Ben Fine observed that a theorem of Magnus on normal closures of elements in free groups is first order expressible and thus holds in every elementarily free group. This classical theorem, vintage 1931, asserts that if two elements in a free group have the same normal closure, then either they are conjugate or one is conjugate to the inverse of the other in the free group. An examination of a s...

Earlier we introduced the notion of a stable set of points (SSP) and showed that a CNF formula is unsatisfiable iff there is a set of points (i.e. complete assignments) that is stable. Experiments showed that SSPs of CNF formulas of practical interest are very big. So computing an SSP of a CNF formula point by point is, in general, infeasible. In this report, we show how an SSP can be computed ...

We examine the conditions under which we can keep simplicity or categoricity after adding a Skolem function to the theory. AMS classification: 03C45 , 03C50

The genotype and phenotype have been separated to make genetic manipulation simpler. Each program is represented as a xed-length integer array and then mapped onto Backus-Naur Form (BNF). The program statements used are shown in Figure 1a. The BNF is designed to minimise the size of the genome that describes a program. The mapping, between the genotype and phenotype, is similar to Grammatical E...

This article covers a theory of dense subsets in general semigroups, including basic algebraic properties of dense and disjunctive subsets, characterizations of dense subset preserving homomorphisms and some remarkable properties of dense elements.

The aim of this paper is to investigate the connections between minimal keys and antikeys for special Sperner-systems by hypergraphs. The BoyceCodd normal form and some related problems are also studied in this paper.

It is known experimentally that there is a threshold for satisfiability in 3-CNF formulre around the value 4.25 for the ratio of variables to clauses. It is also known that the threshold is sharp [Fri99], but that proof does not give a value for the threshold. We use purely combinatorial techniques to count the number of satisfiable boolean formulre given in conjunctive normal form. The intenti...

The topic of this chapter is the study of certain forms of “redundancies” in propositional conjunctive normal forms and generalisations. In Sections 11.1 11.7 we study “minimally unsatisfiable conjunctive normal forms” (and generalisations), unsatisfiable formulas which are irredundant in a strong sense, and in Sections 11.8 11.13 we study “autarkies”, which represent a general framework for co...