Boolean algebra

Abstract: This paper is an attempt at developing a theory of algebraic systems that would correspond in a natural fashion to the No-valued propositional calculus(2). For want of a better name, we shall call these algebraic systems MV-algebras where MV is supposed to suggest many-valued logics. It is known that the classical two-valued logic gives rise to the study of Boolean algebras and, as can be expected, every Boolean algebra will be an MValgebra whereas the converse does not hold. However, many results for Boolean algebras can be appropriately carried over to MV-algebras, although in some cases the proofs become more subtle and delicate. The motivation behind the present study is to find a proof of the completeness of the Novalued logic by using some algebraic results concerning MV-algebras; more specifically, it is known that the completeness of the two-valued logic is a consequence of the Boolean prime ideal theorem and we wish to exploit just some such corresponding result for MV-algebras(3). It will be seen that our effort in duplicating this result is only partially successful. In the first four sections of this paper we present various theorems concerning both the arithmetic in MV-algebras and the structure of these algebras. In the last section we give some applications of our results to the study of completeness of NO-valued logic and some related topics. We point out here that the treatment of MV-algebras as given here is not meant to be complete and exhaustive. 1. Axioms of MV-algebras and some elementary consequences. An MV

Abstract: In this paper we introduce a variant of temporal logic tailored for specifying desired properties of continuous signals. The logic is based on a bounded subset of the real-time logic mitl, augmented with a static mapping from continuous domains into propositions. From formulae in this logic we create automatically property monitors that can check whether a given signal of bounded length and finite variability satisfies the property. A prototype implementation of this procedure was used to check properties of simulation traces generated by Matlab/Simulink.

Abstract: Self-taught mathematician and father of Boolean algebra, George Boole (1815–1864) published An Investigation of the Laws of Thought in 1854. In this highly original investigation of the fundamental laws of human reasoning, a sequel to ideas he had explored in earlier writings, Boole uses the symbolic language of mathematics to establish a method to examine the nature of the human mind using logic and the theory of probabilities. Boole considers language not just as a mode of expression, but as a system one can use to understand the human mind. In the first 12 chapters, he sets down the rules necessary to represent logic in this unique way. Then he analyses a variety of arguments and propositions of various writers from Aristotle to Spinoza. One of history's most insightful mathematicians, Boole is compelling reading for today's student of intellectual history and the science of the mind.

Abstract: Preface Introduction 2 Boolean Algebra 3 L-matrix 4 L-language 5 String Acceptors 6 [omega]-theory: L-automaton/L-process 7 The Selection/Resolution Model 8 Reduction of Verification 9 Structural Induction 10 Binary Decision Diagrams Appendices Bibliography Glossary Index