An interpretation of the propositional Boolean algebra as a k-algebra. Effective calculus

Abstract

We construct in the first part of the paper a Boolean algebra, isomorphic to a propositional Boolean algebra (C, ∀, λ, ¬, →), that is also a k-algebra, and such that the ideals of the Boolean algebra correspond exactly to the ideals of the k-algebra.

An implementation on a Computer Algebra System is given in the second part of the paper. This implementation makes possible, for instance, to compare propositions, or to calculate a minimal base of an ideal or filter.

The use of well-known techniques from Commutative Algebra and Computer Algebra allows us to decide problems such as the ideal membership (in this particular k-algebra) with methods of lower complexity than calculating Gröbner Basis.

Besides, following our approach, results in the k-algebra can be directly translated into the Boolean algebra.

We also show how we try to apply this interpretation to verification of Knowledge Basis.