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Mining the Surface: Witnessing the Low Complexity Theorems of Arithmetic

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Logic, Language, Information, and Computation (WoLLIC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13468))

Abstract

One of the elegant achievements in the history of proof theory is the characterization of the provably total recursive functions of an arithmetical theory by its proof-theoretic ordinal as a way to measure the time complexity of the functions. Unfortunately, the characterization is not sufficiently fine-grained to capture the subclass of bounded functions with a bounded definition. In this paper, we provide a refined characterization to capture the feasibly definable and bounded functions that are provably total in Peano arithmetic. Roughly speaking, we identify the functions as the ones that are computable by a sequence of \(\textrm{PV}\)-provable polynomial time modifications on an initial polynomial time value, where the computational steps are indexed by the ordinals below \(\epsilon _0\), decreasing by the modifications. The machinery we provide is quite general and applicable to any theory for which there is a natural ordinal analysis. However, for the sake of brevity, we limit ourselves only to Peano arithmetic and leave the general case to the future work.

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Acknowledgements

We are grateful to Pavel Pudlák and Neil Thapen to put the connection between the total search problems and the theories of arithmetic into our attention. We are also thankful for the fruitful discussions we had. The support by the FWF project P 33548 is also gratefully acknowledged.

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Authors and Affiliations

  1. University of Groningen, Bernoulliborg, Nijenborgh 9, 9747 AG, Groningen, The Netherlands

    Amirhossein Akbar Tabatabai

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  1. Amirhossein Akbar Tabatabai
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Correspondence to Amirhossein Akbar Tabatabai .

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Editors and Affiliations

  1. Technische Universität Wien, Vienna, Austria

    Agata Ciabattoni

  2. University College London, London, UK

    Elaine Pimentel

  3. Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil

    Ruy J. G. B. de Queiroz

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Akbar Tabatabai, A. (2022). Mining the Surface: Witnessing the Low Complexity Theorems of Arithmetic. In: Ciabattoni, A., Pimentel, E., de Queiroz, R.J.G.B. (eds) Logic, Language, Information, and Computation. WoLLIC 2022. Lecture Notes in Computer Science, vol 13468. Springer, Cham. https://doi.org/10.1007/978-3-031-15298-6_24

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  • DOI: https://doi.org/10.1007/978-3-031-15298-6_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-15297-9

  • Online ISBN: 978-3-031-15298-6

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