Polynominal Time Algorithms for Some Self-Duality Problems

Consider the problem of deciding whether a Boolean formula f is self-dual, i.e. f is logically equivalent to its dual formula f d , deened by f d (x) = f(x). This problem is a well-studied problem in several areas like theory of coteries, database theory, hypergraph theory or computational learning theory. In this paper we exhibit polynomial time algorithms for testing self-duality for several natural classes of formulas where the problem was not known to be solvable. Some of the results are obtained by means of a new characterization of self-dual formulas in terms of its Fourier spectrum. 1 Preliminaries and Deenitions The self-duality problem is a well-studied problem in several areas. It is of particular interest the monotone case, since the problem of deciding whether a monotone DNF formula is self-dual or not is known to be equivalent to several other problems in diierent elds like theory of coteries (used in distributed systems) 6, 8], database theory 14], hypergraph theory 4] and computational learning theory 2]. Before discussing previous and our results, we provide a more detailed dee-nition of the problem and related concepts. We consider Boolean concepts f : f0; 1g n 7 ! f0; 1g depending on n variables fv 1 ; : : :; v n g. We denote by x i the ith bit of vector x 2 f0; 1g n and by x and f the complement of x and f, respectively. A literal is either a variable or its negation. A Boolean concepts is said to be monotone if for every pair of assignments a; b 2 f0; 1g n , a b implies f(a) f(b). Let us discuss rst how to represent Boolean concepts. The most common and widely used representation of a Boolean concept is the Disjunctive Normal Form or DNF. It is well know that every Boolean concept can be expressed as a DNF formula. A DNF formula is a disjunction of terms, where a term is a conjunction of literals. Respectively, we deene the symmetric class of Conjunctive Normal Form or CNF formulas. A CNF is a conjunction of clauses, where a clause is a ? Supported by the Esprit EC program under project 7141 (ALCOM-II) and the Span-ish DGICYT (project PB92-0709).