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Documents authored by van Oostrom, Vincent

Document
DOI: 10.4230/LIPIcs.FSCD.2023.22
α-Avoidance

Authors: Samuel Frontull, Georg Moser, and Vincent van Oostrom

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)

Abstract
When substitutions and bindings interact, there is a risk of undesired side effects if the substitution is applied naïvely. The λ-calculus captures this phenomenon concretely, as β-reduction may require the renaming of bound variables to avoid variable capture. In this paper we introduce α-paths as an estimation for α-avoidance, roughly expressing that α-conversions are not required to prevent variable capture. These paths provide a novel method to analyse and predict the potential need for α in different calculi. In particular, we show how α-path characterises α-avoidance for several sub-calculi of the λ-calculus like (i) developments, (ii) affine/linear λ-calculi, (iii) the weak λ-calculus, (iv) μ-unfolding and (iv) finally the safe λ-calculus. Furthermore, we study the unavoidability of α-conversions in untyped and simply-typed λ-calculi and prove undecidability of the need of α-conversions for (leftmost-outermost reductions) in the untyped λ-calculus. To ease the work with α-paths, we have implemented the method and the tool is publicly available.
Cite as

Samuel Frontull, Georg Moser, and Vincent van Oostrom. α-Avoidance. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 22:1-22:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{frontull_et_al:LIPIcs.FSCD.2023.22,
  author =	{Frontull, Samuel and Moser, Georg and van Oostrom, Vincent},
  title =	{{\alpha-Avoidance}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{22:1--22:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.22},
  URN =		{urn:nbn:de:0030-drops-180068},
  doi =		{10.4230/LIPIcs.FSCD.2023.22},
  annote =	{Keywords: \lambda-calculus, variable capture, \alpha-conversion, developments, safe \lambda-calculus, undecidability}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2021.24
Z; Syntax-Free Developments

Authors: Vincent van Oostrom

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)

Abstract
We present the Z-property and instantiate it to various rewrite systems: associativity, positive braids, self-distributivity, the lambda-calculus, lambda-calculi with explicit substitutions, orthogonal TRSs, .... The Z-property is proven equivalent to Takahashi’s angle property by means of a syntax-free notion of development. We show that several classical consequences of having developments such as confluence, normalisation, and recurrence, can be regained in a syntax-free way, and investigate how the notion corresponds to the classical syntactic notion of development in term rewriting.
Cite as

Vincent van Oostrom. Z; Syntax-Free Developments. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 24:1-24:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{vanoostrom:LIPIcs.FSCD.2021.24,
  author =	{van Oostrom, Vincent},
  title =	{{Z; Syntax-Free Developments}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{24:1--24:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.24},
  URN =		{urn:nbn:de:0030-drops-142620},
  doi =		{10.4230/LIPIcs.FSCD.2021.24},
  annote =	{Keywords: rewrite system, confluence, normalisation, recurrence}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2016.32
Normalisation by Random Descent

Authors: Vincent van Oostrom and Yoshihito Toyama

Published in: LIPIcs, Volume 52, 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)

Abstract
We present abstract hyper-normalisation results for strategies. These results are then applied to term rewriting systems, both first and higher-order. For example, we show hyper-normalisation of the left--outer strategy for, what we call, left--outer pattern rewrite systems, a class comprising both Combinatory Logic and the lambda-calculus but also systems with critical pairs. Our results apply to strategies that need not be deterministic but do have Newman's random descent property: all reductions to normal form have the same length, with Huet and Lévy's external strategy being an example. Technically, we base our development on supplementing the usual notion of commutation diagram with a notion of order, expressing that the measure of its right leg does not exceed that of its left leg, where measure is an abstraction of the usual notion of length. We give an exact characterisation of such global commutation diagrams, for pairs of reductions, by means of local ones, for pairs of steps, we dub Dyck diagrams.
Cite as

Vincent van Oostrom and Yoshihito Toyama. Normalisation by Random Descent. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{vanoostrom_et_al:LIPIcs.FSCD.2016.32,
  author =	{van Oostrom, Vincent and Toyama, Yoshihito},
  title =	{{Normalisation by Random Descent}},
  booktitle =	{1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-010-1},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{52},
  editor =	{Kesner, Delia and Pientka, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2016.32},
  URN =		{urn:nbn:de:0030-drops-59862},
  doi =		{10.4230/LIPIcs.FSCD.2016.32},
  annote =	{Keywords: strategy, hyper-normalisation, commutation, random descent}
}
Document
DOI: 10.4230/LIPIcs.RTA.2013.174
Proof Orders for Decreasing Diagrams

Authors: Bertram Felgenhauer and Vincent van Oostrom

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

Abstract
We present and compare some well-founded proof orders for decreasing diagrams. These proof orders order a conversion above another conversion if the latter is obtained by filling any peak in the former by a (locally) decreasing diagram. Therefore each such proof order entails the decreasing diagrams technique for proving confluence. The proof orders differ with respect to monotonicity and complexity. Our results are developed in the setting of involutive monoids. We extend these results to obtain a decreasing diagrams technique for confluence modulo.
Cite as

Bertram Felgenhauer and Vincent van Oostrom. Proof Orders for Decreasing Diagrams. In 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 21, pp. 174-189, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{felgenhauer_et_al:LIPIcs.RTA.2013.174,
  author =	{Felgenhauer, Bertram and van Oostrom, Vincent},
  title =	{{Proof Orders for Decreasing Diagrams}},
  booktitle =	{24th International Conference on Rewriting Techniques and Applications (RTA 2013)},
  pages =	{174--189},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2013.174},
  URN =		{urn:nbn:de:0030-drops-40616},
  doi =		{10.4230/LIPIcs.RTA.2013.174},
  annote =	{Keywords: involutive monoid, confluence modulo, decreasing diagram, proof order}
}
Document
DOI: 10.4230/LIPIcs.RTA.2012.240
Triangulation in Rewriting

Authors: Vincent van Oostrom and Hans Zantema

Published in: LIPIcs, Volume 15, 23rd International Conference on Rewriting Techniques and Applications (RTA'12) (2012)

Abstract
We introduce a process, dubbed triangulation, turning any rewrite relation into a confluent one. It is more direct than usual completion, in the sense that objects connected by a peak are directly related rather than their normal forms. We investigate conditions under which this process preserves desirable properties such as termination.
Cite as

Vincent van Oostrom and Hans Zantema. Triangulation in Rewriting. In 23rd International Conference on Rewriting Techniques and Applications (RTA'12). Leibniz International Proceedings in Informatics (LIPIcs), Volume 15, pp. 240-255, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{vanoostrom_et_al:LIPIcs.RTA.2012.240,
  author =	{van Oostrom, Vincent and Zantema, Hans},
  title =	{{Triangulation in Rewriting}},
  booktitle =	{23rd International Conference on Rewriting Techniques and Applications (RTA'12)},
  pages =	{240--255},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-38-5},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{15},
  editor =	{Tiwari, Ashish},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2012.240},
  URN =		{urn:nbn:de:0030-drops-34964},
  doi =		{10.4230/LIPIcs.RTA.2012.240},
  annote =	{Keywords: triangulation,codeterminism,completion,(co)confluence,(co)termination}
}
Document
DOI: 10.4230/LIPIcs.RTA.2010.5
Realising Optimal Sharing

Authors: Vincent van Oostrom

Published in: LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010)

Abstract
Realising Optimal Sharing
Cite as

Vincent van Oostrom. Realising Optimal Sharing. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications. Leibniz International Proceedings in Informatics (LIPIcs), Volume 6, pp. 5-6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{vanoostrom:LIPIcs.RTA.2010.5,
  author =	{van Oostrom, Vincent},
  title =	{{Realising Optimal Sharing}},
  booktitle =	{Proceedings of the 21st International Conference on Rewriting Techniques and Applications},
  pages =	{5--6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-18-7},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{6},
  editor =	{Lynch, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.5},
  URN =		{urn:nbn:de:0030-drops-26403},
  doi =		{10.4230/LIPIcs.RTA.2010.5},
  annote =	{Keywords: Optimal Sharing}
}
Document
DOI: 10.4230/LIPIcs.RTA.2010.17
Higher-Order (Non-)Modularity

Authors: Claus Appel, Vincent van Oostrom, and Jakob Grue Simonsen

Published in: LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010)

Abstract
We show that, contrary to the situation in first-order term rewriting, almost none of the usual properties of rewriting are modular for higher-order rewriting, irrespective of the higher-order rewriting format. We show that for the particular format of simply typed applicative term rewriting systems modularity of confluence, normalization, and termination can be recovered by imposing suitable linearity constraints.
Cite as

Claus Appel, Vincent van Oostrom, and Jakob Grue Simonsen. Higher-Order (Non-)Modularity. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications. Leibniz International Proceedings in Informatics (LIPIcs), Volume 6, pp. 17-32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{appel_et_al:LIPIcs.RTA.2010.17,
  author =	{Appel, Claus and van Oostrom, Vincent and Simonsen, Jakob Grue},
  title =	{{Higher-Order (Non-)Modularity}},
  booktitle =	{Proceedings of the 21st International Conference on Rewriting Techniques and Applications},
  pages =	{17--32},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-18-7},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{6},
  editor =	{Lynch, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.17},
  URN =		{urn:nbn:de:0030-drops-26427},
  doi =		{10.4230/LIPIcs.RTA.2010.17},
  annote =	{Keywords: Higher-order rewriting, modularity, termination, normalization}
}
Document
DOI: 10.4230/LIPIcs.RTA.2010.85
Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting

Authors: Joerg Endrullis, Clemens Grabmayer, Dimitri Hendriks, Jan Willem Klop, and Vincent van Oostrom

Published in: LIPIcs, Volume 6, Proceedings of the 21st International Conference on Rewriting Techniques and Applications (2010)

Abstract
We present some contributions to the theory of infinitary rewriting for weakly orthogonal term rewrite systems, in which critical pairs may occur provided they are trivial. We show that the infinitary unique normal form property (UNinf) fails by a simple example of a weakly orthogonal TRS with two collapsing rules. By translating this example, we show that UNinf also fails for the infinitary lambda-beta-eta-calculus. As positive results we obtain the following: Infinitary confluence, and hence UNinf, holds for weakly orthogonal TRSs that do not contain collapsing rules. To this end we refine the compression lemma. Furthermore, we consider the triangle and diamond properties for infinitary developments in weakly orthogonal TRSs, by refining an earlier cluster-analysis for the finite case.
Cite as

Joerg Endrullis, Clemens Grabmayer, Dimitri Hendriks, Jan Willem Klop, and Vincent van Oostrom. Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications. Leibniz International Proceedings in Informatics (LIPIcs), Volume 6, pp. 85-102, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{endrullis_et_al:LIPIcs.RTA.2010.85,
  author =	{Endrullis, Joerg and Grabmayer, Clemens and Hendriks, Dimitri and Klop, Jan Willem and van Oostrom, Vincent},
  title =	{{Unique Normal Forms in Infinitary Weakly Orthogonal Rewriting}},
  booktitle =	{Proceedings of the 21st International Conference on Rewriting Techniques and Applications},
  pages =	{85--102},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-18-7},
  ISSN =	{1868-8969},
  year =	{2010},
  volume =	{6},
  editor =	{Lynch, Christopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2010.85},
  URN =		{urn:nbn:de:0030-drops-26469},
  doi =		{10.4230/LIPIcs.RTA.2010.85},
  annote =	{Keywords: Weakly orthogonal term rewrite systems, unique normal form property, infinitary rewriting, infinitary lambda-beta-eta-calculus,}
}
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