An Improved Tight Closure Algorithm for Integer Octagonal Constraints

Integer octagonal constraints (a.k.a. ``Unit Two Variables Per Inequality'' or ``UTVPI integer constraints'') constitute an interesting class of constraints for the representation and solution of integer problems in the fields of constraint programming and formal analysis and verification of software and hardware systems, since they couple algorithms having polynomial complexity with a relatively good expressive power. The main algorithms required for the manipulation of such constraints are the satisfiability check and the computation of the inferential closure of a set of constraints. The latter is called `tight' closure to mark the difference with the (incomplete) closure algorithm that does not exploit the integrality of the variables. In this paper we present and fully justify an O(n^3) algorithm to compute the tight closure of a set of UTVPI integer constraints.Comment: 15 pages, 2 figure

Deciding Conditional Termination

We address the problem of conditional termination, which is that of defining the set of initial configurations from which a given program always terminates. First we define the dual set, of initial configurations from which a non-terminating execution exists, as the greatest fixpoint of the function that maps a set of states into its pre-image with respect to the transition relation. This definition allows to compute the weakest non-termination precondition if at least one of the following holds: (i) the transition relation is deterministic, (ii) the descending Kleene sequence overapproximating the greatest fixpoint converges in finitely many steps, or (iii) the transition relation is well founded. We show that this is the case for two classes of relations, namely octagonal and finite monoid affine relations. Moreover, since the closed forms of these relations can be defined in Presburger arithmetic, we obtain the decidability of the termination problem for such loops.Comment: 61 pages, 6 figures, 2 table

Exact Join Detection for Convex Polyhedra and Other Numerical Abstractions

Deciding whether the union of two convex polyhedra is itself a convex polyhedron is a basic problem in polyhedral computations; having important applications in the field of constrained control and in the synthesis, analysis, verification and optimization of hardware and software systems. In such application fields though, general convex polyhedra are just one among many, so-called, numerical abstractions, which range from restricted families of (not necessarily closed) convex polyhedra to non-convex geometrical objects. We thus tackle the problem from an abstract point of view: for a wide range of numerical abstractions that can be modeled as bounded join-semilattices --that is, partial orders where any finite set of elements has a least upper bound--, we show necessary and sufficient conditions for the equivalence between the lattice-theoretic join and the set-theoretic union. For the case of closed convex polyhedra --which, as far as we know, is the only one already studied in the literature-- we improve upon the state-of-the-art by providing a new algorithm with a better worst-case complexity. The results and algorithms presented for the other numerical abstractions are new to this paper. All the algorithms have been implemented, experimentally validated, and made available in the Parma Polyhedra Library.Comment: 36 pages, 4 figure

Sparsity Preserving Algorithms for Octagons

Known algorithms for manipulating octagons do not preserve their sparsity, leading typically to quadratic or cubic time and space complexities even if no relation among variables is known when they are all bounded. In this paper, we present new algorithms, which use and return octagons represented as weakly closed difference bound matrices, preserve the sparsity of their input and have better performance in the case their inputs are sparse. We prove that these algorithms are as precise as the known ones

Incrementally Closing Octagons

The octagon abstract domain is a widely used numeric abstract domain expressing relational information between variables whilst being both computationally efficient and simple to implement. Each element of the domain is a system of constraints where each constraint takes the restricted form ±xi±xj≤c. A key family of operations for the octagon domain are closure algorithms, which check satisfiability and provide a normal form for octagonal constraint systems. We present new quadratic incremental algorithms for closure, strong closure and integer closure and proofs of their correctness. We highlight the benefits and measure the performance of these new algorithms

Vérification relationnelle pour des programmes avec des données entières

Les travaux présentés dans cette thèse sont lies aux problèmes de vérification de l'atteignabilité et de la terminaison de programmes qui manipulent des données entières non-bornées. On décrit une nouvelle méthode de vérification basée sur une technique d'accélération de boucle, qui calcule, de manière exacte, la clôture transitive d'une relation arithmétique. D'abord, on introduit un algorithme d'accélération de boucle qui peut calculer, en quelques secondes, des clôtures transitives pour des relations de l'ordre d'une centaine de variables. Ensuite, on présente une méthode d'analyse de l'atteignabilité, qui manipule des relations entre les variables entières d'un programme, et applique l'accélération pour le calcul des relations entrée-sortie des procédures, de façon modulaire. Une approche alternative pour l'analyse de l'atteignabilité, présentée également dans cette thèse, intègre l'accélération avec l'abstraction par prédicats, afin de traiter le problème de divergence de cette dernière. Ces deux méthodes ont été évaluées de manière pratique, sur un nombre important d'exemples, qui étaient, jusqu'a présent, hors de la portée des outils d'analyse existants. Dernièrement, on a étudié le problème de la terminaison pour certaines classes de boucles de programme, et on a montré la décidabilité pour les relations étudiées. Pour ces classes de relations arithmétiques, on présente un algorithme qui s'exécute en temps au plus polynomial, et qui calcule l'ensemble d'états qui peuvent générer une exécution infinie. Ensuite on a intégré cet algorithme dans une méthode d'analyse de la terminaison pour des programmes qui manipulent des données entières.This work presents novel methods for verification of reachability and termination properties of programs that manipulate unbounded integer data. Most of these methods are based on acceleration techniques which compute transitive closures of program loops. We first present an algorithm that accelerates several classes of integer relations and show that the new method performs up to four orders of magnitude better than the previous ones. On the theoretical side, our framework provides a common solution to the acceleration problem by proving that the considered classes of relations are periodic. Subsequently, we introduce a semi-algorithmic reachability analysis technique that tracks relations between variables of integer programs and applies the proposed acceleration algorithm to compute summaries of procedures in a modular way. Next, we present an alternative approach to reachability analysis that integrates predicate abstraction with our acceleration techniques to increase the likelihood of convergence of the algorithm. We evaluate these algorithms and show that they can handle a number of complex integer programs where previous approaches failed. Finally, we study the termination problem for several classes of program loops and show that it is decidable. Moreover, for some of these classes, we design a polynomial time algorithm that computes the exact set of program configurations from which non-terminating runs exist. We further integrate this algorithm into a semi-algorithmic method that analyzes termination of integer programs, and show that the resulting technique can verify termination properties of several non-trivial integer programs.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

Trichotomy for Integer Linear Systems Based on Their Sign Patterns

In this paper, we consider solving the integer linear systems, i.e., given a matrix A in R^{m*n}, a vector b in R^m, and a positive integer d, to compute an integer vector x in D^n such that Ax <= b, where m and n denote positive integers, R denotes the set of reals, and D={0,1,..., d-1}. The problem is one of the most fundamental NP-hard problems in computer science. For the problem, we propose a complexity index h which is based only on the sign pattern of A. For a real r, let ILS_=(r) denote the family of the problem instances I with h(I)=r. We then show the following trichotomy: - ILS_=(r) is linearly solvable, if r < 1, - ILS_=(r) is weakly NP-hard and pseudo-polynomially solvable, if r = 1, and - ILS_=(r) is strongly NP-hard, if r > 1. This, for example, includes the existing results that quadratic systems and Horn systems can be solved in pseudo-polynomial time

Closing the Performance Gap between Doubles and Rationals for Octagons

Octagons have enduring appeal because their domain opera- tions are simple, readily mapping to for-loops which apply max, min and sum to the entries of a Difference Bound Matrix (DBM). In the quest for efficiency, arithmetic is often realised with double-precision floating- point, albeit at the cost of the certainty provided by arbitrary-precision rationals. In this paper we show how Compact DBMs (CoDBMs), which have recently been proposed as a memory refinement for DBMs, enable arithmetic calculation to be short-circuited in various domain operations. We also show how comparisons can be avoided by changing the tables which underpin CoDBMs. From the perspective of implementation, the optimisations are attractive because they too are conceptually simple, following the ethos of Octagons. Yet they can halve the running time on rationals, putting CoDBMs on rationals on a par with DBMs on doubles