ISAAC 1997: Algorithms and Computation pp 112-121

# Two-face horn extensions

• Thomas Eiter
• Toshihide Ibaraki
• Kazuhisa Makino
Session 3B
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)

## Abstract

A partially defined Boolean function (pdBf) (T, F) generalizes a Boolean function, by allowing that the function values on some inputs are unknown, where T, F ⊆ {0,1}n are disjoint sets of true and false vectors, respectively. A pdBf (T, F) often arises in conjunction with data analysis, and in such a case, it is natural and important to decide if (T, F) has an extension f : {0,1}n → {0,1} such that f(v) = 1 (resp., 0) for all χ ε T (resp., χ ε F). It may also be required that such an extension f is Horn, because the false set of a Horn function can be described by Horn rules (whose satisfiability problem can be solved in polynomial time). In this paper, we introduce two interesting restrictions of Horn functions, namely double and bidual Horn functions, because both true and false sets of such functions can be described by Horn-like rules, and furthermore, abduction, one of the basic operation of expert systems, for such functions can be done in polynomial time, whereas abduction for general Horn functions is NP-hard. Our main results show that deciding if a pdBf has extensions of such functions can be done in polynomial time.

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## Authors and Affiliations

• Thomas Eiter
• 1
• Toshihide Ibaraki
• 2
• Kazuhisa Makino
• 3
1. 1.Institut für InformatikUniversität GießenGießenGermany
2. 2.Department of Applied Mathematics and Physics, Graduate School of EngineeringKyoto UniversityKyotoJapan
3. 3.Department of Systems and Human Science, Graduate School of Engineering ScienceOsaka UniversityToyonaka, OsakaJapan