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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 &qu ..."
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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...
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...
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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in this document are those of the author(s) and should not be interpreted as representing the official policies, either
Experiences with specification and verification in LOTOS: a report on two case studies
 IN PROC. WIFT '95
, 1995
"... 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.
Problems in Rewriting Applied to Categorical Concepts By the Example of a Computational Comonad
 Proceedings of the Sixth International Conference on Rewriting Techniques and Applications
, 1995
"... . We present a canonical system for comonads which can be extended to the notion of a computational comonad [BG92] where the crucial point is to find an appropriate representation. These canonical systems are checked with the help of the Larch Prover [GG91] exploiting a method by G. Huet [Hue90a] to ..."
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. We present a canonical system for comonads which can be extended to the notion of a computational comonad [BG92] where the crucial point is to find an appropriate representation. These canonical systems are checked with the help of the Larch Prover [GG91] exploiting a method by G. Huet [Hue90a] to represent typing within an untyped rewriting system. The resulting decision procedures are implemented in the programming language Elf [Pfe89] since typing is directly supported by this language. Finally we outline an incomplete attempt to solve the problem which could be used as a benchmark for rewriting tools. 1 Introduction The starting point of this work was to provide methods for checking the commutativity of diagrams arising in category theory. Diagrams in this context are used as a visual description of equations between morphisms. To check the commutativity of a diagram amounts to check the equality of the morphisms involved. One way to support this task is to solve the uniform wor...
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...