A given Boolean function has its input distributed among many parties. The aim is to determine which parties to tMk to and what information to exchange with each of them in order to evaluate the function while minimizing the total communication. This paper shows that it is possible to obtain the Boolean answer deterministically with only a polynomial increase in communication with respect to the information lower bound given by the nondeterministic communication complexity of the function.

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. --- BQP is low for PP, i.e., PP BQP = PP. --- There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. --- There exists a relativized world where BQP does not have complete sets. --- There exists a relativized world where P = BQP but P 6= UP " coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

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

Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.

PRAMs also approximate the situation where communication to and from shared memory is much more expensive than local operations, for example, where each processor is located on a separate chip and access to shared memory is through a combining network. Not surprisingly, abstract PRAMs can be much more powerful than restricted instruction set PRAMs. THEOREM 21.16 Any function of n variables can be computed by an abstract EROW PRAM in O(log n) steps using n= log 2 n processors and n=2 log 2 n shared memory cells. PROOF Each processor begins by reading log 2 n input values and combining them into one large value. The information known by processors are combined in a binary-tree-like fashion. In each round, the remaining processors are grouped into pairs. In each pair, one processor communicates the information it knows about the input to the other processor and then leaves the computation. After dlog 2 ne rounds, one processor knows all n input values. Then this processor computes th...

We introduce symmetric perfect generic sets. These sets vary from the usual generic sets by allowing limited infinite encoding into the oracle. We then show that the Berman-Hartmanis isomorphism conjecture [BH77] holds relative to any sp-generic oracle, i.e., for any symmetric perfect generic set A, all NP^A-complete sets are polynomial-time isomorphic relative to A. Prior to this work there were no known oracles relative to which the isomorphism conjecture held. As part of our proof that the isomorphism conjecture holds relative to symmetric perfect generic sets we also show that P A = FewP A for any symmetric perfect generic A.

This technical point will prevent the reader from suspecting any measure-theoretic fallacy.

Two extensions of the Linial, Mansour, Nisan AC 0 learning algorithm are presented. The LMN method works when input examples are drawn uniformly. The new algorithms improve on theirs by performing well when given inputs drawn from unknown, mutually independent distributions. A variant of the one of the algorithms is conjectured to work in an even broader setting. 1