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DOI: 10.4230/LIPIcs.FSCD.2024.18

Representation of Peano Arithmetic in Separation Logic

Authors: Sohei Ito and Makoto Tatsuta

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Separation logic is successful for software verification of heap-manipulating programs. Numbers are necessary to be added to separation logic for verification of practical software where numbers are important. However, properties of the validity such as decidability and complexity for separation logic with numbers have not been fully studied yet. This paper presents the translation of Pi-0-1 formulas in Peano arithmetic to formulas in a small fragment of separation logic with numbers, which consists only of the intuitionistic points-to predicate, 0 and the successor function. Then this paper proves that a formula in Peano arithmetic is valid in the standard model if and only if its translation in this fragment is valid in the standard interpretation. As a corollary, this paper also gives a perspective proof for the undecidability of the validity in this fragment. Since Pi-0-1 formulas can describe consistency of logical systems and non-termination of computations, this result also shows that these properties discussed in Peano arithmetic can also be discussed in such a small fragment of separation logic with numbers.

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Sohei Ito and Makoto Tatsuta. Representation of Peano Arithmetic in Separation Logic. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ito_et_al:LIPIcs.FSCD.2024.18,
  author =	{Ito, Sohei and Tatsuta, Makoto},
  title =	{{Representation of Peano Arithmetic in Separation Logic}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{18:1--18:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.18},
  URN =		{urn:nbn:de:0030-drops-203476},
  doi =		{10.4230/LIPIcs.FSCD.2024.18},
  annote =	{Keywords: First order logic, Separation logic, Peano arithmetic, Presburger arithmetic}
}

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DOI: 10.4230/LIPIcs.CSL.2011.538

Non-Commutative Infinitary Peano Arithmetic

Authors: Makoto Tatsuta and Stefano Berardi

Published in: LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Abstract

Does there exist any sequent calculus such that it is a subclassical logic and it becomes classical logic when the exchange rules are added? The first contribution of this paper is answering this question for infinitary Peano arithmetic. This paper defines infinitary Peano arithmetic with non-commutative sequents, called non-commutative infinitary Peano arithmetic, so that the system becomes equivalent to Peano arithmetic with the omega-rule if the the exchange rule is added to this system. This system is unique among other non-commutative systems, since all the logical connectives have standard meaning and specifically the commutativity for conjunction and disjunction is derivable. This paper shows that the provability in non-commutative infinitary Peano arithmetic is equivalent to Heyting arithmetic with the recursive omega rule and the law of excluded middle for Sigma-0-1 formulas. Thus, non-commutative infinitary Peano arithmetic is shown to be a subclassical logic. The cut elimination theorem in this system is also proved. The second contribution of this paper is introducing infinitary Peano arithmetic having antecedent-grouping and no right exchange rules. The first contribution of this paper is achieved through this system. This system is obtained from the positive fragment of infinitary Peano arithmetic without the exchange rules by extending it from a positive fragment to a full system, preserving its 1-backtracking game semantics. This paper shows that this system is equivalent to both non-commutative infinitary Peano arithmetic, and Heyting arithmetic with the recursive omega rule and the Sigma-0-1 excluded middle.

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Makoto Tatsuta and Stefano Berardi. Non-Commutative Infinitary Peano Arithmetic. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 538-552, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{tatsuta_et_al:LIPIcs.CSL.2011.538,
  author =	{Tatsuta, Makoto and Berardi, Stefano},
  title =	{{Non-Commutative Infinitary Peano Arithmetic}},
  booktitle =	{Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL},
  pages =	{538--552},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-32-3},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{12},
  editor =	{Bezem, Marc},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.538},
  URN =		{urn:nbn:de:0030-drops-32551},
  doi =		{10.4230/LIPIcs.CSL.2011.538},
  annote =	{Keywords: proof theory, cut elimination, intuitionistic logic, infinitary logic, recursive omega rules, substructural logic}
}

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Representation of Peano Arithmetic in Separation Logic

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Non-Commutative Infinitary Peano Arithmetic

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