On the Complexity of the Tiden-Arnborg Algorithm for Unification modulo One-Sided Distributivity

We prove that the Tiden and Arnborg algorithm for equational unification modulo one-sided distributivity is not polynomial time bounded as previously thought. A set of counterexamples is developed that demonstrates that the algorithm goes through exponentially many steps.Comment: In Proceedings UNIF 2010, arXiv:1012.455

Unification modulo a partial theory of exponentiation

Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement protocols. In our earlier work, we gave many decidability as well as undecidability results for multiple equational theories, involving various properties of modular exponentiation. Here, we consider a partial subtheory focussing only on exponentiation and multiplication operators. Two main results are proved. The first result is positive, namely, that the unification problem for the above theory (in which no additional property is assumed of the multiplication operators) is decidable. The second result is negative: if we assume that the two multiplication operators belong to two different abelian groups, then the unification problem becomes undecidable.Comment: In Proceedings UNIF 2010, arXiv:1012.455

Terminating Non-Disjoint Combined Unification (Extended Abstract)

International audienc

On Unification Modulo One-Sided Distributivity: Algorithms, Variants and Asymmetry

An algorithm for unification modulo one-sided distributivity is an early result by Tid\'en and Arnborg. More recently this theory has been of interest in cryptographic protocol analysis due to the fact that many cryptographic operators satisfy this property. Unfortunately the algorithm presented in the paper, although correct, has recently been shown not to be polynomial time bounded as claimed. In addition, for some instances, there exist most general unifiers that are exponentially large with respect to the input size. In this paper we first present a new polynomial time algorithm that solves the decision problem for a non-trivial subcase, based on a typed theory, of unification modulo one-sided distributivity. Next we present a new polynomial algorithm that solves the decision problem for unification modulo one-sided distributivity. A construction, employing string compression, is used to achieve the polynomial bound. Lastly, we examine the one-sided distributivity problem in the new asymmetric unification paradigm. We give the first asymmetric unification algorithm for one-sided distributivity

Unital Anti-Unification: Type and Algorithms

Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete, and return tree grammars from which the set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions

Terminating Non-Disjoint Combined Unification

International audienceThe equational unification problem, where the underlying equational theory may be given as the union of component equational theories, appears often in practice in many fields such as automated reasoning, logic programming, declarative programming, and the formal analysis of security protocols. In this paper, we investigate the unification problem in the non-disjoint union of equational theories via the combination of hierarchical unification procedures. In this context, a unification algorithm known for a base theory is extended with some additional inference rules to take into account the rest of the theory. We present a simple form of hierarchical unification procedure. The approach is particularly well-suited for any theory where a unification procedure can be obtained in a syntactic way using transformation rules to process the axioms of the theory. Hierarchical unification procedures are exemplified with various theories used in protocol analysis. Next, we look at modularity methods for combining theories already using a hierarchical approach. In addition, we consider a new complexity measure that allows us to obtain terminating (combined) hierarchical unification procedures

Unification of Higher-order Patterns modulo Simple Syntactic Equational Theories

We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λ øverlinex.F(øverlinex) = λ øverlinex. F(øverlinex^π ) where øverlinex^π is a permutation of øverlinex When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator