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Documents authored by Endrullis, Jörg

Document
DOI: 10.4230/LIPIcs.FSCD.2018.14
Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems

Authors: Jörg Endrullis, Jan Willem Klop, and Roy Overbeek

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Abstract
Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract reduction systems, it is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps -> with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract reduction systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. We also show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Finally, as a background theme, we discuss the logical issue of first-order definability of the notion of confluence.
Cite as

Jörg Endrullis, Jan Willem Klop, and Roy Overbeek. Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{endrullis_et_al:LIPIcs.FSCD.2018.14,
  author =	{Endrullis, J\"{o}rg and Klop, Jan Willem and Overbeek, Roy},
  title =	{{Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.14},
  URN =		{urn:nbn:de:0030-drops-91848},
  doi =		{10.4230/LIPIcs.FSCD.2018.14},
  annote =	{Keywords: confluence, decreasing diagrams, weak diamond property}
}
Document
DOI: 10.4230/LIPIcs.RTA.2015.143
A Coinductive Framework for Infinitary Rewriting and Equational Reasoning

Authors: Jörg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, and Alexandra Silva

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)

Abstract
We present a coinductive framework for defining infinitary analogues of equational reasoning and rewriting in a uniform way. The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.
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Jörg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, and Alexandra Silva. A Coinductive Framework for Infinitary Rewriting and Equational Reasoning. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 143-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{endrullis_et_al:LIPIcs.RTA.2015.143,
  author =	{Endrullis, J\"{o}rg and Hansen, Helle Hvid and Hendriks, Dimitri and Polonsky, Andrew and Silva, Alexandra},
  title =	{{A Coinductive Framework for Infinitary Rewriting and Equational Reasoning}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{143--159},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.143},
  URN =		{urn:nbn:de:0030-drops-51949},
  doi =		{10.4230/LIPIcs.RTA.2015.143},
  annote =	{Keywords: infinitary rewriting, coinduction}
}
Document
DOI: 10.4230/LIPIcs.RTA.2015.160
Proving non-termination by finite automata

Authors: Jörg Endrullis and Hans Zantema

Published in: LIPIcs, Volume 36, 26th International Conference on Rewriting Techniques and Applications (RTA 2015)

Abstract
A new technique is presented to prove non-termination of term rewriting. The basic idea is to find a non-empty regular language of terms that is closed under rewriting and does not contain normal forms. It is automated by representing the language by a tree automaton with a fixed number of states, and expressing the mentioned requirements in a SAT formula. Satisfiability of this formula implies non-termination. Our approach succeeds for many examples where all earlier techniques fail, for instance for the S-rule from combinatory logic.
Cite as

Jörg Endrullis and Hans Zantema. Proving non-termination by finite automata. In 26th International Conference on Rewriting Techniques and Applications (RTA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 36, pp. 160-176, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{endrullis_et_al:LIPIcs.RTA.2015.160,
  author =	{Endrullis, J\"{o}rg and Zantema, Hans},
  title =	{{Proving non-termination by finite automata}},
  booktitle =	{26th International Conference on Rewriting Techniques and Applications (RTA 2015)},
  pages =	{160--176},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-85-9},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{36},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2015.160},
  URN =		{urn:nbn:de:0030-drops-51952},
  doi =		{10.4230/LIPIcs.RTA.2015.160},
  annote =	{Keywords: non-termination, finite automata, regular languages}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2011.16
Infinitary Rewriting Coinductively

Authors: Jörg Endrullis and Andrew Polonsky

Published in: LIPIcs, Volume 19, 18th International Workshop on Types for Proofs and Programs (TYPES 2011)

Abstract
We provide a coinductive definition of strongly convergent reductions between infinite lambda terms. This approach avoids the notions of ordinals and metric convergence which have appeared in the earlier definitions of the concept. As an illustration, we prove the existence part of the infinitary standardization theorem. The proof is fully formalized in Coq using coinductive types. The paper concludes with a characterization of infinite lambda terms which reduce to themselves in a single beta step.
Cite as

Jörg Endrullis and Andrew Polonsky. Infinitary Rewriting Coinductively. In 18th International Workshop on Types for Proofs and Programs (TYPES 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 19, pp. 16-27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{endrullis_et_al:LIPIcs.TYPES.2011.16,
  author =	{Endrullis, J\"{o}rg and Polonsky, Andrew},
  title =	{{Infinitary Rewriting Coinductively}},
  booktitle =	{18th International Workshop on Types for Proofs and Programs (TYPES 2011)},
  pages =	{16--27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-49-1},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{19},
  editor =	{Danielsson, Nils Anders and Nordstr\"{o}m, Bengt},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2011.16},
  URN =		{urn:nbn:de:0030-drops-38971},
  doi =		{10.4230/LIPIcs.TYPES.2011.16},
  annote =	{Keywords: infinitary rewriting, coinduction, lambda calculus, standardization}
}
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