Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions

Abstract

Polynomial time partial information classes are extensions of the class P of languages decidable in polynomial time. A partial information algorithm for a language A computes, for fixed n ∈ ℕ, on input of words x ₁,...,x _n a set P of bitstrings, called a pool, such that χA(x ₁,...,x _n ) ∈ P, where P is chosen from a family $\mathcal{D}$ of pools. A language A is in $P\left[ \mathcal{D} \right]$, if there is a polynomial time partial information algorithm which for all inputs (x ₁,... x _n ) outputs a pool $\mathcal{P} \in \mathcal{D}$ with χa(x ₁,..., x _n ) ∈ P. Many extensions of P studied in the literature, including approximable languages, cheatability, p-selectivity and frequency computations, form a class $P\left[ \mathcal{D} \right]$ for an appropriate family $\mathcal{D}$.

We characterise those families $\mathcal{D}$ for which $P\left[ \mathcal{D} \right]$ is closed under certain polynomial time reductions, namely bounded truth-table, truth-table, and Turing reductions. We also treat positive reductions. A class $P\left[ \mathcal{D} \right]$ is presented which strictly contains the class P-sel of p-selective languages and is closed under positive truth-table reductions.