# 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.

## 63 Citations

A Hierarchy of Algebras for Boolean Subsets

- MathematicsArch. Formal Proofs
- 2020

A common ground for these constructions based on a new equational axiomatisation of Boolean algebras is presented, for example, with the antidomain operation, dynamic negation and Stone algebraes.

A Shortest 2-Basis for Boolean Algebra in Terms of the Sheffer Stroke

- MathematicsJournal of Automated Reasoning
- 2004

In this article, we present a short 2-basis for Boolean algebra in terms of the Sheffer stroke and prove that no such 2-basis can be shorter. We also prove that the new 2-basis is unique (for its…

Automated Discovery of Single Axioms for

- Mathematics
- 2004

We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give…

Arithmetic version of Boolean algebra

- Philosophy, Mathematics2009 2nd IEEE International Conference on Computer Science and Information Technology
- 2009

It is discussed that the logical results in Boolean algebra can equally be derived with ordinary algebraic operations and fundamental logical equivalent propositions via arithmetic versions are established.

On the Two Short Axiomatizations of Ortholattices

- Mathematics
- 2003

In the paper, two short axiom systems for Boolean algebras are introduced. In the first section we show that the single axiom (DN1) proposed in [2] in terms of disjunction and negation characterizes…

Automated discovery of single axioms for ortholattices

- Mathematics
- 2005

Abstract.We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We…

Algebraic Backgrounds for Data Structures Models

- Mathematics, Computer Science
- 2019

The main aim of the Algebraic Aggregate Theory is to present uniformly data structures in mathematics and its applications in terms of Universal Algebra. In the given paper are characterized three…

## 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…