Abstract
In this paper typical properties of large random Boolean AND/OR formulas are investigated. Such formulas with n variables are viewed as rooted binary trees chosen from the uniform distribution of all rooted binary trees with m leaves, where n is fixed and m tends to infinity. The leaves are labeled by literals and the inner nodes by the connectives AND/OR, both uniformly at random. In extending the investigation to infinite trees, we obtain a close relation between the formula size complexity of an arbitrary Boolean function f and the probability of its occurrence under this distribution, i.e., the negative logarithm of this probability differs from the formula size complexity of f only by a polynomial factor.
This research was supported by GA CR, Grant No. 201/95/0976, and by Heinrich-Hertz-Stiftung while visiting Universität Dortmund, FB Informatik, LS II.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
R. B. Boppana: Amplification of Probabilistic Boolean Formulas, Proc. 26th Ann. IEEE Symp. on Foundations of Computer Science, 1985, 20–29.
J. Friedman: Probabilistic Spaces of Boolean Functions of a Given Complexity: Generalities and Random k-SAT Coefficients, CS-TR-387-92, Princeton University, 1992.
A. N. Kolmogorov: Foundations of the Theory of Probability, Chelsea Publishing Company, New York, 1950.
J. B. Paris, A. Vencovská, G. M. Wilmers: A Natural Prior Probability Distribution Derived from the Propositional Calculus, Annals of Pure and Applied Logic 70, 1994, 243–285.
Y. Rabani, Y. Rabinovich and A. Sinclair: A Computational View of Population Genetics, preprint.
Y. Rabinovich, A. Sinclair and A. Wigderson: Quadratic Dynamical Systems, Proc. 33rd Ann. IEEE Symp. on Foundations of Computer Science, 1992, 304–313.
A. A. Razborov: Bounded-depth Formulae over {λ, ⊕ and some Combinatorial Problems. In Complexity of Algorithms and Applied Mathematical Logic (in Russian), Ser. Voprosy Kibernetiky (Problems in Cybernetics), ed.: S. I. Adian, Moscow, 1988, 149–166.
P. Savický: Improved Boolean Formulas for the Ramsey Graphs, to appear in Random Structures and Algorithms.
P. Savický: Complexity and Probability of some Boolean Formulas, TR 538, FB Informatik, Universität Dortmund, 1994.
L. G. Valiant: Short Monotone Formulae for the Majority Function, J. Algorithms 5, 1984, 363–366.
A. Woods, personal communication.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lefmann, H., Savický, P. (1995). Some typical properties of large AND/OR Boolean formulas. In: Wiedermann, J., Hájek, P. (eds) Mathematical Foundations of Computer Science 1995. MFCS 1995. Lecture Notes in Computer Science, vol 969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60246-1_130
Download citation
DOI: https://doi.org/10.1007/3-540-60246-1_130
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60246-0
Online ISBN: 978-3-540-44768-9
eBook Packages: Springer Book Archive