Set Unification

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas--e.g., deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature--and properly revisited and analyzed--and partly novel proposals. In particular, we present a new goal-driven algorithm for general ACI1 unification and a new simpler algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. 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

A correct, precise and efficient integration of set-sharing, freeness and linearity for the analysis of finite and rational tree languages

It is well known that freeness and linearity information positively interact with aliasing information, allowing both the precision and the efficiency of the sharing analysis of logic programs to be improved. In this paper, we present a novel combination of set-sharing with freeness and linearity information, which is characterized by an improved abstract unification operator. We provide a new abstraction function and prove the correctness of the analysis for both the finite tree and the rational tree cases. Moreover, we show that the same notion of redundant information as identified in Bagnara et al. (2000) and Zaffanella et al. (2002) also applies to this abstract domain combination: this allows for the implementation of an abstract unification operator running in polynomial time and achieving the same precision on all the considered observable properties

An Abstract Machine for Unification Grammars

This work describes the design and implementation of an abstract machine, Amalia, for the linguistic formalism ALE, which is based on typed feature structures. This formalism is one of the most widely accepted in computational linguistics and has been used for designing grammars in various linguistic theories, most notably HPSG. Amalia is composed of data structures and a set of instructions, augmented by a compiler from the grammatical formalism to the abstract instructions, and a (portable) interpreter of the abstract instructions. The effect of each instruction is defined using a low-level language that can be executed on ordinary hardware. The advantages of the abstract machine approach are twofold. From a theoretical point of view, the abstract machine gives a well-defined operational semantics to the grammatical formalism. This ensures that grammars specified using our system are endowed with well defined meaning. It enables, for example, to formally verify the correctness of a compiler for HPSG, given an independent definition. From a practical point of view, Amalia is the first system that employs a direct compilation scheme for unification grammars that are based on typed feature structures. The use of amalia results in a much improved performance over existing systems. In order to test the machine on a realistic application, we have developed a small-scale, HPSG-based grammar for a fragment of the Hebrew language, using Amalia as the development platform. This is the first application of HPSG to a Semitic language.Comment: Doctoral Thesis, 96 pages, many postscript figures, uses pstricks, pst-node, psfig, fullname and a macros fil

Verification and Planning Based on Coinductive Logic Programming

Coinduction is a powerful technique for reasoning about unfounded sets, unbounded structures, infinite automata, and interactive computations [6]. Where induction corresponds to least fixed point's semantics, coinduction corresponds to greatest fixed point semantics. Recently coinduction has been incorporated into logic programming and an elegant operational semantics developed for it [11, 12]. This operational semantics is the greatest fix point counterpart of SLD resolution (SLD resolution imparts operational semantics to least fix point based computations) and is termed co- SLD resolution. In co-SLD resolution, a predicate goal p( t) succeeds if it unifies with one of its ancestor calls. In addition, rational infinite terms are allowed as arguments of predicates. Infinite terms are represented as solutions to unification equations and the occurs check is omitted during the unification process. Coinductive Logic Programming (Co-LP) and Co-SLD resolution can be used to elegantly perform model checking and planning. A combined SLD and Co-SLD resolution based LP system forms the common basis for planning, scheduling, verification, model checking, and constraint solving [9, 4]. This is achieved by amalgamating SLD resolution, co-SLD resolution, and constraint logic programming [13] in a single logic programming system. Given that parallelism in logic programs can be implicitly exploited [8], complex, compute-intensive applications (planning, scheduling, model checking, etc.) can be executed in parallel on multi-core machines. Parallel execution can result in speed-ups as well as in larger instances of the problems being solved. In the remainder we elaborate on (i) how planning can be elegantly and efficiently performed under real-time constraints, (ii) how real-time systems can be elegantly and efficiently model- checked, as well as (iii) how hybrid systems can be verified in a combined system with both co-SLD and SLD resolution. Implementations of co-SLD resolution as well as preliminary implementations of the planning and verification applications have been developed [4]. Co-LP and Model Checking: The vast majority of properties that are to be verified can be classified into safety properties and liveness properties. It is well known within model checking that safety properties can be verified by reachability analysis, i.e, if a counter-example to the property exists, it can be finitely determined by enumerating all the reachable states of the Kripke structure

Nominal Unification from a Higher-Order Perspective

Nominal Logic is a version of first-order logic with equality, name-binding, renaming via name-swapping and freshness of names. Contrarily to higher-order logic, bindable names, called atoms, and instantiable variables are considered as distinct entities. Moreover, atoms are capturable by instantiations, breaking a fundamental principle of 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 reduced to a particular fragment of higher-order unification problems: Higher-Order Pattern Unification. This reduction proves that nominal unification can be decided in quadratic deterministic time, using the linear algorithm for Higher-Order Pattern Unification. We also prove that the translation preserves most generality of unifiers

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

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

A Note on Parameterised Knowledge Operations in Temporal Logic

We consider modeling the conception of knowledge in terms of temporal logic. The study of knowledge logical operations is originated around 1962 by representation of knowledge and belief using modalities. Nowadays, it is very good established area. However, we would like to look to it from a bit another point of view, our paper models knowledge in terms of linear temporal logic with {\em past}. We consider various versions of logical knowledge operations which may be defined in this framework. Technically, semantics, language and temporal knowledge logics based on our approach are constructed. Deciding algorithms are suggested, unification in terms of this approach is commented. This paper does not offer strong new technical outputs, instead we suggest new approach to conception of knowledge (in terms of time).Comment: 10 page