Shrinkage of De Morgan Formulae from Quantum Query Complexity

We give a new and improved proof that the shrinkage exponent of De Morgan formulae is 2. Namely, we show that for any Boolean function f : {−1, 1} → {−1, 1}, setting each variable out of x1, . . . , xn with probability 1 − p to a randomly chosen constant, reduces the expected formula size of the function by a factor of O(p). This result is tight and improves the work of H̊astad [H̊as98] by removing logarithmic factors. As a consequence of our results, the function defined by Andreev [And87], A : {−1, 1} → {−1, 1}, which is in P, has formula size at least Ω( n 3 log2 n log3 logn ). This lower bound is tight up to the log log n factor, and is the best known lower bound for functions in P. In addition, the functions defined in [KRT13], hr : {−1, 1} → {−1, 1}, which are also in P, cannot be computed correctly on a fraction greater than 1/2 + 2−r of the inputs, by De Morgan formulae of size at most n 3 r2poly logn , for any parameter r ≤ n . The proof relies on a result from quantum query complexity by [LLS06, HLS07, Rei11]: for any Boolean function f , Q2(f) ≤ O( √ L(f)), where Q2(f) is the bounded-error quantum query complexity of f , and L(f) is the minimal size De Morgan formula computing f . ∗Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. [email protected]. Supported by an Adams Fellowship of the Israel Academy of Sciences and Humanities, by an ISF grant and by the I-CORE Program of the Planning and Budgeting Committee. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 48 (2014)