research-article

Improved witnessing and local improvement principles for second-order bounded arithmetic

Authors:

Arnold Beckmann,

Samuel R. BussAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 15, Issue 1

Article No.: 2, Pages 1 - 35

https://doi.org/10.1145/2559950

Published: 06 March 2014 Publication History

Abstract

This article concerns the second-order systems U¹₂ and V¹₂ of bounded arithmetic, which have proof-theoretic strengths corresponding to polynomial-space and exponential-time computation. We formulate improved witnessing theorems for these two theories by using S¹₂ as a base theory for proving the correctness of the polynomial-space or exponential-time witnessing functions. We develop the theory of nondeterministic polynomial-space computation, including Savitch's theorem, in U¹₂. Kołodziejczyk et al. [2011] have introduced local improvement properties to characterize the provably total NP functions of these second-order theories. We show that the strengths of their local improvement principles over U¹₂ and V¹₂ depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U¹₂ proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically-many rounds is complete for the provably total NP search problems of V¹₂. Related results are obtained for local improvement principles with one improvement round and for local improvement over rectangular grids.

References

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Arnold Beckmann and Samuel R. Buss. 2009. Polynomial local search in the polynomial hierarchy and witnessing in fragments of bounded arithmetic. J. Math. Logic 9, 1, 103--138.

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Samuel R. Buss. 1986. Bounded Arithmetic. Bibliopolis, Naples. Ph.D.

Digital Library

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Cited By

Jain SLi JRobere RXun Z(2024)On Pigeonhole Principles and Ramsey in TFNP2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00033(406-428)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00033
Akbar Tabatabai A(2024)Witnessing flows in arithmeticMathematical Structures in Computer Science10.1017/S0960129524000185(1-37)Online publication date: 20-Sep-2024
https://doi.org/10.1017/S0960129524000185
Kołodziejczyk LThapen N(2022)Approximate counting and NP search problemsJournal of Mathematical Logic10.1142/S021906132250012X22:03Online publication date: 22-Jun-2022
https://doi.org/10.1142/S021906132250012X
Show More Cited By

Index Terms

Improved witnessing and local improvement principles for second-order bounded arithmetic
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Theorem proving algorithms
2. Theory of computation
  1. Logic
    1. Proof theory
  2. Models of computation
    1. Computability

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cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 15, Issue 1

February 2014

279 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2590829

Issue’s Table of Contents

Copyright © 2014 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 06 March 2014

Accepted: 01 February 2013

Received: 01 March 2012

Published in TOCL Volume 15, Issue 1

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Cited By

Jain SLi JRobere RXun Z(2024)On Pigeonhole Principles and Ramsey in TFNP2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00033(406-428)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00033
Akbar Tabatabai A(2024)Witnessing flows in arithmeticMathematical Structures in Computer Science10.1017/S0960129524000185(1-37)Online publication date: 20-Sep-2024
https://doi.org/10.1017/S0960129524000185
Kołodziejczyk LThapen N(2022)Approximate counting and NP search problemsJournal of Mathematical Logic10.1142/S021906132250012X22:03Online publication date: 22-Jun-2022
https://doi.org/10.1142/S021906132250012X
Akbar Tabatabai A(2022)Mining the Surface: Witnessing the Low Complexity Theorems of ArithmeticLogic, Language, Information, and Computation10.1007/978-3-031-15298-6_24(378-394)Online publication date: 9-Sep-2022
https://doi.org/10.1007/978-3-031-15298-6_24
Krajíček J(2019)Proof Complexity10.1017/9781108242066Online publication date: 25-Mar-2019
https://doi.org/10.1017/9781108242066
Buss S(2019)Quasipolynomial size proofs of the propositional pigeonhole principleTheoretical Computer Science10.1016/j.tcs.2015.02.005576:C(77-84)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.1016/j.tcs.2015.02.005
Goldberg PPapadimitriou C(2018)Towards a unified complexity theory of total functionsJournal of Computer and System Sciences10.1016/j.jcss.2017.12.00394(167-192)Online publication date: Jun-2018
https://doi.org/10.1016/j.jcss.2017.12.003
Aisenberg JBonet MBuss SCrăciun AIstrate G(2018)Short proofs of the Kneser–Lovász coloring principleInformation and Computation10.1016/j.ic.2018.02.010261(296-310)Online publication date: Aug-2018
https://doi.org/10.1016/j.ic.2018.02.010
Beckmann ABuss S(2017)The NP Search Problems of Frege and Extended Frege ProofsACM Transactions on Computational Logic10.1145/306014518:2(1-19)Online publication date: 2-Jun-2017
https://dl.acm.org/doi/10.1145/3060145
Beckmann ARazafindrakoto J(2017)Total Search Problems in Bounded Arithmetic and Improved WitnessingLogic, Language, Information, and Computation10.1007/978-3-662-55386-2_3(31-47)Online publication date: 29-Jun-2017
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