Abstract not found

This paper is based on my lecture [26]. It examines the problem of proving non-trivial lower bounds for the length of proofs in propositional logic from the perspective of methods available rather than surveying known partial results (i.e., lower bounds for weaker proof systems). We discuss neither motivations for proving lower bounds for propositional logic nor relations to other problems in logic or complexity theory. The reader is referred to [20] for the background information (as well as for all details missing in this paper). The paper is aimed at curious non-specialists. The style of our exposition is accordingly informal at places and we do not burden the text (especially in the introduction) with exhausting references not directly related to our main objective. The reader starving for details can find them, together with all original references, in [20] (see also expository articles [25, 32]). Introduction

This report supersedes NYSERNet Technical Report 89-12-29Abstract

This thesis investigates terminological representation languages, as used in kl-one-type knowledge representation systems, from an algebraic point of view. Terminological representation languages are based on two primitive syntactic types, called concepts and roles, which are usually interpreted model-theoretically as sets and relations, respectively. I propose an algebraic rather than a model-theoretic approach. I show that terminological representations can be naturally accommodated in equational algebras of sets interacting with relations, and I use equational logic as a vehicle for reasoning about concepts interacting with roles.

In this paper a Boolean minimization algorithm is considered and implemented as an applet in Java. The application is based on the Quine-McCluskey simplification technique with some modifications. The given application can be accessed on line since it is posted on the World Wide Web (WWW), with up to four variables, at the URL

The algebra of logic developed by Boole was not Boolean algebra. In this article we give a natural framework that allows one to easily reconstruct his algebra and see the di#culties it created for his successors.

Abstract. We give several new lattice identities valid in nonmodular lattices such that a uniquely complemented lattice satisfying any of these identities is necessarily Boolean. Since some of these identities are consequences of modularity as well, these results generalize the classical result of Birkhoff and von Neumann that every uniquely complemented modular lattice is Boolean. In particular, every uniquely complemented lattice in M ∨ V(N5), the least nonmodular variety, is Boolean. 1.

introduction

ed, a set closed under three such operations and satisfying an appropriate set of 4 postulates is called a Boolean algebra. Various formulations of the postulate sets are known; two of the earliest are due to Huntington [26], [27] (see also [5], Ch. X, Sect. 4, and [6], Ch. XI, p. 342). The power set P(X) of a set X, under union, intersection, and complementation, is a well-known Boolean algebra; its associated unitary Boolean ring is the set P(X) under the operations of symmetric dierence and intersection, and this example serves as a model of the association of a Boolean ring with a Boolean algebra. On the other hand, if R is a Boolean ring and if operations _, ^, and 0 and dened on R by x_y = x+y xy, x^y = xy, and x 0 = 1 x, then under these three operations R is a Boolean algebra associated with R as a Boolean ring. In [41] Stone showed that the theory of unitary Boolean rings is equivalent to the theory of topological spaces that are compact, Hausdor , and totally disconnec...

Boundary algebra is a new and simple notation for the Boolean algebra 2 and the truth functors. The primary arithmetic [PA] is built up from the atoms, ‘() ’ and the blank page, by enclosure between ‘( ‘ and ‘)’, denoting the primitive notion of distinction, and concatenation. Inserting letters denoting the presence or absence of () into a PA formula yields boundary algebra [BA], a simpler notation for Spencer-Brown’s (1969) primary algebra [pa]. The BA axioms are “()()=()”, and “(()) [=⊥] may be written or erased at will.” Repeated application of these axioms to a PA formula yields a member of B={(),⊥}, its simplification. If (a)b [dually (a(b))] ⇔ a≤b, then ⊥≤() [()≤⊥] follows trivially, so that B is a poset. BA is a self-dual notation for the Boolean algebra 2: (a) ⇔ a′, () ⇔ 1 [0] so that B is the carrier for 2, and ab ⇔ a∪b [a∩b]. The basis abc=bca (Dilworth 1938), a(ab) = a(b) (Bricken 2002), and a(a)=() facilitates clausal reasoning and proof by calculation. BA also simplifies the usual normal forms and Quine’s (1982) truth value analysis. () ⇔ true [false] yields boundary logic.