This paper is motivated by the questions: what are the \Sigma

We show that if SAT does not have small circuits, then there must exist a small number of formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P , then the polynomial-time hierarchy collapses to S 2 2 . Even showing that the hierarchy collapsed to 2 remained open prior to this paper.

In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...

: During the past decade, nine papers have obtained increasingly strong consequences from the assumption that boolean or bounded-query hierarchies collapse. The final four papers of this nine-paper progression actually achieve downward collapse---that is, they show that high-level collapses induce collapses at (what beforehand were thought to be) lower complexity levels. For example, for each k 2 it is now known that if P \Sigma p k [1] = P \Sigma p k [2] then PH = \Sigma p k . This article surveys the history, the results, and the technique---the so-called easy-hard method---of these nine papers. 1. J. Kadin. The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing, 17(6):1263-1282, 1988. Erratum appears in the same journal, 20(2):404. 2. K. Wagner. Number-of-query hierarchies. Technical Report 158, Universitat Augsburg, Institut fur Mathematik, Augsburg, Germany, October 1987. 3. K. Wagner. Number-of-query hierarchies. Technica...

.<F3.858e+05> We study the e#ect of query order on computational power and show that P<F2.809e+05> BH<F2.571e+05> j<F2.809e+05> [1]:BH<F2.571e+05> k<F2.809e+05> [1]<F3.858e+05> ---the languages computable via a polynomial-time machine given one query to the<F3.511e+05><F3.858e+05> jth level of the boolean hierarchy followed by one query to the<F3.511e+05><F3.858e+05> kth level of the boolean hierarchy---equals R<F2.233e+05> p<F2.809e+05><F2.233e+05><F2.86e+05><F2.809e+05><F3.321e+05> j+2k-1-tt<F3.858e+05> (NP) if<F3.511e+05> j<F3.858e+05> is even and<F3.511e+05> k<F3.858e+05> is odd and equals R<F2.233e+05> p<F2.809e+05><F2.233e+05><F3.321e+05> j+2k-tt<F3.858e+05> (NP) otherwise. Thus unless the polynomial hierarchy collapses it holds that, for each 1<F4.135e+05> #<F3.511e+05> j<F4.135e+05> #<F3.511e+05><F3.858e+05> k, P<F2.809e+05> BH<F2.571e+05> j<F2.809e+05> [1]:BH<F2.571e+05> k<F2.809e+05> [1]<F3.858e+05> = P<F2.809e+05> BH<F2.571e+05> k<F2.809e+05> [1]:BH<F2.571e+05> ...

This paper investigates nondeterministic bounded query classes in relation to the complexity of NP-hard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NP-hard approximation problems. The results in this paper take advantage of this machine-based model to prove that in many cases, NP-approximation problems have the upward collapse property. That is, a reduction between NP-approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level. For example, if MaxClique reduces to (log n)-approximating MaxClique using many-one reductions, then the Traveling Salesman Problem (TSP) is equivalent to MaxClique under many-one reductions. Several upward collapse theorems are presented in this paper. The proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes. In fact, these results depend on a surprising connection between the Boolean Hierarchy and nondeterministic bounded query classes.

Downward translation of equality refers to cases where a collapse of some pair of complexity classes would induce a collapse of some other pair of complexity classes that (a priori) one expects are smaller. Recently, the first downward translation of equality was obtained that applied to the polynomial hierarchy---in particular, to bounded access to its levels [HHH97a]. In this paper, we provide a much broader downward translation that subsumes not only that downward translation but also that translation's elegant enhancement by Buhrman and Fortnow [BF96]. Our work also sheds light on previous research on the structure of refined polynomial hierarchies [Sel95, Sel94]. 1 Introduction Does the collapse of low-complexity classes imply the collapse of higher-complexity classes? Does the collapse of high-complexity classes imply the collapse of lower-complexity classes? These questions---known respectively as downward and upward translation of equality---have long been central topics in co...