Date: 02 Aug 2001

Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions

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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.