A detailed exposition of foundations of a logic-algebraic model for reasoning with knowledge bases specified by propositional (Boolean) logic is presented. The model is conceived from the logical translation of usual derivatives on polynomials (on residue rings) which is used to design a new inference rule of algebro-geometric inspiration. Soundness and (refutational) completeness of the rule are proved. Some applications of the tools introduced in the paper are shown.

We show how to express any Hasse-Schmidt derivation of an algebra in terms of a finite number of them under natural hypothesis. As an application, we obtain coefficient fields of the completion of a regular local ring of positive characteristic in terms of Hasse-Schmidt derivations.

Abstract. We present a specialised (polynomial-based) rule for the propositional logic called the Independence Rule, which is useful to compute the conservative retractions of propositional logic theories. In this paper we show the soundness and completeness of the logical calculus based on this rule, as well as other applications. The rule is defined by means of a new kind of operator on propositional formulae. It is based on the boolean derivatives on the polynomial ring F2 [x].

Let k be a perfect field of positive characteristic, k(t)_{per} the perfect closure of k(t) and A=k[[X_1,...,X_n]]. We show that for any maximal ideal N of A'=k(t)_{per}\otimes_k A, the elements in \hat{A'_N} which are annihilated by the "Taylor" Hasse-Schmidt derivations with respect to the X_i form a coefficient field of \hat{A'_N}.Comment: Final versio

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