Abstract.Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal to the kth level of the difference hierarchy over E~. We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to level k, then PH collapses to (P~P_ l)_tt) NP, the class of sets recognized in polynomial time with k-1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has

In this paper, it is shown that many natural counting classes, such as PP, C=P, and MOD#, are at least as computationally hard as PH (the polynomial-time hierarchy) in the following sense: for each K of the counting classes above, every set in K(PH) is polynomial-time randomized many-one reducible to a set in K with two-sided exponentially small error probability. As a consequence of the result, we see that all the counting classes above are computationally harder than PH unless PH collapses to a finite level. Some other consequences are also shown. 'The research reported here was done while the authors visited the

The study of the complexity of sets encompasses two complementary aims: (1) establishing-usually via explicit construction of algorithms-that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP-complete sets and the PSPACE-complete sets). For the study of the complexity of closure properties, a recent flurry of results [GNW90, Ogi90,Tod90b,BHW91,BRS,FR] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynomialtime operation. Previously, no property-natural or unnatural-had been known to have this behavior. We also prove other natural operations hard for the closure properties of #P, SpanP, OptP, and MidP, and we explore the relative complexity of operations that seem not to be #P-hard, such as maximum, minimum, decrement, and median. Moreover, for each of #P, SpanP, OptP, and MidP, we give a natural complete characterization-in terms of the collapse of complexity classes-i-of the conditions under which that class has every feasible closure property. 1. Let f be a function such that, for some i, f maps from N; to N. We say that f is a closure property (of arity i). 'Note that I(x, y)1 depends only npon Ixl and Iyl.

The study of the complexity of sets encompasses two complementary aims:(1) establishing-usually via explicit construction of algorithms-that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP-complete sets and the PSPACE-complete sets ).For the study of the complexity of closure properties, a recent flurry of results [GNW90, Ogi90,Tod90b,BHW91,BRS,FR] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties.In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynomialtime operation. Previously, no property-natural or unnatural-had been known to have this behavior. We also prove other natural operations hard for the closure properties of #P, SpanP, OptP, and MidP, and we explore the relative complexity of operations that seem not to be #P-hard, such as maximum, minimum, decrement, and median. Moreover, for each of #P, SpanP, OptP, and MidP, we give a natural complete characterization-in terms of the collapse of complexity classes-i-of the conditions under which that class has every feasible closure property.

In this paper, it is shown that many natural counting classes, such as PP, C=P, and MOD#, are at least as computationally hard as PH (the polynomial-time hierarchy) in the following sense: for each K of the counting classes above, every set in K(PH) is polynomial-time randomized many-one reducible to a set in K with two-sided exponentially small error probability. As a consequence of the result, we see that all the counting classes above are computationally harder than PH unless PH collapses to a finite level. Some other consequences are also shown.