Context unification is in PSPACE

Contexts are terms with one `hole', i.e. a place in which we can substitute an argument. In context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation. Context unification is a natural subvariant of second-order unification, which is undecidable, and a generalization of word equations, which are decidable, at the same time. It is the unique problem between those two whose decidability is uncertain (for already almost two decades). In this paper we show that the context unification is in PSPACE. The result holds under a (usual) assumption that the first-order signature is finite. This result is obtained by an extension of the recompression technique, recently developed by the author and used in particular to obtain a new PSPACE algorithm for satisfiability of word equations, to context unification. The recompression is based on performing simple compression rules (replacing pairs of neighbouring function symbols), which are (conceptually) applied on the solution of the context equation and modifying the equation in a way so that such compression steps can be in fact performed directly on the equation, without the knowledge of the actual solution.Comment: 27 pages, submitted, small notation changes and small improvements over the previous tex

Unification in the Description Logic EL Without Top Constructor

Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The inexpressive Description Logic EL is of particular interest in this context since, on the one hand, several large biomedical ontologies are defined using EL. On the other hand, unification in EL has recently been shown to be NP-complete, and thus of considerably lower complexity than unification in other DLs of similarly restricted expressive power. However, EL allows the use of the top concept (>), which represents the whole interpretation domain, whereas the large medical ontology SNOMEDCT makes no use of this feature. Surprisingly, removing the top concept from EL makes the unification problem considerably harder. More precisely, we will show that unification in EL without the top concept is PSpace-complete.This is an updated version of the original report that includes Appendix A on locality of unifiers

Undecidability of the unification and admissibility problems for modal and description logics

We show that the unification problem `is there a substitution instance of a given formula that is provable in a given logic?' is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logic with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ)

Recompression: a simple and powerful technique for word equations

In this paper we present an application of a simple technique of local recompression, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is based on local modification of variables (replacing X by aX or Xa) and iterative replacement of pairs of letters appearing in the equation by a `fresh' letter, which can be seen as a bottom-up compression of the solution of the given word equation, to be more specific, building an SLP (Straight-Line Programme) for the solution of the word equation. Using this technique we give a new, independent and self-contained proofs of most of the known results for word equations. To be more specific, the presented (nondeterministic) algorithm runs in O(n log n) space and in time polynomial in log N, where N is the size of the length-minimal solution of the word equation. The presented algorithm can be easily generalised to a generator of all solutions of the given word equation (without increasing the space usage). Furthermore, a further analysis of the algorithm yields a doubly exponential upper bound on the size of the length-minimal solution. The presented algorithm does not use exponential bound on the exponent of periodicity. Conversely, the analysis of the algorithm yields an independent proof of the exponential bound on exponent of periodicity. We believe that the presented algorithm, its idea and analysis are far simpler than all previously applied. Furthermore, thanks to it we can obtain a unified and simple approach to most of known results for word equations. As a small additional result we show that for O(1) variables (with arbitrary many appearances in the equation) word equations can be solved in linear space, i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were simplified, overall presentation improve

A Symbolic Intruder Model for Hash-Collision Attacks

In the recent years, several practical methods have been published to compute collisions on some commonly used hash functions. In this paper we present a method to take into account, at the symbolic level, that an intruder actively attacking a protocol execution may use these collision algorithms in reasonable time during the attack. Our decision procedure relies on the reduction of constraint solving for an intruder exploiting the collision properties of hush functions to constraint solving for an intruder operating on words

Comparing and evaluating extended Lambek calculi

Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expressive logical calculus was needed for many other cases. Therefore, many extensions and variants of the Lambek calculus have been proposed, since the eighties and up until the present day. As a result, there is now a large class of calculi, each with its own empirical successes and theoretical results, but also each with its own logical primitives. This raises the question: how do we compare and evaluate these different logical formalisms? To answer this question, I present two unifying frameworks for these extended Lambek calculi. Both are proof net calculi with graph contraction criteria. The first calculus is a very general system: you specify the structure of your sequents and it gives you the connectives and contractions which correspond to it. The calculus can be extended with structural rules, which translate directly into graph rewrite rules. The second calculus is first-order (multiplicative intuitionistic) linear logic, which turns out to have several other, independently proposed extensions of the Lambek calculus as fragments. I will illustrate the use of each calculus in building bridges between analyses proposed in different frameworks, in highlighting differences and in helping to identify problems.Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona, Spain. 201