Boolean expression

Abstract: We study the satisfiability of random Boolean expressions built from many clauses with K variables per clause (K-satisfiability). Expressions with a ratio alpha of clauses to variables less than a threshold alphac are almost always satisfiable, whereas those with a ratio above this threshold are almost always unsatisfiable. We show the existence of an intermediate phase below alphac, where the proliferation of metastable states is responsible for the onset of complexity in search algorithms. We introduce a class of optimization algorithms that can deal with these metastable states; one such algorithm has been tested successfully on the largest existing benchmark of K-satisfiability.

Abstract: A systematic procedure is presented for writing a Boolean function as a minimum sum of products This procedure is a simplification and extension of the method presented by W V Quine Specific attention is given to terms which can be included in the function solely for the designer's convenience

Abstract: A Boolean network is a logical dynamic system, which has been used to describe cellular networks. Using a new matrix product, called semi-tensor product of matrices, a logical function can be expressed as an algebraic function. This expression can covert the Boolean networks into discrete-time linear dynamic systems. Similarly, the Boolean control networks can also be converted into discrete time bilinear dynamic systems. Under these forms the standard matrix analysis can be used to consider the structure and the control problems of Boolean (control) networks. After the detailed description of this new approach, the controllability of Boolean control networks is considered in the paper as an application.

Abstract: Boolean algebras are those mathematical systems first developed by George Boole in the treatment of logic by symbolic methodsf and since extensively investigated by other students of logic, including Schröder, Whitehead, Sheffer, Bernstein, and Huntington.J Since they embody in abstract form the principal algebraic rules governing the manipulation of classes or aggregates, these systems are of technical interest to the mathematician quite as much as to the logician. It is thus natural to suppose that a study of Boolean algebras by the methods of modern algebra will prove fruitful of important and useful results. Indeed, if one reflects upon various algebraic phenomena occurring in group theory, in ideal theory, and even in analysis, one is easily convinced that a systematic investigation of Boolean algebras, together with still more general systems, is probably essential to further progress in these theories.! The writer's interest in the subject, for example, arose in connection with the spectral theory of symmetric transformations in Hubert space and certain related properties of abstract integrals. In the actual development of the proposed theory of Boolean algebras, there emerged some extremely close connections with general topology which led at once to results of sufficient importance to confirm our a priori views of the probable value of such a theory.|| In the present paper, which is one of a projected series, we shall be concerned primarily with the problem of determining the representation of a