Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-19T03:04:11.810Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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