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DOI: 10.4230/LIPIcs.CSL.2016.34

Semantics for "Enough-Certainty" and Fitting's Embedding of Classical Logic in S4

Authors: Gergei Bana and Mitsuhiro Okada

Published in: LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)

Abstract

In this work we look at how Fitting's embedding of first-order classical logic into first-order S4 can help in reasoning when we are interested in satisfaction "in most cases", when first-order properties are allowed to fail in cases that are considered insignificant. We extend classical semantics by combining a Kripke-style model construction of "significant" events as possible worlds with the forcing-Fitting-style semantics construction by embedding classical logic into S4. We provide various examples. Our main running example is an application to symbolic security protocol verification with complexity-theoretic guarantees. In particular, we show how Fitting's embedding emerges entirely naturally when verifying trace properties in computer security.

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Gergei Bana and Mitsuhiro Okada. Semantics for "Enough-Certainty" and Fitting's Embedding of Classical Logic in S4. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 34:1-34:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bana_et_al:LIPIcs.CSL.2016.34,
  author =	{Bana, Gergei and Okada, Mitsuhiro},
  title =	{{Semantics for "Enough-Certainty" and Fitting's Embedding of Classical Logic in S4}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{34:1--34:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Talbot, Jean-Marc and Regnier, Laurent},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.34},
  URN =		{urn:nbn:de:0030-drops-65746},
  doi =		{10.4230/LIPIcs.CSL.2016.34},
  annote =	{Keywords: first-order logic, possible-world semantics, Fitting embedding, asymptotic probabilities, verification of complexity-theoretic properties}
}

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Invited Talk

DOI: 10.4230/LIPIcs.RTA.2013.4

Husserl and Hilbert on Completeness and Husserl's Term Rewrite-based Theory of Multiplicity (Invited Talk)

Authors: Mitsuhiro Okada

Published in: LIPIcs, Volume 21, 24th International Conference on Rewriting Techniques and Applications (RTA 2013)

Abstract

Hilbert and Husserl presented axiomatic arithmetic theories in different ways and proposed two different notions of 'completeness' for arithmetic, at the turning of the 20th Century (1900-1901). The former led to the completion axiom, the latter completion of rewriting. We look into the latter in comparison with the former. The key notion to understand the latter is the notion of definite multiplicity or manifold (Mannigfaltigkeit). We show that his notion of multiplicity is understood by means of term rewrite theory in a very coherent manner, and that his notion of 'definite' multiplicity is understood as the relational web (or tissue) structure, the core part of which is a 'convergent' term rewrite proof structure. We examine how Husserl introduced his term rewrite theory in 1901 in the context of a controversy with Hilbert on the notion of completeness, and in the context of solving the justification problem of the use of imaginaries in mathematics, which was an important issue in the foundations of mathematics in the period.

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Mitsuhiro Okada. Husserl and Hilbert on Completeness and Husserl's Term Rewrite-based Theory of Multiplicity (Invited Talk). In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 4-19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{okada:LIPIcs.RTA.2013.4,
  author =	{Okada, Mitsuhiro},
  title =	{{Husserl and Hilbert on Completeness and Husserl's Term Rewrite-based Theory of Multiplicity}},
  booktitle =	{24th International Conference on Rewriting Techniques and Applications (RTA 2013)},
  pages =	{4--19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-53-8},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{21},
  editor =	{van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.4},
  URN =		{urn:nbn:de:0030-drops-40524},
  doi =		{10.4230/LIPIcs.RTA.2013.4},
  annote =	{Keywords: History of term rewrite theory, Husserl, Hilbert, proof theory, Knuth-Bendix completion}
}

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Documents authored by Okada, Mitsuhiro

Semantics for "Enough-Certainty" and Fitting's Embedding of Classical Logic in S4

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Husserl and Hilbert on Completeness and Husserl's Term Rewrite-based Theory of Multiplicity (Invited Talk)

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