Since its inception as a student project in 2001, initially just for the handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library has been continuously improved and extended by joining scrupulous research on the theoretical foundations of (possibly non-convex) numerical abstractions to a total adherence to the best available practices in software development. Even though it is still not fully mature and functionally complete, the Parma Polyhedra Library already offers a combination of functionality, reliability, usability and performance that is not matched by similar, freely available libraries. In this paper, we present the main features of the current version of the library, emphasizing those that distinguish it from other similar libraries and those that are important for applications in the field of analysis and verification of hardware and software systems. 1 We restrict ourselves to those libraries that are freely available and provide the services required by applications in static analysis and computer-aided verification. 2

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Abstract. Convex polyhedra constitute the most used abstract domain among those capturing numerical relational information. Since the domain of convex polyhedra admits infinite ascending chains, it has to be used in conjunction with appropriate mechanisms for enforcing and accelerating convergence of the fixpoint computation. Widening operators provide a simple and general characterization for such mechanisms. For the domain of convex polyhedra, the original widening operator proposed by Cousot and Halbwachs amply deserves the name of standard widening since most analysis and verification tools that employ convex polyhedra also employ that operator. Nonetheless, there is an unfulfilled demand for more precise widening operators. In this paper, after a formal introduction to the standard widening where we clarify some aspects that are often overlooked, we embark on the challenging task of improving on it. We present a framework for the systematic definition of new and precise widening operators for convex polyhedra. The framework is then instantiated so as to obtain a new widening operator that combines several heuristics and uses the standard widening as a last resort so that it is never less precise. A preliminary experimental evaluation has yielded promising results.

Convex polyhedra constitute the most used abstract domain among those capturing numerical relational information. Since the domain of convex polyhedra admits infinite ascending chains, it has to be used in conjunction with appropriate mechanisms for enforcing and accelerating convergence of the fixpoint computation. Widening operators provide a simple and general characterization for such mechanisms. For the domain of convex polyhedra, the original widening operator proposed by Cousot and Halbwachs amply deserves the name of standard widening since most analysis and verification tools that employ convex polyhedra also employ that operator. Nonetheless, there is an unfulfilled demand for more precise widening operators. In this paper, after a formal introduction to the standard widening where we clarify some aspects that are often overlooked, we embark on the challenging task of improving on it. We present a framework for the systematic definition of new and precise widening operators for convex polyhedra. The framework is then instantiated so as to obtain a new widening operator that combines several heuristics and uses the standard widening as a last resort so that it is never less precise. A preliminary experimental evaluation has yielded promising results. This work has been partly supported by MURST projects "Aggregate-and numberreasoning for computing: from decision algorithms to constraint programming with multisets, sets, and maps" and "Constraint Based Verification of Reactive Systems".

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Weakly-relational numeric constraints provide a compromise between complexity and expressivity that is adequate for several applications in the field of formal analysis and verification of software and hardware systems. We address the problems to be solved for the construction of full-fledged, efficient and provably correct abstract domains based on such constraints. We first propose to work with semantic abstract domains, whose elements are geometric shapes, instead of the (more concrete) syntactic abstract domains of constraint networks and matrices on which the previous proposals are based. This allows to solve, once and for all, the problem whereby closure by entailment, a crucial operation for the realization of such domains, seemed to impede the realization of proper widening operators. In our approach, the implementation of widenings relies on the availability of an effective reduction procedure for the considered constraint description: one for the domain of bounded difference shapes already exists in the literature; we provide algorithms for the significantly more complex cases of rational and integer octagonal shapes. We also improve upon the state-of-the-art by presenting, along with their proof of correctness, closure by entailment algorithms of reduced complexity for domains based on rational and integer octagonal constraints. The consequences of implementing weakly-relational numerical domains with floating point numbers are also discussed.

Abstract. We discuss the construction of proper widening operators on several weakly-relational numeric abstractions. Our proposal differs from previous ones in that we actually consider the semantic abstract domains, whose elements are geometric shapes, instead of the (more concrete) syntactic abstract domains of constraint networks and matrices. Since the closure by entailment operator preserves geometric shapes, but not their syntactic expressions, our widenings are immune from the divergence issues that could be faced by the previous approaches when interleaving the applications of widening and closure. The new widenings, which are variations of the standard widening for convex polyhedra defined by Cousot and Halbwachs, can be made as precise as the previous proposals working on the syntactic domains. The implementation of each new widening relies on the availability of an effective reduction procedure for the considered constraint description: we provide such an algorithm for the domain of octagonal shapes.

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