Abstract. We give a new practical algorithm to compute, in finite time, a fixpoint (and often the least fixpoint) of a system of equations in the abstract numerical domains of zones and templates used for static analysis of programs by abstract interpretation. This paper extends previous work on the non-relational domain of intervals to relational domains. The algorithm is based on policy iteration techniques-rather than Kleene iterations as used classically in static analysis-and generates from the system of equations a finite set of simpler systems that we call policies. This set of policies satisfies a selection property which ensures that the minimal fixpoint of the original system of equations is the minimum of the fixpoints of the policies. Computing a fixpoint of a policy is done by linear programming. It is shown, through experiments made on a prototype analyzer, compared in particular to analyzers such as LPInv or the Octagon Analyzer, to be in general more precise and faster than the usual Kleene iteration combined with widening and narrowing techniques.

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Abstract. We propose a new approach for the symbolic exploration of timed automata that solves a particular aspect of the combinatory explosion occurring in the widely used clock zone automata, the splitting of symbolic states depending on the order of transition occurrences, even if these transitions concern unrelated components in a parallel system. Our goal is to preserve independence (commutation of transitions) from the original timed automaton to the symbolic level, thus fully avoiding state splitting, yet avoiding problems of previous similar approaches with "maximal bounds abstraction". We achieve this goal by (1) lifting the theory of Mazurkiewicz traces to timed words and symbolic state exploration, (2) examining symbolic path exploration from a formal language point of view, and (3) by splitting the concerns of (abstraction free) successor computation and zone comparison by a new abstraction related to maximal bounds. The theory results in data structures and algorithms that we have experimentally validated, finding good reductions.