On disguised double horn functions and extensions
As a natural restriction of disguised Horn functions (i.e., Boolean functions which become Horn after a renaming (change of polarity) of some of the variables), we consider the class C DH R of disguised double Horn functions, i.e., the functions which and whose complement are both disguised Horn. We investigate the syntactical properties of this class and relationship to other classes of Boolean functions. Moreover, we address the extension problem of partially defined Boolean functions (pdBfs) in C DH R, where a pdBf is a function defined on a subset (rather than the full set) of Boolean vectors. We show that the class C DH R coincides with the class C 1–DL of 1-decision lists, and with the intersections of several well-known classes of Boolean functions. Furthermore, polynomial time algorithms for the recognition of a function in C DH R from Horn formulas and other classes of formulas are provided, while the problem is intractable in general. We also present an algorithm for the extension problem which, properly implemented, runs in linear time.
KeywordsBoolean Function Polynomial Time Algorithm Extension Problem Disjunctive Normal Form Prime Implicant
Unable to display preview. Download preview PDF.
- 1.H. Aizenstein, T. Hegedüs, L. Hellerstein and L. Pitt, Complexity theoretic hardness results for query learning, to appear in Journal of Complexity.Google Scholar
- 3.M. Anthony and N. Biggs, Computational Learning Theory, Cambridge University Press, 1992.Google Scholar
- 6.E. Boros, T. Ibaraki and K. Makino, Error-free and best-fit extensions of partially defined Boolean functions, RUTCOR RRR 14-95, Rutgers University, 1995. Information and Computation, to appear.Google Scholar
- 7.S. Ceri, G. Gottlob, L. Tanca, Logic Programming and Databases, Springer, 1990.Google Scholar
- 11.T. Eiter, T. Ibaraki, and K. Makino, Multi-Face Horn Functions, CD-TR 96/95, CD Lab for Expert Systems, TU Vienna, Austria, iii + 97 pages, 1996.Google Scholar
- 12.T. Eiter, T. Ibaraki, and K. Makino, Double Horn functions, RUTCOR Research Report RRR 18–97, Rutgers University 1997; to appear in Information and Computation.Google Scholar
- 13.T. Eiter, T. Ibaraki, and K. Makino, Two-face Horn extensions, to appear in Proceedings of ISAAC'97, Springer LNCS.Google Scholar
- 16.M. Golumbic, P.L. Hammer, P. Hansen, and T. Ibaraki (eds), Horn Logic, search and satisfiability, Annals of Mathematics and Artificial Intelligence 1, 1990.Google Scholar
- 20.K. Makino, K. Hatanaka and T. Ibaraki, Horn extensions of a partially defined Boolean function, RUTCOR RRR 27-95, Rutgers University, 1995.Google Scholar
- 21.S. Muroga, Threshold Logic and Its Applications, Wiley-Interscience, 1971.Google Scholar
- 22.R. L. Rivest, Learning decision lists, Machine Learning, 2:229–246, 1996.Google Scholar