain of convex polyhedra. In particular, this version of the library implements an instance of the certificate-based widening framework proposed in [BHZ03b].

A finite convergence certificate for an extrapolation operator is a formal way of ensuring that such an operator is indeed a widening on the considered domain. Given a widening operator on the base-level domain, together with the corresponding convergence certificate, the BHZ03 framework shows how it is possible to lift this widening so as to work on the finite powerset domain, while still ensuring convergence in a finite number of iterations.

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

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

Using the Library

A Note on the Implementation of the Operators

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

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

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

On Object-Orientation and Polymorphism: A Disclaimer

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

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

Most operators of the library depend on one or more parameters that are declared ``const'', meaning that they will not be changed by the application of the considered operator. Due to the adoption of lazy computation techniques, in many cases such a const-correctness guarantee only holds at the semantic level, whereas it does not necessarily hold at the implementation level. For a typical example, consider the extraction from a polyhedron of its constraint system representation. While this operation is not going to change the polyhedron, it might actually invoke the internal conversion algorithm and modify the generators representation of the polyhedron object, e.g., by reordering the generators and removing those that are detected as redundant. Thus, any previously computed reference to the generators of the polyhedron (be it a direct reference object or an indirect one, such as an iterator) will no longer be valid. For this reason, code fragments such as the following should be avoided, as they may result in undefined behavior:

// Find a reference to the first point of the non-empty polyhedron `ph'.
const GenSys& gs = ph.generators();
GenSys::const_iterator i = gs.begin();
for (GenSys::const_iterator gs_end = gs.end(); i != gs_end; ++i)
  if (i->is_point())
    break;
const Generator& p = *i;
// Get the constraints of `ph'.
const ConSys& cs = ph.constraints();
// Both the const iterator `i' and the reference `p'
// are no longer valid at this point.
cout << p.divisor() << endl;  // Undefinded behavior!
++i;                          // Undefinded behavior!

As a rule of thumb, if a polyhedron plays any role in a computation (even as a const parameter), then any previously computed reference to parts of the polyhedron may have been invalidated. Note that, in the example above, the computation of the constraint system could have been placed after the uses of the iterator i and the reference p. Anyway, if really needed, it is always possible to take a copy of, instead of a reference to, the parts of interest of the polyhedron; in the case above, one may have taken a copy of the generator system by replacing the second line of code with the following:

GenSys gs = ph.generators();

The same observations, modulo syntactic sugar, apply to the operators defined in the C interface of the library.

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V. Van Dongen H. Le Verge and D. K. Wilde. Loop nest synthesis using the polyhedral library. Publication interne 830, IRISA, Campus de Beaulieu, Rennes, France, 1994.

[HPR94]

N. Halbwachs, Y.-E. Proy, and P. Raymond. Verification of linear hybrid systems by means of convex approximations. In B. Le Charlier, editor, Static Analysis: Proceedings of the 1st International Symposium, volume 864 of Lecture Notes in Computer Science, pages 223-237, Namur, Belgium, 1994. Springer-Verlag, Berlin.

[HPR97]

N. Halbwachs, Y.-E. Proy, and P. Roumanoff. Verification of real-time systems using linear relation analysis. Formal Methods in System Design, 11(2):157-185, 1997.

[HPWT01]

T. A. Henzinger, J. Preussig, and H. Wong-Toi. Some lessons from the hytech experience. In Proceedings of the 40th Annual Conference on Decision and Control, pages 2887-2892. IEEE Computer Society Press, 2001.

[Jea02]

B. Jeannet. Convex Polyhedra Library, release 1.1.3c edition, March 2002. Documentation of the ``New Polka'' library available at http://www.irisa.fr/prive/Bertrand.Jeannet/newpolka.html.

[Kuh56]

H. W. Kuhn. Solvability and consistency for linear equations and inequalities. American Mathematical Monthly, 63:217-232, 1956.

[Le 92]

92 H. Le Verge. A note on Chernikova's algorithm. Publication interne 635, IRISA, Campus de Beaulieu, Rennes, France, 1992.

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V. Loechner. PolyLib: A library for manipulating parameterized polyhedra. Available at http://icps.u-strasbg.fr/~loechner/polylib/, March 1999. Declares itself to be a continuation of [Wil93].

[LW97]

V. Loechner and D. K. Wilde. Parameterized polyhedra and their vertices. International Journal of Parallel Programming, 25(6):525-549, 1997.

[Mas92]

F. Masdupuy. Array operations abstraction using semantic analysis of trapezoid congruences. In Proceedings of the 6th ACM International Conference on Supercomputing, pages 226-235, Washington, DC, USA, 1992. ACM Press.

[Mas93]

F. Masdupuy. Array Indices Relational Semantic Analysis Using Rational Cosets and Trapezoids. Thèse d'informatique, École Polytechnique, Palaiseau, France, December 1993.

[MRTT53]

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[NR00]

S. P. K. Nookala and T. Risset. A library for Z-polyhedral operations. Publication interne 1330, IRISA, Campus de Beaulieu, Rennes, France, 2000.

[NW88]

G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. Wiley Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, 1988.

[Sch99]

A. Schrijver. Theory of Linear and Integer Programming. Wiley Interscience Series in Discrete Mathematics and Optimization. John Wiley & Sons, 1999.

[Sri93]

D. Srivastava. Subsumption and indexing in constraint query languages with linear arithmetic constraints. Annals of Mathematics and Artificial Intelligence, 8(3-4):315-343, 1993.

[SW70]

J. Stoer and C. Witzgall. Convexity and Optimization in Finite Dimensions I. Springer-Verlag, Berlin, 1970.

[Wey35]

H. Weyl. Elementare theorie der konvexen polyeder. Commentarii Mathematici Helvetici, 7:290-306, 1935. English translation in [Wey50].

[Wey50]

H. Weyl. The elementary theory of convex polyhedra. In H. W. Kuhn, editor, Contributions to the Theory of Games - Volume I, number 24 in Annals of Mathematics Studies, pages 3-18. Princeton University Press, Princeton, New Jersey, 1950. Translated from [Wey35] by H. W. Kuhn.

[Wil93]

D. K. Wilde. A library for doing polyhedral operations. Master's thesis, Oregon State University, Corvallis, Oregon, December 1993. Also published as IRISA Publication interne 785, Rennes, France, 1993.

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