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Unnecessary inferences in associative-commutative completion procedures

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Abstract

Theoretical results for identifying unnecessary inferences are discussed in the context of the use of a completion-procedure-based approach toward automated reasoning. The notion of a general superposition is introduced and it is proved that in a completion procedure, once a general superposition is considered, all its instances are unnecessary inferences and, thus, do not have to be considered. It is also shown that this result can be combined with another criterion, called the prime superposition criterion, proposed by Kapur, Musser, and Narendran, thus implying that prime and general superpositions are sufficient. These results should be applicable to other approaches toward automated reasoning, too. These criteria can be effectively implemented, and their implementation has resulted in automatically proving instances of Jacobson's theorem (also known as the ring commutativity problems) usingRRL (Rewrite Rule Laboratory), a theorem prover based on rewriting techniques and completion.

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Authors and Affiliations

  1. Department of Computer Science, The University of Iowa, 52242, Iowa City, IA, USA

    Hantao Zhang

  2. Institute of Programming and Logics, Department of Computer Science, State University of New York at Albany, 12222, Albany, NY, USA

    Deepak Kapur

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  1. Hantao Zhang
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A preliminary version of this paper appeared in a paper entitled “Consider only general superpositions in completion procedures” in theProceedings of the Third International Conference on Rewriting Techniques and Applications, Chapel Hill, NC, April, 1989, Lecture Notes in Computer Science, Vol. 355, Springer-Verlag, Berlin, pp. 513–527. Part of the work of Hantao Zhang was done at the Rensselaer Polytechnic Institute, New York, and he was partially supported by National Science Foundation Grant No. CCR-8408461; also affiliated with Institute of Programming and Logics at SUNY, Albany, NY, and RPI. Deepak Kapur was partially supported by National Science Foundation Grantr Nos. CCR-8408461 and CCR-8906678.

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Zhang, H., Kapur, D. Unnecessary inferences in associative-commutative completion procedures. Math. Systems Theory 23, 175–206 (1990). https://doi.org/10.1007/BF02090774

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