## Solution of the Robbins Problem (1997)

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Venue: | Journal of Automated Reasoning |

Citations: | 132 - 3 self |

### BibTeX

@ARTICLE{Mccune97solutionof,

author = {William Mccune},

title = {Solution of the Robbins Problem},

journal = {Journal of Automated Reasoning},

year = {1997},

volume = {19},

pages = {263--276}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this article we show that the three equations known as commutativity, associativity, and the Robbins equation are a basis for the variety of Boolean algebras. The problem was posed by Herbert Robbins in the 1930s. The proof was found automatically by EQP, a theorem-proving program for equational logic. We present the proof and the search strategies that enabled the program to find the proof. Key words: Associative-commutative unification, Boolean algebra, EQP, equational logic, paramodulation, Robbins algebra, Robbins problem. 1. Introduction This article contains the answer to the Robbins question of whether all Robbins algebras are Boolean. The answer is yes, all Robbins algebras are Boolean. The proof that answers the question was found by EQP, an automated theoremproving program for equational logic. In 1933, E. V. Huntington presented the following three equations as a basis for Boolean algebra [6, 5]: x + y = y + x, (commutativity) (x + y) + z = x + (y + z), (associativit...

### Citations

509 |
word problems in universal algebras
- Knuth, Bendix
- 1970
(Show Context)
Citation Context ...uses the given clause algorithm to drive the search. paper.tex; 4/03/1997; 10:03; no v.; p.3 4 WILLIAM McCUNE (AC) unification and matching. The search algorithm is similar to Knuth-Bendix completion =-=[9], with the-=- option of using the "basic" restriction [1, 13]. We describe here some of EQP's features that were used in the proof of Lemma 3. See [11] for details on these and the other features of EQP.... |

305 | Otter 3.0 Reference Manual and Guide
- McCune
- 1994
(Show Context)
Citation Context ...to equational logic and can perform associative-commutative 3 EQP automatically rearranges AC subterms and renames variables. 4 EQP is similar in many ways to our more well known theorem prover Otter =-=[10]-=-; the most important differences are that Otter (1) does not have associative-commutative unification or matching, (2) applies to full first-order statements, (3) has fewer paramodulation options, and... |

168 |
Canonical forms and unification
- Hullot
- 1980
(Show Context)
Citation Context ... the most important differences are that Otter (1) does not have associative-commutative unification or matching, (2) applies to full first-order statements, (3) has fewer paramodulation options, and =-=(4)-=- always uses the given clause algorithm to drive the search. paper.tex; 4/03/1997; 10:03; no v.; p.3 4 WILLIAM McCUNE (AC) unification and matching. The search algorithm is similar to Knuth-Bendix com... |

79 |
A Unification Algorithm for Associative-Commutative Functions
- Stickel
- 1981
(Show Context)
Citation Context ...e some of EQP's features that were used in the proof of Lemma 3. See [11] for details on these and the other features of EQP. 3.1. AC Unification and Matching Associative-commutative (AC) unification =-=[15]-=- builds the properties of associativity and commutativity of a binary operation into the inference process so that the corresponding equations need not be present as explicit axioms. Two terms are AC ... |

73 | Theorem Proving with Ordering and Equality Constrained Clauses
- Nieuwenhuis, Rubio
- 1995
(Show Context)
Citation Context ...aper.tex; 4/03/1997; 10:03; no v.; p.3 4 WILLIAM McCUNE (AC) unification and matching. The search algorithm is similar to Knuth-Bendix completion [9], with the option of using the "basic" re=-=striction [1, 13]-=-. We describe here some of EQP's features that were used in the proof of Lemma 3. See [11] for details on these and the other features of EQP. 3.1. AC Unification and Matching Associative-commutative ... |

64 |
Basic paramodulation and superposition
- Bachmair, Ganzinger, et al.
- 1992
(Show Context)
Citation Context ...aper.tex; 4/03/1997; 10:03; no v.; p.3 4 WILLIAM McCUNE (AC) unification and matching. The search algorithm is similar to Knuth-Bendix completion [9], with the option of using the "basic" re=-=striction [1, 13]-=-. We describe here some of EQP's features that were used in the proof of Lemma 3. See [11] for details on these and the other features of EQP. 3.1. AC Unification and Matching Associative-commutative ... |

26 |
New sets of independent postulates for the algebra of logic”, Trans
- Huntington
- 1933
(Show Context)
Citation Context ... that answers the question was found by EQP, an automated theoremproving program for equational logic. In 1933, E. V. Huntington presented the following three equations as a basis for Boolean algebra =-=[6, 5]-=-: x + y = y + x, (commutativity) (x + y) + z = x + (y + z), (associativity) n(n(x) + y) + n(n(x) + n(y)) = x. (Huntington equation) The unary operation n can be read as complement. (Boolean algebra is... |

24 | 33 basic test problems: A practical evaluation of some paramodulation strategies
- McCune
- 1997
(Show Context)
Citation Context ...rom Lemma 1, Lemma 3, and the observation that the Robbins equation is valid in all Boolean algebras. 3. The Theorem Prover EQP The theorem prover that found the proof of Lemma 3 is EQP, presented in =-=[11]-=-. 4 EQP is restricted to equational logic and can perform associative-commutative 3 EQP automatically rearranges AC subterms and renames variables. 4 EQP is similar in many ways to our more well known... |

18 |
Boolean algebra. A correction
- Huntington
- 1933
(Show Context)
Citation Context ... that answers the question was found by EQP, an automated theoremproving program for equational logic. In 1933, E. V. Huntington presented the following three equations as a basis for Boolean algebra =-=[6, 5]-=-: x + y = y + x, (commutativity) (x + y) + z = x + (y + z), (associativity) n(n(x) + y) + n(n(x) + n(y)) = x. (Huntington equation) The unary operation n can be read as complement. (Boolean algebra is... |

18 |
RRL: Rewrite rule laboratory user's manual
- Kapur, Zhang
- 1989
(Show Context)
Citation Context ...ic failed to find a proof. Although AC matching is a special case of AC unification, we use a different algorithm, because (1) AC matching is less complicated (conceptually and practi5 The RRL system =-=[8]-=- has a strategy similar to our super-0 strategy. 6 With the benchmark problem to find the AC unifiers of the pair of terms x + x + x and y+z+u+v,EQP ordinarily returns 1,044,569 unifiers; with the sup... |

16 |
algebra: Conditions that make a near-Boolean algebra Boolean
- Robbins
- 1990
(Show Context)
Citation Context ...+ x = c). Winker then proved (by hand) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker =-=[16, 17]-=-). A Robbins algebra satisfying 9c9d(c + d = c) is a Boolean algebra. Lemma 2 (S. Winker [16, 17]). A Robbins algebra satisfying 9c9d(n(c+d) = n(c)) is a Boolean algebra. Appendix B contains a compute... |

15 |
Only prime superpositions need be considered in the Knuth-Bendix procedure
- Kapur, Musser, et al.
- 1985
(Show Context)
Citation Context ...the marks are inherited during paramodulation. paper.tex; 4/03/1997; 10:03; no v.; p.5 6 WILLIAM McCUNE predicate calculus with equality [1, 13], and it is related to the idea of prime superpositions =-=[7]. The &quo-=-t;basic" restriction imposes an order on derivations, but, like most other strategies, its use can interfere with searches as well as help them. It appears to have been a key strategy in finding ... |

15 |
Absorption and idempotency criteria for a problem in near-Boolean algebras
- Winker
- 1992
(Show Context)
Citation Context ...+ x = c). Winker then proved (by hand) several weaker conditions sufficient. The two such conditions that play a role in the present work are contained in the following two lemmas. Lemma 1 (S. Winker =-=[16, 17]-=-). A Robbins algebra satisfying 9c9d(c + d = c) is a Boolean algebra. Lemma 2 (S. Winker [16, 17]). A Robbins algebra satisfying 9c9d(n(c+d) = n(c)) is a Boolean algebra. Appendix B contains a compute... |

6 | Undecidable properties of finite sets of equations
- McNulty
- 1976
(Show Context)
Citation Context ...of Energy, under Contract W-31-109-Eng-38. 2 WILLIAM McCUNE associativity, and the Robbins equation? (No algorithm exists for deciding whether a finite set of equations is a basis for Boolean algebra =-=[12]-=-.) Robbins and Huntington could not find a proof or counterexample, and the problem later became a favorite of Alfred Tarski, who gave it to many of his students and colleagues [2], [3, p. 245]. Algeb... |

6 |
Automated Reasoning: Introduction and Applications, 2nd edition
- Wos, Overbeek, et al.
- 1992
(Show Context)
Citation Context ...s AC unification and matching. 3.2. Paramodulation and Demodulation The search for a proof uses paramodulation (an inference rule for equality) and demodulation (simplification of inferred equations) =-=[18]-=-. A simplification orderingson terms is used to orient equations and to guarantee termination of demodulation. The term ordering is t 1st 2 if length(t 1 ) ? length(t 2 ) and no variable has more occu... |

6 |
Automated Reasoning: Introduction and Applications, revised edition
- Wos, Overbeek, et al.
- 1992
(Show Context)
Citation Context ...ails on EQP's AC unification and matching. 3.2. Paramodulation and Demodulation The search uses paramodulation (an inference rule for equality) and demodulation (simplification of inferred equations) =-=[15]. A simp-=-lification ordering �� on terms is used to orient equations and to guarantee termination of demodulation. The term ordering is t 1 �� t 2 if length(t 1 ) ? length(t 2 ) and no variable has mor... |

1 |
Phone conversation
- Robbins
- 1996
(Show Context)
Citation Context ...Robbins problem arose from Huntington's mistake, in particular, that the error in Huntington's paper is that the Robbins equation appears in place of the Huntington equation [6]. According to Robbins =-=[14]-=-, the Robbins problem is not related to Huntington's error. 2 Proof. With the Robbins equation, let x be n(x), complement both sides of the equation, and simplify with n(n(x)) = x to obtain the Huntin... |

1 | 11:32; no v - tex |