Mateu Villaret

Nominal Unification from a Higher-Order Perspective

Nominal logic is an extension of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to lambda-terms, in nominal terms, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of the lambda-calculus. Despite these differences, nominal unification can be seen from a higher-order perspective. From this view, we show that nominal unification can be quadratically reduced to a particular fragment of higher-order unification problems: higher-order pattern unification. We also prove that the translation preserves most generality of unifiers.

An Efficient Nominal Unification Algorithm

Nominal Unification is an extension of first-order unification where terms can contain binders and unification is performed modulo α equivalence. Here we prove that the existence of nominal unifiers can be decided in quadratic time. First, we linearly-reduce nominal unification problems to a sequence of freshness and equalities between atoms, modulo a permutation, using ideas as Paterson and Wegman for first-order unification. Second, we prove that solvability of these reduced problems may be checked in quadratic time. Finally, we point out how using ideas of Brown and Tarjan for unbalanced merging, we could solve these reduced problems more efficiently

Building Automated Theorem Provers for Infinitely-Valued Logics with Satisfiability Modulo Theory Solvers

There is a relatively large number of papers dealing with complexity and proof theory issues of infinitely-valued logics. Nevertheless, little attention has been paid so far to the development of efficient solvers for such logics. In this paper we show how the technology of Satisfiability Modulo Theories (SMT) can be used to build efficient automated theorem provers for relevant infinitely-valued logics, including Lukasiewicz, Gödel and Product logics. Moreover, we define a test suite for those logics, and report on an experimental investigation that evaluates the practical complexity of Lukasiewicz and Gödel logics, and provides empirical evidence of the good performance of SMT technology for automated theorem proving on infinitely-valued logics.

The Complexity of Monadic Second-Order Unification

Monadic second-order unification is second-order unification where all function constants occurring in the equations are unary. Here we prove that the problem of deciding whether a set of monadic equations has a unifier is NP-complete, where we use the technique of compressing solutions using singleton context-free grammars. We prove that monadic second-order matching is also NP-complete.

Well-nested context unification

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.

Stratified context unification is NP-complete

Context Unification is the problem to decide for a given set of second-order equations E where all second-order variables are unary, whether there exists a unifier, such that for every second-order variable X, the abstraction λx. r instantiated for X has exactly one occurrence of the bound variable x in r. Stratified Context Unification is a specialization where the nesting of second-order variables in E is restricted. It is already known that Stratified Context Unification is decidable, NP-hard, and in PSPACE, whereas the decidability and the complexity of Context Unification is unknown. We prove that Stratified Context Unification is in NP by proving that a size-minimal solution can be represented in a singleton tree grammar of polynomial size, and then applying a generalization of Plandowskis polynomial algorithm that compares compacted terms in polynomial time. This also demonstrates the high potential of singleton tree grammars for optimizing programs maintaining large terms. A corollary of our result is that solvability of rewrite constraints is NP-complete.

Currying Second-Order Unification Problems

The Curry form of a term, like f(a, b), allows us to write it, using just a single binary function symbol, as @(@(f, a), b). Using this technique we prove that the signature is not relevant in second-order unification, and conclude that one binary symbol is enough. By currying variable applications, like X(a), as @(X, a), we can transform second-order terms into first-order terms, but we have to add beta-reduction as a theory. This is roughly what it is done in explicit unification. We prove that by currying only constant applications we can reduce second-order unification to second-order unification with just one binary function symbol. Both problems are already known to be undecidable, but applying the same idea to context unification, for which decidability is still unknown, we reduce the problem to context unification with just one binary function symbol.We also discuss about the difficulties ofapplying the same ideas to third or higher order unification.

Anti-unification for Unranked Terms and Hedges

We study anti-unification for unranked terms and hedges that may contain term and hedge variables. The anti-unification problem of two hedges

A system for solving constraint satisfaction problems with SMT

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Solving constraint satisfaction problems with SAT modulo theories

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