We provide an effective procedure for deciding the existence of off-line guessing attacks on security protocols, for a bounded number of sessions. The procedure consists of a constraint solving algorithm for determining satisfiability and equivalence of a class of second-order E-unification problems, where the equational theory E is presented by a convergent subterm rewriting system. To the best of our knowledge, this is the first decidability result to use the generic definition of off-line guessing attacks due to Corin et al. based on static equivalence in the applied pi calculus.

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−→

Context unification is a natural variant of second order unification that represents a generalization of word unification at the same time. While second order unification is wellknown to be undecidable and word unification is decidable it is currently open if solvability of context equations is decidable. We show that solvability of systems of context equations with two context variables is decidable. The context variables may have an arbitrary number of occurrences, and the equations may contain an arbitrary number of individual variables as well. The result holds under the assumption that the first order background signature is finite.

Abstract. There is a close relationship between word unification and second-order unification. This similarity has been exploited for instance for proving decidability of monadic second-order unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higher-order unification procedures) when variables are restricted to occur at most twice. Hence a well-known open question was the decidability of second-order unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid E-unification to second-order unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid E-unification is decidable. This happens, for example, for one-variable problems where the variable occurs at most twice (because rigid E-unification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable second-order unification problems with variable occurrence restrictions. 1

this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of -terms in . Quite apart from the new light it sheds on fi-reduction, such an analysis turns out to have several other benefits

Abstract. Context unification (CU) is the open problem of solving context equations for trees. We distinguish a new decidable variant of CU– well-nested CU – and present a new unification algorithm that solves well-nested context equations in non-deterministic polynomial time. We show that minimal well-nested solutions of context equations can be composed from the material present in the equation (see Theorem 1). This property is wishful when modeling natural language ellipsis in CU. 1

Linear Second-Order Unification and Context Unification are closely related problems. However, their equivalence was never formally proved. Context unification is a restriction of linear second-order unification. Here we prove that linear second-order unification can be reduced to context unification with tree-regular constraints. Decidability of context unification is still an open question. We comment on the possibility that linear second-order unification is decidable, if context unification is, and how to get rid of the tree-regular constraints. This is done by reducing rank-bound tree-regular constraints to word-regular constraints.

. The second-order matching problem is the problem of determining, for a finite set {#t i , s i # | i # I} of pairs of a second-order term t i and a first-order closed term s i , called a matching expression, whether or not there exists a substitution # such that t i # = s i for each i # I . It is well-known that the second-order matching problem is NP-complete. In this paper, we introduce the following restrictions of a matching expression: k-ary, k-fv , predicate, ground , and function-free. Then, we show that the second-order matching problem is NP-complete for a unary predicate, a unary ground, a ternary function-free predicate, a binary function-free ground, and an 1-fv predicate matching expressions, while it is solvable in polynomial time for a binary function-free predicate, a unary function-free, a k-fv function-free (k # 0), and a ground predicate matching expressions. 1 Introduction The unification problem is the problem of determining whether or not any two ter...

Abstract. Context unification was originally defined by H. Comon in ICALP’92, as the problem of finding a unifier for a set of equations containing first-order variables and context variables. These context variables have arguments, and can be instantiated by contexts. In other words, they are second-order variables that are restricted to be instantiated by linear terms (a linear term is a λ-expression λx1 ···λxn.t where every xi occurs exactly once in t). In this paper, we prove that, if the so called rank-bound conjecture is true, then the context unification problem is decidable. This is done reducing context unification to solvability of traversal equations (a kind of word unification modulo certain permutations) and then, reducing traversal equations to word equations with regular constraints. 1

Abstract. Bounded Second-Order Unification is the problem of deciding, for a given second-order equation t? = u and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F, the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-Order Unification is undecidable. We prove that Bounded Second-Order Unification is NP-complete, provided that m is given in unary encoding, by proving that a size-minimal solution can be represented in polynomial space, and then applying a generalization of Plandowski’s polynomial algorithm that compares compacted terms in polynomial time. 1