Precise Widening Operators for Convex Polyhedra

Abstract

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.

Abstract acceleration in linear relation analysis

“…where ∇ is the standard widening, is also a widening (the operator proposed in [6] is slightly more complicated), and for any P 1 , P 2 ,…”

“…• Classic Linear Relation Analysis, without widening upto, without the new path algorithm, with two widenings: the standard widening ( [39]) and the one proposed in [6]. The first analysis is implemented in Aspic and the StInG tool ([56]) can deal with both of them.…”

Widening as Abstract Domain

“…We update t P to t H = t[p U 3 l] P with t H (p) = l and t H (q) = t(q) for q T = p. In abuse of notation we use [p U 3 : : :] to construct a new mapping and Y for the empty map. We enforce the invariant that all tests p P Pred are redundant in the child domain by applying lter : ¢ g 3 which is de ned as lter(t; c) = [p U 3 t(p) j p P dom(t) [ [dom(t 2 ) Again, applying lter removes thresholds that are not redundant in c 1 t g c 2 . Given the collected thresholds, the widening ht 1 ; c 1 i r l ht 2 ; c 2 i is now able to re ne the widened child state c 1 r l g c 2 by applying those predicates e 1 0;…”

“…It is widely acknowledged that computing a few iterations of a loop without widening can improve the precision of the computed xpoint [2]. For instance, the program in Fig.…”

“…One might argue that the modular design itself creates the need for propagation which is unnecessary when using a monolithic domain such as an o -the-shelf polyhedra package [2]. However, combining several simple domains allows for a more exible trade-o between e ciency and precision by adjusting the interaction between domains [16].…”