We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes.

help us decide where and how to put our efforts into solving We show that MAEXP, the exponential time version of problems in complexity theory. It is still true that virtually the Merlin-Arthur class, does not have polynomial size cir- all of the theorems in computational complexity theory that cuits. This significantly improves the previous known result have reasonable relativizations do relativize (see [For94]). due to Kannan since we furthermore show that our result But we do have a small number of exceptions that arise does not relativize. This is the first separation result in com- from the area of interactive proofs. These results have preplexity theory that does not relativize. As a corollary to our viously always taken the form of collapses such as IP= separation result we also obtain that PEXP, the exponen- PSPACE [LFKN92, Sha92], MIP=NEXP [BFL91] and tial time version of PP is not in P=poly. PCP(O(1);O(logn))=NP [ALM+92]. In this paper we give the first reasonable nonrel-1

We give the first nontrivial model-independent time-space tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general random-access nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for log-space uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomial-time hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...

The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unrelativized case. Although this paper is not the first to provide a counterexample to the Random Oracle Hypothesis, it does provide a most compelling counterexample by showing that for almost all oracles A, IP A 6= PSPACE A . If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, co-NP A 6` IP A . These results extend to the multi-prover proof systems of Ben-Or, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the definition. A class IPP, similar to IP, is defined. Surprisingly, the IPP = PSPACE result holds for all oracle worlds. 1 Department of Computer Science, Cornell University, Ithaca, NY 14853, U.S.A...