Verification of properties expressed in the two-variable fragment of first-order logic FO 2 has been investigated in a number of contexts. The satisfiability problem for FO 2 over arbitrary structures is known to be NEXPTIME-complete, with satisfiable formulas having exponential-sized models. Over words, where FO 2 is known to have the same expressiveness as unary temporal logic, satisfiability is again NEXPTIME-complete. Over finite labelled ordered trees FO 2 has the same expressiveness as navigational XPath, a popular query language for XML documents. Prior work on XPath and FO 2 gives a 2EXPTIME bound for satisfiability of FO 2 over trees. This work contains a comprehensive analysis of the complexity of FO 2 on trees, and on the size and depth of models. We show that different techniques are required depending on the vocabulary used, whether the trees are ranked or unranked, and the encoding of labels on trees. We also look at a natural restriction of FO 2 , its guarded version, GF 2 . Our results depend on an analysis of types in models of FO 2 formulas, including techniques for controlling the number of distinct subtrees, the depth, and the size of a witness to satisfiability for FO 2 sentences over finite trees.
Abstract. We study the control reachability problem in the Dolev-Yao model of cryptographic protocols when principals are represented by tail recursive processes with generated names. We propose a conservative approximation of the problem by reduction to a non-standard collapsed operational semantics and we introduce checkable syntactic conditions entailing the equivalence of the standard and the collapsed semantics. Then we introduce a conservative and decidable set-based analysis of the collapsed operational semantics and we characterize a situation where the analysis is exact.
We settle the complexity bounds of the model checking problem for the ambient calculus with public names against the ambient logic. We show that if either the calculus contains replication or the logic contains the guarantee operator, the problem is undecidable. In the case of the replication-free calculus and guarantee-free logic we prove that the problem is PSPACE-complete. For the complexity upper-bound, we devise a new representation of processes that remains of polynomial size during process execution; this allows us to keep the model checking procedure in polynomial space. Moreover, we prove PSPACE-hardness of the problem for several quite simple fragments of the calculus and the logic; this suggests that there are no interesting fragments with polynomial-time model checking algorithms.
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