Abstract
We show in this paper that the application of the resolution principle to a set of clauses can be regarded as the construction of a term rewriting system confluent on valid formulas. This result allows the extension of standard properties and methods of equational theories (such as Birkhoff’s theorem and Knuth and Bendix completion algorithm) to quantifier-free first order predicate calculus.
These results are extended to first order predicate calculus in an equational theory, as studied by Plotkin [15], Slagle [17] and Lankford [12].
This paper is a continuation of the work of Hsiang [5], who has already shown that rewrite methods can be used in first order predicate calculus. The main difference is that Hsiang applies rewrite methods only as a refutational proof technique, trying to generate the equation TRUE=FALSE. We generalize these methods to satisfiable theories; in particular, we show that the concept of confluent rewriting system, which is the main tool for studying equational theories, can be extended to any quantifier-free first order theory. Furthermore, we show that rewrite methods can be used even if formulas are kept in clausal form.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BIRKHOFF G.: On the structure of abstract algebras. Proc.Cambridge Phil.Soc. 31, pp 433–454 (1935).
BUCKEN H.: Reduction systems and small cancellation theory. Proc. Fourth Workshop on Automated Deduction, 53–59 (1979)
CHANG C.L. and LEE R.C.: Symbolic logic and mechanical theorem proving. Academic Press, New-York (1973)
FAGES F.: Formes canoniques dans les algebres booleennes, et application a la demonstration automatlque en logique du premier ordre. These de 3me cycle, Universite Pierre et Marie Curie, Juin 1983.
HSIANG J. and DERSHOWITZ N.: Rewrite methods for clausal and non-clausal theorem proving. ICALP 83, Spain.(1983).
HUET G.: A complete proof of correctness of the KNUTH-BENDIX completion algorithm. INRIA, Rapport de recherche No 25, Juillet 1980.
HUET G., OPPEN D.C., Equations and rewrite rules: a survey, Technical Report CSL-11, SRI International, Jan. 1980.
HULLOT J.M., Canonical form and unification. Proc. of the Fifth Conference on Automated Deduction, Les Arcs. (July 1980).
JOUANNAUD J.P., KIRCHNER C. and KIRCHNER H.: Incremental unification in equational theories. Proc. of the 21th Allerton Conference (1982).
JOUANNAUD J.P.: Confluent and coherent sets of reduction with equations. Application to proofs in data types. Proc. 8th Colloquium on Trees in Algebra and Programming (1983).
KNUTH D. and BENDIX P., Simple word problems in universal algebra Computational problems in abstract algebra, Ed. Leech J., Pergamon Press, 1970, 263–297.
LANKFORD D.S.: Canonical inference, Report ATP-32, Department of Mathematics and Computer Science, University of Texas at Austin, Dec. 1975.
LEE R.C.: A completeness theorem and a computer program for finding theorems derivable for given axioms. Ph.D diss. in eng., U. of California, Berkeley, Calif., 1967.
PETERSON G. and STICKEL M.: Complete sets of reductions for some equationnal theories. JACM, Vol. 28, No2, Avril 1981, pp 233–264.
PLOTKIN G.: Building-in Equational Theories. Machine Intelligence, pp 73–90. (1972)
ROBINSON J.A: A machine-oriented logic based on the resolution principle. JACM, Vol.12, No1, Janvier 1965, pp 23–41
SLAGLE J.R: Automatic theorem proving for theories with simplifiers, commutativity and associativity. JACM 21, pp. 622–642. (1974)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1984 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Paul, E. (1984). A New Interpretation of the Resolution Principle. In: Shostak, R.E. (eds) 7th International Conference on Automated Deduction. CADE 1984. Lecture Notes in Computer Science, vol 170. Springer, New York, NY. https://doi.org/10.1007/978-0-387-34768-4_20
Download citation
DOI: https://doi.org/10.1007/978-0-387-34768-4_20
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-96022-7
Online ISBN: 978-0-387-34768-4
eBook Packages: Springer Book Archive