Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. In 1975, Simon proved that for unboundederror polynomial-time machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class, BPP path , and, more generally, we study the structural properties of BPP path . We prove that BPP path contains both NP BPP and P NP[log] , and that BPP path is contained in P \Sigma p 2 [log] , BPP NP , and PP. We conclude that, unless the polynomial hierarchy collapses, bounded-error threshold computation is strictly more powerful than bounded-error probabilistic computation. We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length. We show, for both BPP and BPP path , that if there is any database for which this formalization of security differs from the security given by oblivious database access, then P 6= PSPACE. It follows that if any set lacking small circuits can be securely accepted, then P 6= PSPACE.

This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Lof. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to results like: P = NP if and only if there exists an algorithmically random set that is P btt -hard for NP.

Circuit-size complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspace-random oracle A, and relative to almost every oracle A 2 ESPACE.

Via competing provers, we show that if a language A is self-reducible and has polynomial-size circuits then S 2 = S 2 . Building on this, we strengthen the Kamper-- AFK Theorem, namely, we prove that if NP coNP)/poly then the polynomial hierarchy collapses to S 2 . We also strengthen Yap's Theorem, namely, we prove that if NP coNP/poly then the polynomial hierarchy collapses to S 2 . Under the same assumptions, the best previously known collapses were to ZPP respectively ([KW98, BCK 94], building on [KL80, AFK89, Kam91, Yap83]). It is known that S 2 [Cai01]. That result and its relativized version show that our new collapses indeed improve the previously known results. Since the Kamper-- AFK Theorem and Yap's Theorem are used in the literature as bridges in a variety of results---ranging from the study of unique solutions to issues of approximation---our results implicitly strengthen all those results.

We continue the study of robust reductions initiated by Gavald`a and Balc'azar. In particular, a 1991 paper of Gavald`a and Balc'azar [GB91] claimed an optimal separation between the power of robust and nondeterministic strong reductions. Unfortunately, their proof is invalid. We re-establish their theorem. Generalizing robust reductions, we note that robustly strong reductions are built from two restrictions, robust underproductivity and robust overproductivity, both of which have been separately studied before in other contexts. By systematically analyzing the power of these reductions, we explore the extent to which each restriction weakens the power of reductions. We show that one of these reductions yields a new, strong form of the Karp-Lipton Theorem. 1 Introduction The study of the relative power of reductions has long been one of central importance in computational complexity theory. Reductions are the key tools used in complexity theory to compare the difficulty of problems. ...

We study the question of whether every P set has an easy (i.e., polynomialtime computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P 1 ` FP, where #P 1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P PH 1 function can be computed in FP #P #P 1 1 . Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption #P 1 ` FP implies P = BPP and PH ` MOD k P for each k 2, which provides further evidence that not all sets in P have an easy census function. We also relate a set's property of having an easy census function to other well-studied properties of sets, such as rankability and scalability (the closure of the rankable sets under P-isomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be n ff -e...

Cai and Selman [CS99] defined a modification of Levin's notion of average polynomial time and proved, for every P-bi-immune language L and every polynomial-time computable distribution with infinite support, that L is not recognizable in polynomial time on the -average. We call such languages distributionally-hard. Pavan and Selman [PS00] proved that there exist distributionally-hard sets that are not P-bi-immune if and only P contains P-printable-immune sets. We extend this characterizion to include assertions about several traditional questions about immunity, about finding witnesses for NP-machines, and about existence of one-way functions. Similarly, we address the question of whether NP contains sets that are distributionally hard. Several of our results are implications for which we cannot prove whether or not their converse holds. In nearly all such cases we provide oracles relative to which the converse fails.

Although tally sets are generally considered to be weak when used as oracles, it is shown here that in relativizing certain complexity classes, they are in fact no less powerful than any other class of sets and are more powerful than the class of recursive sets. More specifically, relativizations of the classes of P-printable sets and sets with small Generalized Kolmogorov complexity (SGK) are studied. It is shown here that all sparse sets are P TALLY -printable and are in SGK TALLY , and that there are self-P-printable sets that are neither P REC -printable nor in SGK REC . There are also sets that are P REC -printable and in SGK REC that are not self-P-printable. Relativizations to various subrecursive oracles are also presented. A restriction on the number of oracle queries is also presented, with the result that relativizing SGK to any oracle with at most O(log n) queries results in a set that is still in SGK. 1 Introduction Tally sets have generally been considered wea...