On polynomially D verbose sets

Abstract

A general framework is presente for the study of complexity classes that are defined via polynomial time algorithms that compute partial information about the characteristic function of a given input. Given n ∈ N and a family D of sets D⊂-{0,1}^*, a language A is polynomially D-verbose (or: A ∈ P [D]) iff there is a polynomial time algorithm that on input (x ₁,..., x _n ) outputs a D ∈ D such that the characteristic string χa(x ₁,..., x _n ) is in D. Also the variant where only pairwise distinct input words are allowed is studied. p-selective sets, p-verbose sets, easily p-countable sets, sets that allow a polynomial time frequency computation, and cheatable sets are special cases of this definition. It is shown that it suffices to study families that are in a certain normal form. An algorithm is presented that decides for given families D ₁, D ₂ whether P [D ₁] ⊂-P [D ₂]. The classes P [D] are, except for trivial cases, not closed under union, intersection or join. The classes closed under complement are characterized, as well as those closed under ≤ ^p _m - and ≤ ^p_1−tt -reductions. For a given family of sets D the class of polynomially D-verbose languages contains non-recursive languages iff it contains all p-selective languages. The families D for which a D-verbose set can be non-recursive are fully characterized by a simple combinatorial property. It is also shown that for fixed n the classes form a distributive lattice. A diagram that shows this lattice for n=2 is presented.