Kolmogorov complexity and degrees of tally sets
type: Item
instance of: scholarly article
title: Kolmogorov complexity and degrees of tally sets
zbMATH Open document ID: 0704.03023
DOI: 10.1016/0890-5401(90)90052-J
author: Osamu Watanabe
author: Eric Allender
published in: Information and Computation
publication date: 1990-01-01T00:00:00Z
review text: This paper is related to the questions left open by \textit{S. Tang} and \textit{R. Book} [Lect. Notes Comput. Sci. 317, 591--599 (1988; Zbl 0658.03025)]: Is \(E^p_m(TALLY)=E^p_{1-tt}(TALLY)?\) Is \(E^p_m(TALLY)=E^p_{btt}(TALLY)?\) Is there some \(k\) such that \(E^p_{k-tt}(TALLY)=E^p_{k+1-tt}(TALLY)?\) The authors introduce the notions of \(NE\) predicate (a binary predicate \(R\) such that, for some nondeterministic Turing machine \(M\) which runs in time \(2^{O(n)}\), \(R(x,y) \Leftrightarrow y\) is an accepting computation for \(x\) on \(M\)) and \(E\)-solvability of an \(NE\) predicate and shows that: \(E=E^{NP}\Rightarrow\) every \(NE\) predicate is \(E\)-solvable \(\Rightarrow\) \(E=NE\). Some characterizations of \(E\)-solvability of \(NE\) predicates are given, and the following main results are proved: 1. If all \(NE\) predicates are \(E\)-solvable, then \(E^p_{btt}(TALLY)=E^p_m(TALLY)\). 2. If not all \(NE\) predicates are \(E\)-solvable, then \(E^p_m(TALLY)\subset E^p_{1-tt}(TALLY)\). 3. If not all \(NE\) predicates are \(E\)-solvable, then for all \(k\), \(E^p_{k-tt}(TALLY)\subset E^p_{k+1-tt}(TALLY)\). The techniques used in the proofs are those provided by the machinery of generalized Kolmogorov complexity.
reviewed by: Gheorghe Grigoraş
Mathematics Subject Classification ID: 03D15
Mathematics Subject Classification ID: 68Q15
Mathematics Subject Classification ID: 68Q30
zbMATH DE Number: 4154425
zbMATH Keywords: tally set
zbMATH Keywords: polynomial time
zbMATH Keywords: exponential time
zbMATH Keywords: NE predicate
zbMATH Keywords: generalized Kolmogorov complexity
MaRDI profile type: Publication
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full work available at URL: 0890-5401(90)90052-j
OpenAlex ID: W2090493738
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description: scientific article; zbMATH DE number 4154425