. In this article we show that the three equations known as commutativity, associativity, and the Robbins equation are a basis for the variety of Boolean algebras. The problem was posed by Herbert Robbins in the 1930s. The proof was found automatically by EQP, a theorem-proving program for equational logic. We present the proof and the search strategies that enabled the program to find the proof. Key words: Associative-commutative unification, Boolean algebra, EQP, equational logic, paramodulation, Robbins algebra, Robbins problem. 1. Introduction This article contains the answer to the Robbins question of whether all Robbins algebras are Boolean. The answer is yes, all Robbins algebras are Boolean. The proof that answers the question was found by EQP, an automated theoremproving program for equational logic. In 1933, E. V. Huntington presented the following three equations as a basis for Boolean algebra [6, 5]: x + y = y + x, (commutativity) (x + y) + z = x + (y + z), (associativit...

We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easy-to-find proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ

We present a new approach for proving termination of rewrite systems by innermost termination. From the resulting abstract criterion we derive concrete conditions, based on critical peak properties, under which innermost termination implies termination (and confluence). Finally, we show how to apply the main results for providing new sufficient conditions for the modularity of termination.

We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...

The inductionless induction (also called proof by consistency) approach for proving equations by induction from an equational theory, requires a consistency check for equational theories. A new method using test sets for checking consistency of an equational theory is proposed. Using this method, a variation of the Knuth-Bendix completion procedure can be used for automatically proving equations by induction. The method does not suffer from limitations imposed by the methods proposed by Musser as well as by Huet and Hullot, and is as powerful as Jouannaud and Kounalis' method based on ground-reducibility. A theoretical comparison of the test set method with Jouannaud and Kounalis' method is given showing that the test set method is generally much better. Both the methods have been implemented in RRL, Rewrite Rule Laboratory, a theorem proving environment based on rewriting techniques and completion. In practice also, the test set method is faster than Jouannaud and Kounalis' ...

: In this report we give an overview of the development of our new Waldmeister prover for equational theories. We elaborate a systematic stepwise design process, starting with the inference system for unfailing Knuth--Bendix completion and ending up with an implementation which avoids the main diseases today's provers suffer from: overindulgence in time and space. Our design process is based on a logical three--level system model consisting of basic operations for inference step execution, aggregated inference machine, and overall control strategy. Careful analysis of the inference system for unfailing completion has revealed the crucial points responsible for time and space consumption. For the low level of our model, we introduce specialized data structures and algorithms speeding up the running system and cutting it down in size --- both by one order of magnitude compared with standard techniques. Flexible control of the mid--level aggregation inside the resulting prover is made po...

We present a new criterion for confluence of (possibly) non-terminating leftlinear term rewriting systems. The criterion is based on certain strong joinability properties of parallel critical pairs . We show how this criterion relates to other well-known results, consider some special cases and discuss some possible extensions. 1 Introduction and Overview Computation formalisms which are based on rewriting systems heavily rely on the fundamental properties of termination and confluence. For terminating and confluent systems normal forms exist and are unique, irrespective of the computation (rewriting) strategy. For non-terminating but confluent systems, normal forms need not exist, however, if a normal form exists, it is still unique. More generally, any (possibly infinite) diverging computations can be joined again. In some cases, non-termination is inherently unavoidable, in other cases it may be very difficult to verify this property. Hence the problem of proving confluence (with o...

We introduce the distributed theorem prover Peers-mcd for networks of workstations. Peers-mcd is the parallelization of the Argonne prover EQP, according to our Clause-Diffusion methodology for distributed deduction. The new features of Peers-mcd include the AGO (Ancestor-Graph Oriented) heuristic criteria for subdividing the search space among parallel processes. We report the performance of Peers-mcd on several experiments, including problems which require days of sequential computation. In these experiments Peersmcd achieves considerable, sometime super-linear, speed-up over EQP. We analyze these results by examining several statistics produced by the provers. The analysis shows that the AGO criteria partitions the search space effectively, enabling Peers-mcd to achieve super-linear speed-up by parallel search. 1 Introduction Distributed deduction is concerned with the problem of proving difficult theorems by distributing the work among networked computers. The motivation is to st...

. The inefficiency of AC-completion is mainly due to the doubly exponential number of AC-unifiers and thereby of critical pairs generated. We present AC-complete E-unification, a new technique whose goal is to reduce the number of AC-critical pairs inferred by performing unification in a extension E of AC (e.g. ACU, Abelian groups, Boolean rings, ...) in the process of normalized completion [21]. The idea is to represent complete sets of AC-unifiers by (smaller) sets of E-unifiers. Not only the theories E used for unification have exponentially fewer most general unifiers than AC, but one can remove from a complete set of E-unifiers those solutions which have no E-instance which is an AC-unifier. First, we define AC-complete E-unification and describe its fundamental properties. We show how AC-complete E-unification can be done in the elementary case, and how the known combination techniques for unification algorithms can be reused for our purposes. Finally, we give some evidence of t...