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  1. Transactions on Computation Theory (TOCT)
  2. ACM Transactions on Computation Theory (TOCT) : Volume 9
  3. Issue 1, December 2016
  4. Small Depth Proof Systems
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ACM Transactions on Computation Theory (TOCT) : Volume 9
Issue 1, December 2016
On Hardness of Multilinearization and VNP-Completeness in Characteristic 2
Small Depth Proof Systems
Noncommutative Valiant's Classes: Structure and Complete Problems
Relative Discrepancy Does Not Separate Information and Communication Complexity
Nondeterminism and An Abstract Formulation of Nečiporuk’s Lower Bound Method
ACM Transactions on Computation Theory (TOCT) : Volume 8
ACM Transactions on Computation Theory (TOCT) : Volume 7
ACM Transactions on Computation Theory (TOCT) : Volume 6
ACM Transactions on Computation Theory (TOCT) : Volume 5
ACM Transactions on Computation Theory (TOCT) : Volume 4
ACM Transactions on Computation Theory (TOCT) : Volume 3
ACM Transactions on Computation Theory (TOCT) : Volume 2
ACM Transactions on Computation Theory (TOCT) : Volume 1

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Small Depth Proof Systems

Content Provider ACM Digital Library
Author Sreenivasaiah, Karteek Krebs, Andreas Limaye, Nutan Mahajan, Meena
Copyright Year 2016
Description Author Affiliation: Max Planck Institute for Informatics, Saarbrücken, Germany(University of tübingen, Germany (Krebs, Andreas; The institute of mathematical sciences, Chennai, India (Mahajan, Meena); Indian institute of technology, Bombay, India (Limaye, Nutan); Sreenivasaiah, Karteek))
Abstract A proof system for a language $\textit{L}$ is a function $\textit{f}$ such that $Range(\textit{f})$ is exactly $\textit{L}.$ In this article, we look at proof systems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is proof systems that can be computed by bounded fanin circuits of constant depth $(NC^{0})$ or of $\textit{O}(log log \textit{n})$ depth but with $\textit{O}(1)$ alternations (poly log $AC^{0}).$ Each output bit depends on very few input bits; thus such proof systems correspond to a kind of local error correction on a theorem-proof pair. We identify exactly how much power we need for proof systems to capture all regular languages. We show that all regular languages have poly log $AC^{0}$ proof systems, and from a previous result (Beyersdorff et al. [2011a], where $NC^{0}$ proof systems were first introduced), this is tight. Our technique also shows that Maj has poly log $AC^{0}$ proof system. We explore the question of whether Taut has $NC^{0}$ proof systems. Addressing this question about 2TAUT, and since 2TAUT is closely related to reachability in graphs, we ask the same question about Reachability. We show that if Directed reachability has $NC^{0}$ proof systems, then so does 2TAUT. We then show that both Undirected Reachability and Directed UnReachability have $NC^{0}$ proof systems, but Directed Reachability is still open. In the context of how much power is needed for proof systems for languages in NP, we observe that proof systems for a good fraction of languages in NP do not need the full power of $AC^{0};$ they have $SAC^{0}$ or $coSAC^{0}$ proof systems.
Starting Page 1
Ending Page 26
Page Count 26
File Format PDF
ISSN 19423454
e-ISSN 19423462
DOI 10.1145/2956229
Volume Number 9
Issue Number 1
Journal ACM Transactions on Computation Theory (TOCT)
Language English
Publisher Association for Computing Machinery (ACM)
Publisher Date 2016-10-21
Publisher Place New York
Access Restriction One Nation One Subscription (ONOS)
Subject Keyword Proof complexity Circuit complexity Proof circuits Small depth proofs
Content Type Text
Resource Type Article
Subject Computational Theory and Mathematics Theoretical Computer Science
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