Optimality in Goal-Dependent Analysis of Sharing

We face the problems of correctness, optimality and precision for the static analysis of logic programs, using the theory of abstract interpretation. We propose a framework with a denotational, goal-dependent semantics equipped with two unification operators for forward unification (calling a procedure) and backward unification (returning from a procedure). The latter is implemented through a matching operation. Our proposal clarifies and unifies many different frameworks and ideas on static analysis of logic programming in a single, formal setting. On the abstract side, we focus on the domain Sharing by Jacobs and Langen and provide the best correct approximation of all the primitive semantic operators, namely, projection, renaming, forward and backward unification. We show that the abstract unification operators are strictly more precise than those in the literature defined over the same abstract domain. In some cases, our operators are more precise than those developed for more complex domains involving linearity and freeness. To appear in Theory and Practice of Logic Programming (TPLP

Nominal C-Unification

Nominal unification is an extension of first-order unification that takes into account the \alpha-equivalence relation generated by binding operators, following the nominal approach. We propose a sound and complete procedure for nominal unification with commutative operators, or nominal C-unification for short, which has been formalised in Coq. The procedure transforms nominal C-unification problems into simpler (finite families) of fixpoint problems, whose solutions can be generated by algebraic techniques on combinatorics of permutations.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854

Unification in the Description Logic EL

The Description Logic EL has recently drawn considerable attention since, on the one hand, important inference problems such as the subsumption problem are polynomial. On the other hand, EL is used to define large biomedical ontologies. Unification in Description Logics has been proposed as a novel inference service that can, for example, be used to detect redundancies in ontologies. The main result of this paper is that unification in EL is decidable. More precisely, EL-unification is NP-complete, and thus has the same complexity as EL-matching. We also show that, w.r.t. the unification type, EL is less well-behaved: it is of type zero, which in particular implies that there are unification problems that have no finite complete set of unifiers.Comment: 31page

Unification and Logarithmic Space

We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof theory and more specifically linear logic and Geometry of Interaction. We show how unification can be used to build a model of computation by means of specific subalgebras associated to finite permutations groups. We then prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. We also show that the construction can naturally represent pointer machines, an intuitive way of understanding logarithmic space computing

Unification and Matching on Compressed Terms

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called Singleton Tree Grammars (STG), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present paper is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and Singleton Tree Grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of Singleton Tree Grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.Comment: This paper is posted at the Computing Research Repository (CoRR) as part of the process of submission to the journal ACM Transactions on Computational Logic (TOCL)

Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534

Preface Volume 66, Issue 5

AbstractThis part of the volume contains the papers accepted for presentation at the workshop on Unification in Non-Classical Logics (UNCL), co-located with ICALP 2002, which took place on July 12, 2002 in M\'alaga, Spain.The workshop was concerned with one of the most promising areas of research on non-classical logics and its applications. Unification in non-classical logics, with various approaches to handling generalised terms, has drawn more and more attention in recent years. So far, most popular lines of research include fuzzy unification of (conventional) databases and the use of fuzzy concepts in information retrieval.This workshop was conceived as a forum for the exchange of ideas relevant for the concept of unification in non-classical logics, including, but not limited to, the topics of: •Unification in multiple-valued and fuzzy logic programming.•Unification based on similarities and fuzzy equivalence relations.•Categorical unification.•Practical use of non-classical unification, e.g. in expert systems and information retrieval.The program committee selected six papers after a reviewing process in which each submitted paper received at least two reviews. Considerable effort was devoted for the evaluation of the submissions and to providing the authors with helpful feedback. The criteria for selection were originality, quality, and relevance to the topic of the workshop.Alsinet et al reviewed and compared two models which extend first order possibilistic logic in order to enable fuzzy unification. The extension considers mainly fuzzy constants, and in form of restrictions on existential quantifiers.Banerjee and Bujosa presented a non-classical interpretion of classical unification in terms of geometrical constructions over a suitable R-module M. The main result is that unification of two terms can be seen as the intersection of their corresponding affine varieties on M. This paves the way of using methods from computer algebra in the field of unification.In Eklund et al, substitutions and unifiers appear as constructs in Kleisli categories related to particular composed powerset term monads. It is shown that an often used similarity-based approach to fuzzy unification is compatible with the categorical approach, and can be adequately extended.Kutsia presented a unification procedure for a theory with individual and sequence variables, free fixed and flexible arity function symbols and patterns. These theories have been used in different contexts such as databases, rewriting, programming languages, or theorem proving.Medina et al introduced a formal model for similarity-based fuzzy unification in multi-adjoint logic programs. On this computational model, a similarity-based unification approach which provides a semantic framework for logic programming with different notions of similarity was constructed.Virtanen introduced unification in similarity-based logic programming. One of the crucial points is the definition of similarity degrees between sets, giving rise to [lambda]-interpretations. The selection of so called most significant terms again is one of the cornerstones of the paper.We would like to thank all those who submitted papers for consideration, the authors of accepted papers for their interesting discussions during the workshop, the additional referees for their careful work, and Inma Fortes from the local organising committee for her assistance