Regular Article
Double Horn Functions,☆☆

https://doi.org/10.1006/inco.1998.2713Get rights and content
Under an Elsevier user license
open archive

Abstract

In this paper, we define double Horn functions, which are the Boolean functionsfsuch that bothfand its complement (i.e., negation)fare Horn, and investigate their semantical and computational properties. Double Horn functions embody a balanced treatment of positive and negative information in the course of the extension problem of partially defined Boolean functions (pdBfs), where a pdBf is a pair (T, F) of disjoint setsT, F⊆{0, 1}nof true and false vectors, respectively, and an extension of (T, F) is a Boolean functionfthat is compatible withTandF. We derive syntactic and semantic characterizations of double Horn functions, and determine the number of such functions. The characterizations are then exploited to give polynomial time algorithms (i) that recognize double Horn functions from Horn DNFs (disjunctive normal forms), and (ii) that compute the prime DNF from an arbitrary formula, as well as its complement and its dual. Furthermore, we consider the problem of determining a double Horn extension of a given pdBf. We describe a polynomial time algorithm for this problem and moreover an algorithm that enumerates all double Horn extensions of a pdBf with polynomial delay. However, finding a shortest double Horn extension (in terms of the size of a formulaϕrepresenting it) is shown to be intractable.

Cited by (0)

The major part of this research was conducted while the first author visited Kyoto University in 1995, by the support of the Scientific Grant in Aid by the Ministry of Education, Science and Culture of Japan (Grant 06044112).

☆☆

K. W. Ng

f1

E-mail: [email protected], [email protected], [email protected]