We show that MA EXP , the exponential time version of the Merlin-Arthur class, does not have polynomial size circuits. This significantly improves the previous known result due to Kannan since we furthermore show that our result does not relativize. This is the first separation result in complexity theory that does not relativize. As a corollary to our separation result we also obtain that PEXP, the exponential time version of PP is not in P=poly.

The correlation between two Boolean functions of n inputs is de ned as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper, we compute, in closed form, the correlation between any t wo symmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean function having an odd period has an exponentially small correlation (in n) with the parity function. This improves a result of Smolensky 12] restricted to symmetric Boolean functions: the correlation between parity a n d a n y circuit consisting of a M o d q gate over AND-gates of small fan-in, where q is odd and the function computed by the sum of the AND-gates is symmetric, is bounded by 2 ; (n) .In addition, we nd that for a large class of symmetric functions the correlation with parity i s identically zero for in nitely many n. W e c haracterize exactly those symmetric Boolean functions having this property.

No abstract

We show that the bipartite perfect matching problem is in quasi-NC 2 . That is, it has uniform circuits of quasi-polynomial size n O(log n) , and O(log 2 n) depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

A modular query consists of asking how many (modulo m) of k strings belong to a fixed NP language. Modular queries provide a form of restricted access to an NP oracle. For each k and m, we consider the class of languages accepted by NP machines that ask a single modular query. Han and Thierauf [HT95] showed that these classes coincide with levels of the Boolean hierarchy when m is even or k ≤ 2m, and they determined the exact levels. Until now, the remaining case -odd m and large k -looked quite difficult. We pinpoint the level in the Boolean hierarchy for the remaining case; thus, these classes coincide with levels of the Boolean hierarchy for every k and m.In addition we characterize the classes obtained by using an NP(l) acceptor in place of an NP acceptor (NP(l) is the lth level of the Boolean hierarchy). As before, these all coincide with levels in the Boolean hierarchy.