On solving the equality problem in theories defined by Horn clauses
We propose in this paper a slight modification of the Knuth and Bendix algorithm for solving the equality problem in non-equational theories defined by a set of Horn clauses. We prove that the completeness property of the algorithm is then preserved, provided that a weak axiomatization of boolean calculus and equality has been given to the algorithm. In particular, we need only the reflexivity axiom for equality.
This algorithm can also be interpreted as the extension to Horn clauses of the resolution/narrowing algorithm proposed by Lankford [LA], which applies only when the equality predicate does not occur positively in non-unit clauses. We give some examples of theorems proved by this method, which show its efficiency in comparison with other paramodulation-based methods.
Another application of the Knuth and Bendix algorithm is the proof by induction (Musser [MU], Huet-Hullot [HH], and others). We show that our version of the completion algorithm can be used for proving universally quantified formulas (not only equations) in the initial model defined by a set of Horn clauses.
KeywordsNormal Form Inference Rule Equational Theory Critical Pair Valid Consequence
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- [CH]Chang C.L.: The unit proof and the input proof in theorem proving. JACM 17,4 (1970), pp 698–707.Google Scholar
- [CL]Chang C.L. and Lee R.C.: Symbolic logic and mechanical theorem proving. Academic Press, New-York (1973).Google Scholar
- [FR1]Fribourg L.: oriented equational clauses as a programming language. To appear in Proc. ICALP 84.Google Scholar
- [FR2]Fribourg L.: a narrowing procedure for theories with constructors. Proc. CADE-7, Napa, California, 1984, in Springer-Verlag, Lecture Notes in Computer Science 170, edited by R.E. Shostak.Google Scholar
- [HD]Hsiang J. and Dershowitz N.: Rewrite methods for clausal and non-clausal theorem proving. Proc. ICALP 83, Spain (1983).Google Scholar
- [HH]Huet G. and Hullot J.M.: Proofs by induction in equational theories with constructors. JCSS 25-2 (1982).Google Scholar
- [HO]Huet G., Oppen D.C., Equations and rewrite rules: a survey. In Formal Languages: Perspectives and Open Problems, Edition R.Book, Academic Press (1980).Google Scholar
- [HU]Huet G.: A complete proof of correctness of the Knuth-Bendix completion algorithm. JCSS 23,1, pp 11–21 (1981).Google Scholar
- [JK]Jouannaud J.P. and Kirchner H.: Completion of a set of rules modulo a set of equations. Proc of POPL (1984).Google Scholar
- [KB]Knuth D. and Bendix P.: Simple word problems in universal algebra. Computational problems in abstract algebra, Ed. Leech J., Pergamon Press, 1970, 263–297.Google Scholar
- [KI]Kirchner H. A general inductive completion algorithm and application to abstract data types. Proc. CADE-7, Napa, California, 1984, in Springer-Verlag, Lecture Notes in Computer Science 170, edited by R.E.Shostak.Google Scholar
- [LA]Lankford D.S.: Canonical inference, Report ATP-32, Department of Mathematics and Computer Science, University of Texas at Austin, Dec.1975.Google Scholar
- [MOW]McCharen J.D., Overbeek R.A. and Wos L.: Problems and Experiments for and with Automated Theorem-Proving Programs. IEEE Transactions on Computers, vol. C-25, No 8, August 1976.Google Scholar
- [MU]Musser D.R.: On proving inductive properties of abstract data types. Proc. 7th POPL Conference, Las Vegas (1980).Google Scholar
- [PA1]Paul E.: A new interpretation of the resolution principle. Proc. CADE-7, Napa, California, 1984, in Springer-Verlag, Lecture Notes in Computer Science 170, edited by R.E.Shostak. To be published in "Journal of Symbolic Computation"Google Scholar
- [PA2]Paul E.: Proof by induction in equational theories with relations between constructors. Proc. 9th Colloquium on trees in algebra and programming, Bordeaux (March 1984). Cambridge University Press, edited by B.Courcelle.Google Scholar
- [PL]Plotkin G.: Building-in equational theories. Machine Intelligence, pp 73–90 (1972). Edinburgh Publisher.Google Scholar
- [RE]Remy J.L.: Etude des systemes de reecriture conditionnels et application aux types abstraits algebriques. These docteur es sciences, Centre de recherche en informatique de Nancy, July 1982.Google Scholar