Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that (*) Graph Connectivity is not expressible in existential monadic second-order logic (MonNP), even in the presence of a built-in linear order, (*) Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n^o(1), and (*) the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation.

Binary NP consists of all sets of finite structures which are expressible in existential second order logic with second order quantification restricted to relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of relations. We consider mainly three types of classes of relations: unary functions, order relations and graphs with degree bounds. We show that many of these restrictions have the same expressive power and establish a 4-level strict hierarchy, represented by sets, permutations, unary functions and arbitrary binary relations, respectively.

. A set of finite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quantification is over relations of arity 2. In [DLS95] subclasses of Binary NP were considered, in which the second order quantifiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in a three-level linear hierarchy, the levels of which are represented by bijections, successor relations and unary functions respectively. In this paper it is shown that -- Graph Connectivity is expressible by bijections, thereby showing that the two lower levels of the hierarchy coincide; -- the set of graphs with exactly as many vertices as arcs is expressible by unary functions but not by bijections. This shows that level 3 is strictly stronger than the other two levels. 1 Introduction Fagin [Fag74] showed that NP coincides with the class of sets of finite structures that are characteri...

Binary NP is existential second order logic where second order quantification is over relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of relations. We consider mainly three types of classes of relations: unary functions, order relations and graphs with degree bounds. We show that many of these restrictions have the same expressive power and that they all can be ordered in a linear hierarchy. 1 Introduction It is a well-known fact that many results, tools, and techniques of mathematical logic break down when only finite structures are considered as models, cf. [Gur88]. Although this restriction to finite structures is of little interest in most areas of traditional mathematics the situation is quite different in computer science. It was therefore mainly with applications to computer science in mind that finite model theory was developed in the last few years, and a number of fascinating connection...

In this paper, an efficient algorithm to compute the set of prime implicants of a prepositional formula in Conjunctive Normal Form (CNF) is presented. The proposed algorithm uses a concept of representing the formula as a binary matrix and computing paths through the matrix as implicants. The algorithm finds the prime implicants as the prime paths using the divide-and-conquer technique. The proposed algorithm can be used for knowledge compilation, Clause Maintenance Systems where the knowledge base is prepositional formulae. Moreover, the algorithm is easily adaptable to the incremental mode of computation where an earlier formula is updated by a set of clauses.

is the graph with the same vertex set as G, and two vertices are adjacent in G i the distance between them in G is at most k. It is well-known that G is Hamiltonian for any connected graph G. Also, G is Hamiltonian for any 2-connected graph and for any connected claw-free graph G.