This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to def...

New foundations for probability theory are suggested and investigated, derived from the formalized systems theory and using also some other branches of mathematical logic. The probability measures are supposed to take their values not in the unit interval of reals, but in a special kind of Boolean algebras. Some analogies as well as differences of this approach with respect to the classical Kol...

THEORY OF ABELIAN OPERATOR ALGEBRAS: AN APPLICATION OF FORCING BY THOMAS J. JECH1 Abstract. The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models. The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models.

Basic results in combinatorial mathematics provide the foundation for a theory and calculus for reasoning about sequential behavior. A key concept of the theory is a generalization of Boolean implicant in which one now deals with statements of the form: A sequence of Boolean expressions α is an implicant of a set of sequences of Boolean expressions A This notion of a generalized implicant takes...

The concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by Sch. oning (Lecture Notes in Computer Science, Vol. 211, Springer, Berlin, 1985). Informally, a set is low (high resp.) for a relativizable class K of languages if it does not add (adds maximal resp.) power to K when used as an oracle. In this paper, we introduce the notions of...

Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on the study of applications of Boolean functions in coding theory has been made. Two generic constructions of binary linear codes with Boolean functions have bee...

Written by prominent experts in the field, this monograph provides the first comprehensive and unified presentation of the structural, algorithmic, and applied aspects of the theory of Boolean functions. The book focuses on algebraic representations of Boolean functions, especially disjunctive and conjunctive normal form representations. It presentswithin this framework the fundamental elements...

Abstract On a very natural conception of sets, every set has an absolute complement. The ordinary cumulative hierarchy dismisses this idea outright. But we can rectify this, whilst retaining classical logic. Indeed, develop boolean algebra sets arranged in well-ordered levels. I show by presenting Boolean Level Theory, which fuses Theory (from Part 1) with ideas due to Thomas Forster, Alonzo Ch...