Abstract
Consider the problem of identifying min T(f) and max F(f) of a positive (i.e., monotone) Boolean function f, by using membership queries only, where min T(f) (max F(f)) denotes the set of minimal true vectors (maximal false vectors) of f. As the existence of a polynomial total time algorithm (i.e., polynomial time in the length of input and output) for this problem is still open, we consider here a restricted problem: given an unknown positive function f of n variables, decide whether f is 2-monotonic or not, and if f is 2-monotonic, output both min T(f) and max. F(f). For this problem, we propose a simple algorithm, which is based on the concept of maximum latency, and show that it uses O(n 2m) time and O(n 2m) queries, where m=¦ min T(f)¦+¦ max F(f)¦. This answers affirmatively the conjecture raised in [3, 4], and is an improvement over the two algorithms discussed therein: one uses O(n 3m) time and O(n 3m) queries, and the other uses O(nm 2+n2m) time and O(nm) queries.
Preview
Unable to display preview. Download preview PDF.
References
M. Anthony and N. Biggs, Computational Learning Theory, Cambridge University Press, 1992.
J. C. Bioch and T. Ibaraki, Complexity of identification and dualization of positive Boolean functions, RUTCOR Research Report RRR 25-93, Rutgers University, 1993; to appear in Information and Computation.
E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Identifying 2-monotonic positive Boolean functions in polynomial time, ISA'91 Algorithms, edited by W. L. Hsu and R. C. T. Lee, LNCS 557 (1991) 104–115.
E. Boros, P. L. Hammer, T. Ibaraki and K. Kawakami, Polynomial time recognition of 2-monotonic positive Boolean functions given by an oracle, RUTCOR Research Report RRR 10-95, Rutgers University, 1995; to appear in SIAM J. Computing.
Y. Crama, P. L. Hammer and T. Ibaraki, Cause-effect relationships and partially defined boolean functions, Annals of Operations Research, 16 (1988) 299–326.
M. Fredman and L. Khachiyan, On the complexity of dualization of monotone disjunctive normal forms, Technical Report LCSR-TR-225, Department of Computer Science, Rutgers University, 1994.
D. N. Gainanov, On one criterion of the optimality of an algorithm for evaluating monotonic Boolean functions, U.S.S.R. Computational Mathematics and Mathematical Physics, 24 (1984) 176–181.
T. Ibaraki and T. Kameda, A Theory of coteries: Mutual exclusion in distributed systems, IEEE Trans. on Parallel and Distributed Systems, 4 (1993) 779–794.
D. S. Johnson, M. Yannakakis and C. H. Papadimitriou, On generating all maximal independent sets, Information Processing Letters, 27 (1988) 119–123.
K. Makino and T. Ibaraki, The maximum latency and identification of positive Boolean functions, ISAAC'94 Algorithms and Computation, edited by D. Z. Du and X. S. Zhang, LNCS 834 (1994) 324–332.
K. Makino and T. Ibaraki, A fast and simple algorithm for identifying 2-monotonic positive Boolean functions, Technical Report of IEICE, COMP94-46 (1994) 11–20.
S. Muroga, Threshold Logic and Its Applications, Wiley-Interscience, 1971.
J. S. Provan and M. O. Ball, Efficient recognition of matroids and 2-monotonic systems, Applications of Discrete Mathematics, edited by R. Ringeisen and F. Roberts, SIAM, Philadelphia (1988) 122–134.
L. G. Valiant, A theory of the learnable, Communications of the ACM, 27 (1984) 1134–1142.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Makino, K., Ibaraki, T. (1995). A fast and simple algorithm for identifying 2-monotonic positive Boolean functions. In: Staples, J., Eades, P., Katoh, N., Moffat, A. (eds) Algorithms and Computations. ISAAC 1995. Lecture Notes in Computer Science, vol 1004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015434
Download citation
DOI: https://doi.org/10.1007/BFb0015434
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60573-7
Online ISBN: 978-3-540-47766-2
eBook Packages: Springer Book Archive