On the Shrinkage Exponent for Read-Once Formulae

We prove that the size of any read-once de Morgan formula reduces on average by a factor of at least pα − o(1) when all but a fraction p of the input variables are randomly assigned to {0,1} (here α α llog2(√5 − 1) ≈ 3.27). This resolves in the affirmative a conjecture of Paterson and Zwick. The bound is shown to be tight up to a polylogarithmic factor for all p ⩾ n− 1α.

[1]  Uri Zwick,et al.  Shrinkage of de Morgan formulae under restriction , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[2]  Johan Håstad.  The shrinkage exponent is 2 , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[3]  J. Håstad.  Computational limitations of small-depth circuits , 1987 .

[4]  Ravi B. Boppana,et al.  Amplification of probabilistic boolean formulas , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[5]  Moshe Dubiner,et al.  Amplification and Percolation , 1992, FOCS 1992.

[6]  Leslie G. Valiant,et al.  Short Monotone Formulae for the Majority Function , 1984, J. Algorithms.

[7]  V. M. Khrapchenko.  Method of determining lower bounds for the complexity of P-schemes , 1971 .

[8]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[9]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[10]  Noam Nisan,et al.  The Effect of Random Restrictions on Formula Size , 1993, Random Struct. Algorithms.

[11]  Uri Zwick,et al.  How Do Read-Once Formulae Shrink? , 1994, Combinatorics, Probability and Computing.

[12]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .