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How Do Read-Once Formulae Shrink?

Published online by Cambridge University Press:  12 September 2008

Moshe Dubiner  and
Uri Zwick
Moshe Dubiner
Affiliation:
The School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv university, Tel Aviv 69978, Israel e-mail: zwick@math.tau.ac.il
Uri Zwick
Affiliation:
The School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv university, Tel Aviv 69978, Israel e-mail: zwick@math.tau.ac.il
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Abstract

Let f be a de Morgan read-once function of n variables. Let fε be the random restriction obtained by independently assigning to each variable of f, the value 0 with probability (1 -ε)/2, the value 1 with the same probability, and leaving it unassigned with probability ε. We show that fε depends, on the average, on only O(εαn + εn1/α) variables, where . This result is asymptotically the tightest possible. It improves a similar result obtained recently by Håstad, Razborov and Yao.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 4 , December 1994 , pp. 455 - 469
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Boppana, R. B. (1989) Amplification of probabilistic Boolean formulas. Advances in Computer Research, Vol. 5: Randomness and Computation, JAI Press, Greenwich, Conn.Google Scholar
[2]Boppana, R. B. and Sipser, M. (1990) The complexity of finite functions. In: Handbook of Theoretical Computer Science, Vol. A, North-Holland 757–804.Google Scholar
[3]Dubiner, M. and Zwick, U. (1992) Amplification and Percolation. Proc. 33rd FOCS, Pittsburgh258–267.Google Scholar
[4]Dubiner, M. and Zwick, U. (submitted) Amplification by read-once formulae.Google Scholar
[5]Dunne, P. E. (1988) The complexity of Boolean networks, Academic Press.Google Scholar
[6]Furst, M., Saxe, J. B. and Sipser, M. (1984) Parity, circuits and the polynomial time hierarchy. Math. Syst. Theory 17 13–27.CrossRefGoogle Scholar
[7]Håstad, J. (1986) Computational limitations on Small Depth Circuits, PhD thesis, MIT.Google Scholar
[8]Håstad, J. (1993) The shrinkage exponent is 2. Proc. 34nd FOCS, Palo Alto 114–123.Google Scholar
[9]Håstad, J., Razborov, A. and Yao, A. (submitted) On the Shrinkage Exponent for Read-Once Formulae.Google Scholar
[10]Karchmer, M. and Wigderson, A. (1990) Monotone circuits for connectivity require super-logarithmic depth. SIAM J. Disc. Math. 3 718–727.CrossRefGoogle Scholar
[11]Moore, E. F. and Shannon, C. E. (1956) Reliable circuits using less reliable relays. Journal of the Franklin Institute 262 191–208, 281–297.Google Scholar
[12]Nisan, N. and Impagliazzo, R. (1993) The effect of random restrictions on formulae size. Random Structures & algorithms 4 121–133.Google Scholar
[13]Paterson, M. S. and Zwick, U. (1993) Shrinkage of de Morgan formulae under restriction. Random Structures & algorithms 4 135–150.Google Scholar
[14]Valiant, L. G. (1984) Short monotone formulae for the majority function. J. Algorithms 5 363–366.CrossRefGoogle Scholar
[15]Wegener, I., (1987) The complexity of Boolean functions, Wiley–Teubner Series in Computer Science.Google Scholar
[16]Yao, A. (1985) Separating the polynomial-time hierarchy by oracles. Proc. 26th FOCS, Portland1–10.Google Scholar
[17]Аhдрееb, A. E. (1987) O меtoде пoлyчеhия бoлее чеm kbaдpatичhыx hижhиx oцehok для cлoжhoctи π-cxеm. bеcmnun МΔY, cep. ℳameℳ u ℳexan. 42 (1) 70–73. (Andreev, A. E. (1987) On a method for obtaining more than quadratic effective lower bounds for the complexity of π-schemes. Moscow Univ. Math. Bull. 42 (1) 63–66.)Google Scholar
[18]Cyббotobckя, Б. A. (1961) O peaлизции лиhheйыx φyhkций φopmyлamи b бaзиce &, v,–. ДAH CCCP 136 (3) 553–555. (Subbotovskaya, B. A. (1961) Realizations of linear functions by formulas using +,*,−. Soviet Mathematics Doklady 2 110–112.)Google Scholar
[19]Xpaпчеhko, B. M. (1971) O cлoжhoctи peaлизaци лиheйhoй φyhkции b kлacce Π-cxem. Mameℳ. зaℳemku 9 (1) 35–40. (Khrapchenko, V. M. (1971) Complexity of the realization of a linear function in the class of π-circuits. Math. Notes Acad. Sciences USSR 9 21–23.)Google Scholar
[20]Xpaпчеhko, B. M. (1971) Oб oдhom metoде пoлyчеhия hижhиx oцеhok cлoжhoctи П-cxem. Mameℓ. зaℓemku 10 (1) 83–92. (Khrapchenko, V. M. (1971) A method of determining lower bounds for the complexity of Π-schemes. Math. Notes Acad. Sciences USSR 10 474–479.)Google Scholar