Linear algebra is introduced by Faugère in F4 to speed up the reduction procedure during Gröbner basis computations. Linear algebra has also been used in fast implementations of F5 and other signature-based Gröbner basis algorithms. To use linear algebra for reductions, an important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function...

Boolean semantics is a version of formal semantics in which models are supposed to have algebraic, preferably Boolean, structure (Keenan and Faltz 1985). Thus for any category C there is a corresponding denotational Boolean algebra DC of possible denotations of expressions of category C. Furthermore, given that most categories are functionally related (in principle all ”major” categories are Bo...

The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we show that his notion of overlap morphism corresponds classically to that of map preserving arbitrary joins. Moreover we prove that the power collection of a set is the free overlap algebra join-generated from the set. Then, we generalize the concept of overlap algebra and o...

In this paper we define Kleene algebra with tests in a slightly more general way than Kozen’s definition. Then we give an explicit construction of the free Kleene algebra with tests generated by a pair of sets. Moreover we show that our free Kleene algebra with tests is isomorphic to Kozen and Smith’s Kleene algebra with tests if their construction available, that is, a generator of the Boolean...

We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λ-theories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest of a systematic study of the lattice λT of λ-theories grows out of several open problems on lamb...

We give a brief overview of expressiveness and complexity results for a hierarchy of subclasses of quantified Boolean formulas with close connections to Boolean circuits and minimal unsatisfiability.

Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion of any abstract algebra is the extended Stone dual space of the Boolean algebra of recognisable subsets of the abstract algebra endowed with certain residuat...

Neural networks have been applied in various domain including science, commerce, medicine, and industry. However, The knowledge learned by a trained neural network is difficult to understand. This paper proposes a Boolean algebra based algorithm to extract comprehensible Boolean rules from supervised feed-forward neural networks to uncover the black-boxed knowledge. This algorithm is called the...