. 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 superposition-based theorem prover E. E is a sound and complete...

Abstract not found

This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

This note answers questions on whether three identities known to hold for orthomodular lattices are true also for ortholattices. One identity is shown to fail by MACE, a program that searches for counterexamples, an the other two are proved to hold by EQP, an equational theorem prover. The problems, from work in quantum logic, were given to us by Norman Megill. Keywords: Automatic theorem proving, ortholattice, quantum logic, theory of computation. 1 Introduction An ortholattice is an algebra with a binary operation (join) and a unary operation 0 (complement) satisfying the following (independent) set of identities. x y = (x 0 y 0 ) 0 (definition of meet) x y = y x (x y) z = x (y z) x (x y) = x x 00 = x x (y y 0 ) = y y 0 Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. From these identities one can...

This paper presents a taxonomy of parallel theorem-proving methods based on the control of search (e.g., master-slaves versus peer processes), the granularity of parallelism (e.g., fine, medium and coarse grain) and the nature of the method (e.g., ordering-based versus subgoalreduction) . We analyze how the di#erent approaches to parallelization a#ect the control of search: while fine and medium-grain methods, as well as master-slaves methods, generally do not modify the sequential search plan, parallel-search methods may combine sequential search plans (multi-search) or extend the search plan with the capability of subdividing the search space (distributed search). Precisely because the search plan is modified, the latter methods may produce radically di#erent searches than their sequential base, as exemplified by the first distributed proof of the Robbins theorem generated by the Modified Clause-Di#usion prover Peers-mcd. An overview of the state of the field and directions...

In saturation-based theorem provers, the reasoning process consists in constructing the closure of an axiom set under inferences.

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...

In the last years, the development of automated theorem provers has been advancing in a so to speak Olympic spirit, following the motto "faster, higher, stronger"; and the Waldmeister system has been a part of that endeavour. We will survey the concepts underlying this prover, which implements Knuth-Bendix completion in its unfailing variant. The system architecture is based on a strict separation of active and passive facts, and is realized via speci cally tailored representations for each of the central data structures: indexing for the active facts, set-based compression for the passive facts, successor sets for the conjectures. In order to cope with large search spaces, specialized redundancy criteria are employed, and the empirically gained control knowledge is integrated to ease the use of the system. We conclude with a discussion of strengths and weaknesses, and a view of future prospects.

Introduction We present recent developments within the equational theorem prover Waldmeister, an implementation of unfailing Knuth-Bendix completion [BDP89] with re nements towards ordered completion. The new developments rely on a novel organization of the underlying saturation-based proof procedure into a system architecture. As is known, the saturation process tends to quickly ll the memory available unless preventive measures are employed. To overcome this problem, our new \Waldmeister loop" features a highly compact representation of the search state, exploiting its inherent structure. The implementation just being available, the cost and the bene ts of the concept now can exactly be measured. Indeed are our expectations met concerning the drastic cut-down of memory usage with only moderate overhead of time. In addition it has turned out that the revealed structure of the search state paves the way to an easily implemented parallelization of the prover with modest communicati