Polynomial-time inference of all valid implications for Horn and related formulae

• Published in: Annals of mathematics and artificial intelligence Vol. 1; no. 1-4; pp. 21 - 32
• Main Authors: Boros, E., Crama, Y., Hammer, P. L.
• Format: Journal Article Web Resource
• Language: English
• Published: Springer Science & Business Media B.V 01.09.1990
• Subjects: complexity; Computer science; Engineering, computing & technology; Horn formulae; inference; Ingénierie, informatique & technologie; Mathematics; Mathématiques; Physical, chemical, mathematical & earth Sciences; Physique, chimie, mathématiques & sciences de la terre; Propositional logic; resolution; Sciences informatiques
• ISSN: 1012-2443; 1573-7470; 1573-7470
• DOI: 10.1007/BF01531068

This paper investigates the complexity of a general inference problem: given a propositional formula in conjunctive normal form, find all prime implications of the formula. Here, a prime implication means a minimal clause whose validity is implied by the validity of the formula. We show that, under some reasonable assumptions, this problem can be solved in time polynomially bounded in the size of the input and in the number of prime implications. In the case of Horn formulae, the result specializes to yield an algorithm whose complexity grows only linearly with the number of prime implications. The result also applies to a class of formulae generalizing both Horn and quadratic formulae.