Short Single Axioms for Boolean Algebra
@article{McCune2004ShortSA,
title={Short Single Axioms for Boolean Algebra},
author={William McCune and Robert Veroff and Branden Fitelson and Kenneth Harris and Andrew Feist and Larry Wos},
journal={Journal of Automated Reasoning},
year={2004},
volume={29},
pages={1-16}
}We present short single equational axioms for Boolean algebra in terms of disjunction and negation and in terms of the Sheffer stroke. Previously known single axioms for these theories are much longer than the ones we present. We show that there is no shorter axiom in terms of the Sheffer stroke. Automated deduction techniques were used in several parts of the work.
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References
SHOWING 1-10 OF 35 REFERENCES
Single axioms for Boolean algebra.
- Physics
- 2000
Explicit single axioms are presented for Boolean algebra in terms of (1) the Sheffer stroke; (2) disjunction and negation; (3) disjunction, conjunction, and negation; and (4) disjunction,…
Single axioms for groups
- MathematicsJournal of Automated Reasoning
- 2004
Some lower bounds on the complexity of equations of the form (α=x) which are single axioms for group theory are proved, showing that McCune's examples are the shortest possible and refuting a conjecture of Neumann.
Solution of the Robbins Problem
- MathematicsJournal of Automated Reasoning
- 2004
This article shows that the three equations known as commutativity, association, and the Robbins equation are a basis for the variety oflean algebras and presents the proof and the search strategies that enabled the program EQP to find the proof.
A set of five independent postulates for Boolean algebras, with application to logical constants
- Mathematics
- 1913
Postulate-sets for determining the class of Boolean algebrasf have been given by Schröder,^ Whttehead,§ and Huntington.|| Schroder's set of ten postulates assumes—in addition to an undefined class K,…
MACE 2.0 Reference Manual and Guide
- Computer ScienceArXiv
- 2001
MACE is a program that searches for finite models of first-order statements, first translated to clauses, then to relational clauses, and finally for the given domain size, the ground instances are constructed.
A fascinating country in the world of computing: your guide to automated reasoning
- Computer Science
- 1999
The menu, the map, and the magic learning logic by example automated reasoning in full logic circuit design logic circuit validation research in mathematics research in formal logic the formal…
SEM: a System for Enumerating Models
- Computer ScienceIJCAI
- 1995
This paper describes SEM, a System for Enumerating finite Models of first-order many-sorted theories, and shows that general purpose finite model generators are indeed useful in many applications.
Searching for circles of pure proofs
- MathematicsJournal of Automated Reasoning
- 2004
The methodology used to answer the cited specific question of the existence of a circle of pure proofs for the thirteen shortest single axioms for equivalential calculus, subject to the requirement that condensed detachment be used as the rule of inference is presented.
Short 2-bases for Boolean algebra in terms of the She er stroke
- Physics
- 2000
A system for heating a supply solution containing scale-forming ingredients without depositing encrusted scale on the heating surface thereof includes a heating chamber, a separate pressurizing…