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Documents authored by Hetzl, Stefan

Document
DOI: 10.4230/LIPIcs.CSL.2015.110
Tree Grammars for the Elimination of Non-prenex Cuts

Authors: Stefan Hetzl and Sebastian Zivota

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Abstract
Recently a new connection between proof theory and formal language theory was introduced. It was shown that the operation of cut elimination for proofs with prenex Pi_1-cuts in classical first-order logic corresponds to computing the language of a particular type of tree grammars. The present paper extends this connection to arbitrary (i.e. non-prenex) cuts without quantifier alternations. The key to treating non-prenex cuts lies in using a new class of tree grammars, constraint grammars, which describe the relationship of the applicability of its productions by a propositional formula.
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Stefan Hetzl and Sebastian Zivota. Tree Grammars for the Elimination of Non-prenex Cuts. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 110-127, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{hetzl_et_al:LIPIcs.CSL.2015.110,
  author =	{Hetzl, Stefan and Zivota, Sebastian},
  title =	{{Tree Grammars for the Elimination of Non-prenex Cuts}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{110--127},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.110},
  URN =		{urn:nbn:de:0030-drops-54103},
  doi =		{10.4230/LIPIcs.CSL.2015.110},
  annote =	{Keywords: proof theory, cut-elimination, Herbrand's theorem, tree grammars}
}
Document
DOI: 10.4230/LIPIcs.TLCA.2015.1
Herbrand Disjunctions, Cut Elimination and Context-Free Tree Grammars

Authors: Bahareh Afshari, Stefan Hetzl, and Graham E. Leigh

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)

Abstract
Recently a new connection between proof theory and formal language theory was introduced. It was shown that the operation of cut elimination for proofs in first-order predicate logic involving Pi_1-cuts corresponds to computing the language of a particular class of regular tree grammars. The present paper expands this connection to the level of Pi_2-cuts. Given a proof pi of a Sigma_1 formula with cuts only on Pi_2 formulæ, we show there is associated to pi a natural context-free tree grammar whose language is finite and yields a Herbrand disjunction for pi.
Cite as

Bahareh Afshari, Stefan Hetzl, and Graham E. Leigh. Herbrand Disjunctions, Cut Elimination and Context-Free Tree Grammars. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{afshari_et_al:LIPIcs.TLCA.2015.1,
  author =	{Afshari, Bahareh and Hetzl, Stefan and Leigh, Graham E.},
  title =	{{Herbrand Disjunctions, Cut Elimination and Context-Free Tree Grammars}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{1--16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.1},
  URN =		{urn:nbn:de:0030-drops-51516},
  doi =		{10.4230/LIPIcs.TLCA.2015.1},
  annote =	{Keywords: Classical logic, Context-free grammars, Cut elimination, First-order logic, Herbrand's theorem, Proof theory}
}
Document
DOI: 10.4230/LIPIcs.CSL.2012.183
A Systematic Approach to Canonicity in the Classical Sequent Calculus

Authors: Kaustuv Chaudhuri, Stefan Hetzl, and Dale Miller

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Abstract
The sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps-such as instantiating a block of quantifiers-by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means of abstracting away the details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that make the foci as parallel as possible are canonical. Moreover, such proofs are isomorphic to expansion proofs - a well known, minimalistic, and parallel generalization of Herbrand disjunctions - for classical first-order logic. This technique is a systematic way to recover the desired essence of any sequent proof without abandoning the sequent calculus.
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Kaustuv Chaudhuri, Stefan Hetzl, and Dale Miller. A Systematic Approach to Canonicity in the Classical Sequent Calculus. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 183-197, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{chaudhuri_et_al:LIPIcs.CSL.2012.183,
  author =	{Chaudhuri, Kaustuv and Hetzl, Stefan and Miller, Dale},
  title =	{{A Systematic Approach to Canonicity in the Classical Sequent Calculus}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{183--197},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.183},
  URN =		{urn:nbn:de:0030-drops-36723},
  doi =		{10.4230/LIPIcs.CSL.2012.183},
  annote =	{Keywords: Sequent Calculus, Canonicity, Classical Logic, Expansion Trees}
}
Document
DOI: 10.4230/LIPIcs.CSL.2012.320
Herbrand-Confluence for Cut Elimination in Classical First Order Logic

Authors: Stefan Hetzl and Lutz Straßburger

Published in: LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

Abstract
We consider cut-elimination in the sequent calculus for classical first-order logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cut-free proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbrand-disjunction of a cut-free proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the non-erasing reduction lead to the same Herbrand-disjunction.
Cite as

Stefan Hetzl and Lutz Straßburger. Herbrand-Confluence for Cut Elimination in Classical First Order Logic. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 320-334, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{hetzl_et_al:LIPIcs.CSL.2012.320,
  author =	{Hetzl, Stefan and Stra{\ss}burger, Lutz},
  title =	{{Herbrand-Confluence for Cut Elimination in Classical First Order Logic}},
  booktitle =	{Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL},
  pages =	{320--334},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-42-2},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{16},
  editor =	{C\'{e}gielski, Patrick and Durand, Arnaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.320},
  URN =		{urn:nbn:de:0030-drops-36815},
  doi =		{10.4230/LIPIcs.CSL.2012.320},
  annote =	{Keywords: proof theory, first-order logic, tree languages, term rewriting, semantics of proofs}
}
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