Solution of the Robbins Problem

In this article we show that the three equations known as commutativity,associativity, and the Robbins equation are a basis for the variety ofBoolean algebras. The problem was posed by Herbert Robbins in the 1930s. Theproof was found automatically by EQP, a theorem-proving program forequational logic. We present the proof and the search strategies thatenabled the program to find the proof.

[1]  S Winker Absorption and idempotency criteria for a problem in near-Boolean algebras , 1992 .

[2]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[3]  William McCune,et al.  33 basic test problems: a practical evaluation of some paramodulation strategies , 1997 .

[4]  Paliath Narendran,et al.  Only Prime Superpositions Need be Considered in the Knuth-Bendix Completion Procedure , 1988, J. Symb. Comput..

[5]  A. P. Bowran A Boolean Algebra , 1965 .

[6]  Wayne Snyder,et al.  Basic Paramodulation and Superposition , 1992, CADE.

[7]  Jean-Marie Hullot,et al.  Canonical Forms and Unification , 1980, CADE.

[8]  Mark E. Stickel,et al.  A Unification Algorithm for Associative-Commutative Functions , 1981, JACM.

[9]  Albert Rubio,et al.  Theorem Proving with Ordering and Equality Constrained Clauses , 1995, J. Symb. Comput..

[10]  E. V. Huntington A New Set of Independent Postulates for the Algebra of Logic with Special Reference to Whitehead and Russell's Principia Mathematica. , 1932, Proceedings of the National Academy of Sciences of the United States of America.

[11]  George F. McNulty Undecidable Properties of Finite Sets of Equations , 1976, J. Symb. Log..

[12]  E. V. Huntington Boolean Algebra. A Correction , 1933 .

[13]  Larry Wos,et al.  Automated Reasoning: Introduction and Applications , 1984 .

[14]  William McCune,et al.  OTTER 3.0 Reference Manual and Guide , 1994 .

[15]  Steven K. Winker,et al.  Robbins algebra: Conditions that make a near-boolean algebra boolean , 1990, Journal of Automated Reasoning.

[16]  Larry Wos,et al.  Automated reasoning (2nd ed.): introduction and applications , 1992 .