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

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/

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

Constrained completion: Theory, implementation, and results

The Knuth-Bendix completion procedure produces complete sets of reductions but can not handle certain rewrite rules such as commutativity. In order to handle such theories, completion procedure were created to find complete sets of reductions modulo an equational theory. The major problem with this method is that it requires a specialized unification algorithm for the equational theory. Although this method works well when such an algorithm exists, these algorithms are not always available and thus alternative methods are needed to attack problems. A way of doing this is to use a completion procedure which finds complete sets of constrained reductions. This type of completion procedure neither requires specialized unification algorithms nor will it fail due to unorientable identities. We present a look at complete sets of reductions with constraints, developed by Gerald Peterson, and the implementation of such a completion procedure for use with HIPER - a fast completion system. The completion procedure code is given and shown correct along with the various support procedures which are needed by the constrained system. These support procedures include a procedure to find constraints using the lexicographic path ordering and a normal form procedure for constraints. The procedure has been implemented for use under the fast HIPER system, developed by Jim Christian, and thus is quick. We apply this new system, HIPER- extension, to attack a variety of word problems. Implementation alternatives are discussed, developed, and compared with each other as well as with the HIPER system. Finally, we look at the problem of finding a complete set of reductions for a ternary boolean algebra. Given are alternatives to attacking this problem and the already known solution along with its run in the HIPER-extension system --Abstract, page iii

The Role of Term Symmetry in E-Unification and E-Completion

A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix [070] or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied. The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings. The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply plugging in a unification module for the new theory

Dynamic Order-Sorted Term-Rewriting Systems

This thesis considers the problems of order-sorted equational logic and its operational interpretation, order-sorted term rewriting

Adventures in Associative-Commutative Unification

We have discovered an efficient algorithm for matching and unification in associative-commutative (AC) equational theories. In most cases of AC unification our method obviates the need for solving diophantine equations, and thus avoids one of the bottlenecks of other associative-commutative unification techniques. The algorithm efficiently utilizes powerful constraints to eliminate much of the search involved in generating valid substitutions. Moreover, it is able to generate solutions lazily, enabling its use in an SLD-resolution-based environment like Prolog. We have found the method to run much faster and use less space than other associative-commutative unification procedures on many commonly encountered AC problems. 1 Introduction Associative-commutative (AC) equational theories surface in a number of computer science applications, including term rewriting, automatic theorem proving, software verification, and database retrieval. As a simple example, consider trying to find a sub..