Proving Ramsey's theorem by the cover set induction: A case and comparison study

An experiment of the cover set induction method inRRL is presented with a mechanical proof of Ramsey's theorem in graph theory. The proof is similar to the proof obtained by Kaufmann using the Boyer-Moore theorem prover. We show that this similarity is not unusual, because there is a close relationship between the Boyer-Moore logic and the algebraic specification of abstract data types on which the cover set induction method is based. (This implies that many proofs done by the Boyer-Moore theorem prover can be reproduced byRRL.) Our experiment shows thatRRL can automatically prove all the lemmas in Ramsey's theorem, while the Boyer-Moore theorem prover needs several user's hints and takes much longer (CPU time) to finish.

[1]  Hantao Zhang,et al.  Proving the Chinese Remainder Theorem by the Cover Set Induction , 1992, CADE.

[2]  Hantao Zhang,et al.  Implementing Contextual Rewriting , 1992, CTRS.

[3]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[4]  Michael J. O'Donnell,et al.  Equational Logic as a Programming Language , 1985, Logic of Programs.

[5]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .

[6]  Gérard P. Huet,et al.  Proofs by induction in equational theories with constructors , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[7]  David Basin,et al.  The Boyer-Moore Prover and Nuprl: an experimental comparison , 1991 .

[8]  Deepak Kapur,et al.  First-Order Theorem Proving Using Conditional Rewrite Rules , 1988, CADE.

[9]  Paliath Narendran,et al.  Proof by Induction Using Test Sets , 1986, CADE.

[10]  D. Kapur,et al.  Reduction, superposition and induction: automated reasoning in an equational logic , 1988 .

[11]  Robert S. Boyer,et al.  Computational Logic , 1990, ESPRIT Basic Research Series.

[12]  Deepak Kapur,et al.  Proof by Consistency , 1987, Artif. Intell..

[13]  Michaël Rusinowitch,et al.  Mechanizing inductive reasoning , 1990, Bull. EATCS.

[14]  Matt Kaufmann.  An Interactive Enhancement to the Boyer-Moore Theorem Prover , 1988, CADE.

[15]  Deepak Kapur,et al.  An Overview of Rewrite Rule Laboratory (RRL) , 1989, RTA.

[16]  Nachum Dershowitz,et al.  Termination of Rewriting , 1987, J. Symb. Comput..

[17]  David R. Musser,et al.  On proving inductive properties of abstract data types , 1980, POPL '80.

[18]  Robert S. Boyer,et al.  A computational logic handbook , 1979, Perspectives in computing.

[19]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[20]  Robert S. Boyer,et al.  Overview of a Theorem-Prover for A Computational Logic , 1986, CADE.

[21]  Paliath Narendran,et al.  On sufficient-completeness and related properties of term rewriting systems , 1987, Acta Informatica.

[22]  James J. Horning,et al.  The algebraic specification of abstract data types , 1978, Acta Informatica.

[23]  Jean-Pierre Jouannaud,et al.  Proofs by induction in equational theories without constructors , 1985, Bull. EATCS.

[24]  Gérard P. Huet,et al.  Proofs by Induction in Equational Theories with Constructors , 1980, FOCS.

[25]  Deepak Kapur,et al.  A Mechanizable Induction Principle for Equational Specifications , 1988, CADE.