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Solution of the Robbins Problem
 Journal of Automated Reasoning
, 1997
"... . 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 theoremproving program for equationa ..."
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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 theoremproving program for equational logic. We present the proof and the search strategies that enabled the program to find the proof. Key words: Associativecommutative 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...
OTTER 3.3 Reference Manual
"... by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any a ..."
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by the United States Government and operated by The University of Chicago under the provisions of a contract with the Department of Energy. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor The University of Chicago, nor any of their employees or officers, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privatelyowned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of document authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, Argonne National Laboratory, or The University of Chicago. ii
33 Basic Test Problems: A Practical Evaluation of Some Paramodulation Strategies
, 1996
"... 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 "ou ..."
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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, easytofind 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
Automating (Specification = Implementation) using Equational Reasoning and LOTOS
 TAPSOFT '93: Theory and Practice of Software Development, LNCS 668
, 1995
"... 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 automati ..."
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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 nontrivial. 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...
The Role of Automated Reasoning in Integrated System Verification Environments
, 1992
"... in this document are those of the author(s) and should not be interpreted as representing the official policies, either ..."
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Cited by 3 (2 self)
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in this document are those of the author(s) and should not be interpreted as representing the official policies, either
A UNITYbased Algorithm Design Assistant
, 1995
"... We address the problem of the automatic verification of reactive systems. For such algorithms, parallelism, nondeterminism 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 fr ..."
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We address the problem of the automatic verification of reactive systems. For such algorithms, parallelism, nondeterminism 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...
A Case Study for the ERIL Project
, 1992
"... 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 Introduc ..."
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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...
Routing in Regular Networks Using Rewriting
, 1994
"... 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 ..."
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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 grouptheor...
Experiences with Specification and Verification in LOTOS: A Report on Two Case Studies
"... We consider the problems of verifying properties of LOTOS specifications with specific reference to two case studies, one of which was proposed by an industrial collaborator. The case studies present quite different verification requirements and we study a range of verification and validation techni ..."
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We consider the problems of verifying properties of LOTOS specifications with specific reference to two case studies, one of which was proposed by an industrial collaborator. The case studies present quite different verification requirements and we study a range of verification and validation techniques, based on various behavioural congruences and preorders, which may be applied, also using some mechanised tool support. We consider the implications of the (formal) proofs which succeed or fail with respect to our desired properties, and draw some conclusions about the verification process. 1 Introduction Over the last few years we have been studying some of the problems of verifying properties of formal specifications written in LOTOS 1 , the ISO standardised language ([ISO:8807]) for concurrent, distributed, and nondeterministic systems. Some of the issues tackled include: ffl which kinds of verification are needed, particularly for real case studies, ffl how verification can be...