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FORWARD ANALYSIS FOR WSTS, PART I: COMPLETIONS
, 2009
"... Wellstructured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretic ..."
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Cited by 14 (8 self)
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Wellstructured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downwardclosed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed new light on the notion of adequate domains of limits.
Forward analysis for WSTS, part II: Complete WSTS
 In ICALP’09, volume 5556 of LNCS
, 2009
"... Abstract. We describe a simple, conceptual forward analysis procedure for ∞complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show ..."
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Cited by 8 (5 self)
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Abstract. We describe a simple, conceptual forward analysis procedure for ∞complete WSTS S. This computes the clover of a state s0, i.e., a finite description of the closure of the cover of s0. When S is the completion of a WSTS X, the clover in S is a finite description of the cover in X. We show that this applies exactly when X is an ω 2WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized KarpMiller procedure on extensions of Petri nets. We characterize the WSTS where our procedure terminates as those that are cloverflattable. Finally, we apply this to wellstructured counter systems. 1
Ideal Abstractions for WellStructured Transition Systems
"... Many infinite state systems can be seen as wellstructured transition systems (WSTS), i.e., systems equipped with a wellquasiordering on states that is also a simulation relation. WSTS are an attractive target for formal analysis because there exist generic algorithms that decide interesting veri ..."
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Cited by 1 (1 self)
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Many infinite state systems can be seen as wellstructured transition systems (WSTS), i.e., systems equipped with a wellquasiordering on states that is also a simulation relation. WSTS are an attractive target for formal analysis because there exist generic algorithms that decide interesting verification problems for this class. Among the most popular algorithms are accelerationbased forward analyses for computing the covering set. Termination of these algorithms can only be guaranteed for flattable WSTS. Yet, many WSTS of practical interest are not flattable and the question whether any given WSTS is flattable is itself undecidable. We therefore propose an analysis that computes the covering set and captures the essence of accelerationbased algorithms, but sacrifices precision for domain builds on the ideal completion of the wellquasiordered state space, and a widening operator that mimics acceleration and controls the loss of precision of the analysis. We present instances of our framework for various classes of WSTS. Our experience with a prototype implementation indicates that, despite the inherent precision loss, our analysis often computes the precise covering set of the analyzed system.
Noetherian spaces in verification
 In ICALP’10
, 2010
"... Abstract. Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinitestate verification problems, and show how this stimulated the search for infinite procedures à la KarpMiller. 1 ..."
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Cited by 1 (1 self)
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Abstract. Noetherian spaces are a topological concept that generalizes well quasiorderings. We explore applications to infinitestate verification problems, and show how this stimulated the search for infinite procedures à la KarpMiller. 1
A Constructive Proof of the Topological Kruskal Theorem
"... Abstract. We give a constructive proof of Kruskal’s Tree Theorem— precisely, of a topological extension of it. The proof is in the style of a constructive proof of Higman’s Lemma due to Murthy and Russell (1990), and illuminates the role of regular expressions there. In the process, we discover an e ..."
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Cited by 1 (0 self)
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Abstract. We give a constructive proof of Kruskal’s Tree Theorem— precisely, of a topological extension of it. The proof is in the style of a constructive proof of Higman’s Lemma due to Murthy and Russell (1990), and illuminates the role of regular expressions there. In the process, we discover an extension of Dershowitz ’ recursive path ordering to a form of cyclic terms which we call µterms. This all came from recent research on Noetherian spaces, and serves as a teaser for their theory. 1
Forward analysis for Petri nets with name creation ⋆
"... Abstract. Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining νAPNs and proved that they are strictly well structured (WSTS), so that coverability and ..."
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Abstract. Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining νAPNs and proved that they are strictly well structured (WSTS), so that coverability and boundedness are decidable. Here we use the framework recently developed by Finkel and GoubaultLarrecq for forward analysis for WSTS, in the case of νAPNs, to compute the cover, that gives a good over approximation of the set of reachable markings. We prove that the least complete domain containing the set of markings is effectively representable. Moreover, we prove that in the completion we can compute least upper bounds of simple loops. Therefore, a forward KarpMiller procedure that computes the cover is applicable. However, we prove that in general the cover is not computable, so that the procedure is nonterminating in general. As a corollary, we obtain the analogous result for Transfer Data nets and Data Nets. Finally, we show that a slight modification of the forward analysis yields decidability of a weak form of boundedness called widthboundedness. 1
Accelerations for the coverability set of Petri nets with names
, 2001
"... Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining νPNs and proved that they are strictly well structured (WSTS), so that coverability and boundednes ..."
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Pure names are identifiers with no relation between them, except equality and inequality. In previous works we have extended P/T nets with the capability of creating and managing pure names, obtaining νPNs and proved that they are strictly well structured (WSTS), so that coverability and boundedness are decidable. Here we use the framework recently developed by Finkel and GoubaultLarrecq for forward analysis for WSTS, in the case of νPNs, to compute the cover, that gives a good over approximation of the set of reachable markings. We prove that the least complete domain containing the set of markings is effectively representable. Moreover, we prove that in the completion we can compute least upper bounds of simple loops. Therefore, a forward KarpMiller procedure that computes the cover is applicable. However, we prove that in general the cover is not computable, so that the procedure is nonterminating in general. As a corollary, we obtain the analogous result for Transfer Data nets and Data Nets. Finally, we show that a slight modification of the forward analysis yields decidability of a weak form of boundedness called widthboundedness, and identify a subclass of νPN that we call dwbounded νPN, for which the cover is computable.
Approximating Markov Processes By Averaging
"... Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable function ..."
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Normally, one thinks of probabilistic transition systems as taking an initial probability distribution over the state space into a new probability distribution representing the system after a transition. We, however, take a dual view of Markov processes as transformers of bounded measurable functions. This is very much in the same spirit as a “predicatetransformer ” view, which is dual to the statetransformer view of transition systems. We redevelop the theory of labelled Markov processes from this view point, in particular we explore approximation theory. We obtain three main results: (i) It is possible to define bisimulation on general measure spaces and show that it is an equivalence relation. The logical characterization of bisimulation can be done straightforwardly and generally. (ii) A new and flexible approach to approximation based on averaging can be given. This vastly generalizes and streamlines the idea of using conditional expectations to compute approximations. (iii) We show that there is a minimal process bisimulationequivalent to a given process, and this minimal process is obtained as the limit of the finite approximants.
www.stacsconf.org FORWARD ANALYSIS FOR WSTS, PART I: COMPLETIONS
, 2009
"... ABSTRACT. Wellstructured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the ..."
Abstract
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ABSTRACT. Wellstructured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downwardclosed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed new light on the notion of adequate domains of limits. 1.