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The function claw #P lacks a crucial closure property:it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We show that mast previously studied counting classes, including PP, C,P, and ModkP, are "gapdefinable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gapdefinable class, SPP, which is still large enough to contain Few. We also show that SPP consists of exactly thoee languages low for GapP, and thus SPP languages are low for any gapdefinable class. These results unify and improve earlier disparate results of Beigel & Gill [3], Cai & Hemachandra [5], and Kobler, et al. [9]. We show further that any countable collection of languages is contained in a unique minimum gapdefinable class, which implies that the g a p definable classes form a lattice under inclusion. Subtraction seems neceessry for this result, since nothing similar is known for the #P-definable classes.

Berman and Hartmanis BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young JY85] gave a structural de nition of a class of NP-complete sets|the k-creative sets|and de ned a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman{Hartmanis and Joseph{Young conjectures cannot both be correct.We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scrambling functions, we show that much more powerful one-way functions|the annihilating functions|exist relative to a random oracle.

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This paper deals with forcing in arithmetic (as first introduced by Feferman [2]) and its connections with recursive function theory. We define for each n ≥ 1 the class of weakly n-generic sets. We prove that these classes merge with the classes of n-generic sets to form the hierarchy suggested by the terminology. Our notation is the same as that of Jockusch [5].