Opening the AC-Unification Race

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems

Service-Oriented Architecture Modeling: Bridging the Gap between Structure and Behavior

International audienceModel-driven development of large-scale software systems is highly likely to produce models that describe the systems from many diverse perspectives using a variety of modeling languages. Checking and maintaining consistency of information captured in such multi-modeling environments is known to be challenging. In this paper we describe an approach to systematically synchronize multi-models. The approach specifically addresses the problem of synchronizing business processes and domain models in a Service-oriented Architecture development environment. In the approach, the human effort required to synchronize independently developed models is supplemented with significant automated support. This process is used to identify concept divergences, that is, a concept in one model which cannot be matched with concepts in the other model. We automate the propagation of divergence resolution decisions across the conflicting models. We illustrate the approach using models developed for a Car Crash Crisis Management System (CCCMS), a case study problem used to assess Aspect-oriented Modeling approaches

Expectations for Associative-Commutative Unification Speedups in a Multicomputer Environment

An essential element of automated deduction systems is unification algorithms which identify general substitutions and when applied to two expressions, make them identical. However, functions which are associative and commutative, such as the usual addition and multiplication functions, often arise in term rewriting systems, program verification, the theory of abstract data types and logic programming. The introduction to the associative and commutative equality axioms together with standard unification brings with it problems of termination and unreasonably large search spaces. One way around these problems is to remove the troublesome axioms from the system and to employ a unification algorithm which unifies modulo the axioms of associativity and commutativity. Unlike standard unification, the associative-commutative (AC) unification of two expressions can lead to the formation of many most general unifiers. A report is presented on a hybrid AC unification algorithm which has been implemented to run in parallel on an Intel iPSC/

Software Development Support for Shared Sensing Infrastructures: A Generative and Dynamic Approach

International audienceSensors networks are the backbone of large sensing infras-tructures such as Smart Cities or Smart Buildings. Classical approaches suffer from several limitations hampering developers' work (e.g., lack of sensor sharing, lack of dynamicity in data collection policies, need to dig inside big data sets, absence of reuse between implementation platforms). This paper presents a tooled approach that tackles these issues. It couples (i) an abstract model of developers' requirements in a given infrastructure to (ii) timed automata and code generation techniques, to support the efficient deployment of reusable data collection policies on different infrastructures. The approach has been validated on several real-world scenarios and is currently experimented on an academic campus

Unification in the union of disjoint equational theories : combining decision procedures

Most of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms which compute finite complete sets of unifiers. Thus the developed combination methods usually cannot be used to combine decision procedures, i.e., algorithms which just decide solvability of unification problems without computing unifiers. In this paper we describe a combination algorithm for decision procedures which works for arbitrary equational theories, provided that solvability of so-called unification problems with constant restrictions--a slight generalization of unification problems with constants--is decidable for these theories. As a consequence of this new method, we can for example show that general A-unifiability, i.e., solvability of A-unification problems with free function symbols, is decidable. Here A stands for the equational theory of one associative function symbol. Our method can also be used to combine algorithms which compute finite complete sets of unifiers. Manfred Schmidt-Schauß\u27 combination result, the until now most general result in this direction, can be obtained as a consequence of this fact. We also get the new result that unification in the union of disjoint equational theories is finitary, if general unification--i.e., unification of terms with additional free function symbols--is finitary in the single theories

Labelled natural deduction for substructural logics

In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units - formulas labelled according to a labelling algebra. In the system described here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is defined which associates a labelled natural deduction style "structural derivation" with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are defined to solve the constraints generated within a derivation, and their termination conditions discussed. A natural deduction derivation is correct with respect to a given substructural logic, if, under the condition that the algorithmic procedures terminate, the associated set of constraints is satisfied with respect to the underlying labelling algebra. This is shown by proving that the natural deduction system is sound and complete with respect to the LKE tableaux system

Unification in Commutative Theories, Hilbert's Basis Theorem, and Gröbner Bases

Unification in a commutative theory E may be reduced to solving linear equations in the corresponding semiring S(E) (Nutt (1988)). The unification type of E can thus be characterized by algebraic properties of S(E). The theory of abelian groups with n commuting homomorphisms corresponds to the semiring Z[X1,...,Xn]. Thus Hilbert’s Basis Theorem can be used to show that this theory is unitary. But this argument does not yield a unification algorithm. Linear equations in Z[X1,..,Xn] can be solved with the help of Gröbner Base methods, which thus provide the desired algorithm. The theory of abelian monoids with a homomorphism is of type zero (Baader (1988)). This can also be proved by using the fact that the corresponding semiring, namely N[X], is not noetherian. An other example of a semiring (even ring), which is not noetherian, is the ring Z, where X1, ..., Xn ( n > 1 ) are non-commuting indeterminates. This semiring corresponds to the theory of abelian groups with n non-commuting homomorphisms. Surprisingly, by construction of a Gröbner Base algorithm for right ideals in Z, it can be shown that this theory is unitary unifying

A Parallel Implementation of Stickel\u27s AC Unification Algorithm in a Message-Passing Environment

Unification algorithms are an essential component of automated reasoning and term rewriting systems. Unification finds a set of substitutions or unifiers that, when applied to variables in two or more terms, make those terms identical or equivalent. Most systems use Robinson\u27s unification algorithm or some variant of it. However, terms containing functions exhibiting properties such as associativity and commutativity may be made equivalent without appearing identical. Systems employing Robinson\u27s unification algorithm must use some mechanism separate from the unification algorithm to reason with such functions. Often this is done by incorporating the properties into a rule base and generating equivalent terms which can be unified by Robinson\u27s algorithm. However, rewriting the terms in this manner can generate large numbers of useless terms in the problem space of the system. If the properties of the functions are incorporated into the unification algorithm itself, there is no need to rewrite the terms such that they appear identical. Stickel developed an algorithm to unify two terms containing associative and commutative functions. The unifiers (there may be more than one) are found by creating a homogeneous linear Diophantine equation with integer coefficients from the terms being unified. The unifiers can be constructed from solutions to this equation. The unifiers generated from one solution of the Diophantine equation are independent of any other solution to the equation. Therefore, once the Diophantine equation has been solved, the unifiers can be calculated from the solutions in parallel. We have implemented Stickel\u27s AC unification algorithm to run in parallel on a local area network of Sun 4/110 workstations in an effort to improve the speed of AC unification