Stephan Foldes

Consensus algorithms for the generation of all maximal bicliques

We describe a new algorithm for generating all maximal bicliques (i.e. complete bipartite, not necessarily induced subgraphs) of a graph. The algorithm is inspired by, and is quite similar to, the consensus method used in propositional logic. We show that some variants of the algorithm are totally polynomial, and even incrementally polynomial. The total complexity of the most efficient variant of the algorithms presented here is polynomial in the input size, and only linear in the output size. Computational experiments demonstrate its high efficiency on randomly generated graphs with up to 2000 vertices and 20,000 edges.

Equational characterizations of Boolean function classes

Abstract Several noteworthy classes of Boolean functions can be characterized by algebraic identities (e.g. the class of positive functions consists of all functions f satisfying the identity f( x )∨f( y )∨f( x ∨ y )=f( x ∨ y ) ). We give algebraic identities for several of the most frequently analyzed classes of Boolean functions (including Horn, quadratic, supermodular, and submodular functions) and proceed then to the general question of which classes of Boolean functions can be characterized by algebraic identities. We answer this question for function classes closed under addition of inessential (irrelevant) variables. Nearly all classes of interest have this property. We show that a class with this property has a characterization by algebraic identities if and only if the class is closed under the operation of variable identification. Moreover, a single identity suffices to characterize a class if and only if the number of minimal forbidden identification minors is finite. Finally, we consider characterizations by general first-order sentences, rather than just identities. We show that a class of Boolean functions can be described by an appropriate set of such first-order sentences if and only if it is closed under permutation of variables.

On Closed Sets of Relational Constraints and Classes of Functions Closed under Variable Substitutions

Abstract.Pippenger’s Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set A and taking values in a possibly different set B, where any or both of A and B may be finite or infinite.

Post classes characterized by functional terms

Abstract The classes of Boolean functions closed under classical compositions form an algebraic lattice which was completely described in 1941 in a pioneering work of Post (Ann. Math. Stud. (5) (1941)). These classes and the lattice are often referred to as the Post Classes and the Post Lattice, respectively. There are several approaches to present a Post class. Being a set of functions it can be characterized by a traditional set-theoretic description of its members. Since the Post classes are closed under certain operations, they are also often presented by sets of generators. A remarkable approach, which has been widely used in Universal Algebra is a characterization of classes by means of preservation of polyrelations (Kibernetika (3)(Pt1) (1969) 1, (5)(PtII) (1969) 1). Recently, there appeared several new methods for the characterization of classes of logic functions. These methods are based on special formal expression, which in general define a much larger variety of classes particularly including all Post classes. One such result is by Ekin et al. (Discrete Math. 211 (2000) 27), which presents the characterization of classes by a set (possibly infinite) of certain equational identities. The approach developed in [Ekin et al. (2000)] was soon extended to several directions. Pippenger (Discrete Math. 254 (2002) 405) presented the classes through pairs of relations (constrains) in the setting of a Galois Theory. Pogosyan (Multiple Valued Logic, Gordon and Breach, London, 2001, pp. 417–448, Vol. 7) has defined each such class by one functional term (possibly of infinite length), and has introduced the notion of rank for a term as well as for a class. In (Algebra Universalis 44 (2000) 309) established a connection with the Birkhoff-Tarski HSP Theorem. This paper presents a complete characterization of the Post Classes by means of functional terms (as in [Pogosyan, 2001]). We also give a constructive criterion which establishes the minimal ranks for all Post classes.

Submodularity, Supermodularity, and Higher-Order Monotonicities of Pseudo-Boolean Functions

Classes of set functions defined by the positivity or negativity of the higher-order derivatives of their pseudo-Boolean polynomial representations generalize those of monotone, supermodular, and submodular functions. In this paper, these classes are characterized by functional inequalities and are shown to be closed both under algebraic closure conditions and a local closure criterion. It is shown that for everym = 1, in addition to the class of all set functions, there are only three other classes satisfying these algebraic and local closure conditions: those having positive, respectively negative,mth-order derivatives, and those having a polynomial representation of degree less thanm.

Disjunctive and conjunctive normal forms of pseudo-Boolean functions

After showing that every pseudo-Boolean function (i.e. real-valued function with binary variables) can be represented by a disjunctive normal form (essentially the maximum of several weighted monomials), the concepts of implicants and of prime implicants are analyzed in the pseudo-Boolean context, and a consensus-type method is presented for finding all the prime implicants of a pseudo-Boolean function. In a similar way the concepts of conjunctive normal form, implicates and prime implicates, as well as the resolution method are examined in the case of pseudo-Boolean functions.

A Property of Connected Subsets of an Ordered Set

We prove a planarity-independent analogue of a theorem of Nowakowski, Rival and Urrutia concerning lattices contained in orders respresentable as blocking relations.RésuméUn théorème de Nowakowski, Rival et Urrutia concernant les treillis contenus dans des ordres planaires admet un analogue indépendant de toute considération de planarité.

The Lorentz group and its finite field analogs: Local isomorphism and approximation

Finite Lorentz groups acting on four-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real four-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary precision to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields.

Definability of Boolean function classes by linear equations over GF(2)

Necessary and sufficient conditions are provided for a class of Boolean functions to be definable by a set of linear functional equations over the two-element field. The conditions are given both in terms of closure with respect to certain functional compositions and in terms of definability by relational constraints.

Maximal compatible extensions of partial orders

We give a complete description of maximal compatible partial orders on the mono-unary algebra