# Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions

• Arfst Nickelsen
• Till Tantau
Conference paper

DOI: 10.1007/3-540-44669-9_29

Part of the Lecture Notes in Computer Science book series (LNCS, volume 2138)
Cite this paper as:
Nickelsen A., Tantau T. (2001) Closure of Polynomial Time Partial Information Classes under Polynomial Time Reductions. In: Freivalds R. (eds) Fundamentals of Computation Theory. FCT 2001. Lecture Notes in Computer Science, vol 2138. Springer, Berlin, Heidelberg

## 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 x1,...,xn a set P of bitstrings, called a pool, such that χA(x1,...,xn) ∈ 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 (x1,... xn) outputs a pool $$\mathcal{P} \in \mathcal{D}$$ with χa(x1,..., xn) ∈ 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.

### Keywords

structural complexity partial information polynomial time reductions verboseness p-selectivity positive reductions