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. 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...

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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 explore some of the problems of verification by trying to prove that some sort of relationship holds between a given specification and implementation. We are particularly interested in the decisions taken in the process of establishing and formalising the verification requirements and of automating the proof. Despite the apparent simplicity of the original problem, the verification is non-trivial. The example chosen is an abstraction of a real communications problem. We use the formal description technique LOTOS [8] for specification and implementation, and equational reasoning, automated by the RRL term rewriting system [9], for the proof. 1 Introduction The last few years has seen an increase in the use of formal methods in the design and analysis of computer systems. This has many benefits; one of which is being able to verify that certain properties hold of a system (or not, as the case may be). However, although formal methods are popular for specification, formal verification...

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We address the problem of the automatic verification of reactive systems. For such algorithms, parallelism, non-determinism and distribution, lead to frequent design flaws and make debugging difficult. Proving programs with respect to their specification may solve both these problems. In this framework, we describe the implementation of an algorithm design assistant based upon the UNITY formalism. A theorem prover and a Presburger formulas calculator are used to perform the underlying proofs. We illustrate the main difficulties encountered with representative examples. Key words: Program verification, reactive programs, UNITY formalism, parallelism, distribution, theorem proving. I Introduction Concurrency and distribution generate two further difficulties with respect to sequential programming. Concurrency leads to a drastic increase in program states and distribution results in a knowledge loss of both any global state and time. Therefore, program debugging becomes especia...

Recently the use of formal methods in describing and analysing the behaviour of (computer) systems has become more common. This has resulted in the proliferation of a wide variety of different specification formalisms, together with analytical techniques and methodologies for specification development. The particular specification formalism adopted for this study is LOTOS, an ISO standard formal description technique. Although there are many works dealing with how to write LOTOS specifications and how to develop a LOTOS specification from the initial abstract requirements specification to concrete implementation, relatively few works are concerned with the problems of expressing and proving the correctness of LOTOS specifications, i.e. verification. The main objective of this thesis is to address this shortfall by investigating the meaning of verification as it relates to concurrent systems in general, and in particular to those systems described using LOTOS. Further goals are to autom...

We examine an abstraction of a real communications problem in order to identify some of the problems involved in satisfying the verification requirements of such a system. The example is described using LOTOS[ISO88], and the RRL term rewriting system[KZ87] is used to carry out the proofs. 1 Introduction A common problem in the verification of real systems is: given a specification and an implementation, prove that some sort of relationship holds between the two. Often the problem is further complicated by the implementation having been written with little or no reference to the specification, rather than having been formally derived from it. The main aim of our work on this case study is to discover, and hopefully solve, some of the problems encountered in this process. The automation of the proofs required was a further aim. The example presented in the case study is an abstraction of a real communications problem involving three communicating processes at OSI Network level. The exam...

D. Fortin C. Kirchner P. Strogova May 13, 1994 Abstract We describe an approach for finding a shortest path between two nodes in a regular network using rewriting techniques. The minimal length is guaranteed by a minimal presentation for a finite group corresponding to a network. A rewriting method is described for giving a presentation for an arbitrary group. Our technique may be generalized to a simultaneous routing in regular networks. 1 Introduction A routing problem consists of finding a shortest path between two points in a network. This problem has many applications. We can cite the correspondence problem in a big town (how to choose a journey with a public transports to have a minimum of correspondences), the delivery problem (how to minimize the delivery delays), etc. We are rather interested by applications concerning computer architectures. A routing algorithm is the heart of all methods of analyzing or designing interconnection networks. We employ the standard group-theor...