A Complexity Theory for Feasible Closure Properties

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.