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

We distinguish between undefined terms as used in lambda definability
of partial recursive functions and meaningless terms as used in
infinite lambda calculus for the infinitary terms models that
generalise the Bohm model. While there are uncountable many known
sets of meaningless terms, there are four known sets of undefined
terms. Two of these four are sets of meaningless terms.
In this paper we first present set of sufficient conditions for a set
of lambda terms to serve as set of undefined terms in lambda
definability of partial functions. The four known sets of undefined
terms satisfy these conditions.
Next we locate the smallest set of meaningless terms satisfying these
conditions. This set sits very low in the lattice of all sets of
meaningless terms. Any larger set of meaningless terms than this
smallest set is a set of undefined terms. Thus we find uncountably
many new sets of undefined terms.
As an unexpected bonus of our careful analysis of lambda definability
we obtain a natural modification, strict lambda-definability, which
allows for a Barendregt style of proof in which the representation of
composition is truly the composition of representations.

Fer-Jan de Vries. On Undefined and Meaningless in Lambda Definability. In 1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 52, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{devries:LIPIcs.FSCD.2016.18, author = {de Vries, Fer-Jan}, title = {{On Undefined and Meaningless in Lambda Definability}}, booktitle = {1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)}, pages = {18:1--18:15}, 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.2016.18}, URN = {urn:nbn:de:0030-drops-59785}, doi = {10.4230/LIPIcs.FSCD.2016.18}, annote = {Keywords: lambda calculus, lambda definability, partial recursive function, undefined term, meaningless term} }

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**Published in:** LIPIcs, Volume 35, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)

The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible nontermination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekic lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is regarded as a final coalgebra of a many-sorted polynomial functor and can be seen as a limit of finite approximants. As an application, we prove correctness of a generic function that calculates the approximants on a large class of data types.

Alexander Kurz, Alberto Pardo, Daniela Petrisan, Paula Severi, and Fer-Jan de Vries. Approximation of Nested Fixpoints – A Coalgebraic View of Parametric Dataypes. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 205-220, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kurz_et_al:LIPIcs.CALCO.2015.205, author = {Kurz, Alexander and Pardo, Alberto and Petrisan, Daniela and Severi, Paula and de Vries, Fer-Jan}, title = {{Approximation of Nested Fixpoints – A Coalgebraic View of Parametric Dataypes}}, booktitle = {6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)}, pages = {205--220}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-84-2}, ISSN = {1868-8969}, year = {2015}, volume = {35}, editor = {Moss, Lawrence S. and Sobocinski, Pawel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.205}, URN = {urn:nbn:de:0030-drops-55351}, doi = {10.4230/LIPIcs.CALCO.2015.205}, annote = {Keywords: coalgebra, Bekic lemma, infinite data, functional programming, type theory} }

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

In this paper we study meaningless sets in infinitary combinatory logic. So far only a handful of meaningless sets were known. We show that there are uncountably many meaningless sets. As an application to the semantics of finite combinatory logics, we show that there exist uncountably many combinatory algebras that are not a lambda algebra. We also study ways of weakening the axioms of meaningless sets to get, not only sufficient, but also necessary conditions
for having confluence and normalisation.

Paula Severi and Fer-Jan de Vries. Meaningless Sets in Infinitary Combinatory Logic. In 23rd International Conference on Rewriting Techniques and Applications (RTA'12). Leibniz International Proceedings in Informatics (LIPIcs), Volume 15, pp. 288-304, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{severi_et_al:LIPIcs.RTA.2012.288, author = {Severi, Paula and de Vries, Fer-Jan}, title = {{Meaningless Sets in Infinitary Combinatory Logic}}, booktitle = {23rd International Conference on Rewriting Techniques and Applications (RTA'12)}, pages = {288--304}, 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.2012.288}, URN = {urn:nbn:de:0030-drops-34993}, doi = {10.4230/LIPIcs.RTA.2012.288}, annote = {Keywords: Infinitary Rewriting, Combinatory Logic, Meaningless Sets, Confluence, Normalisation} }

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**Published in:** LIPIcs, Volume 10, 22nd International Conference on Rewriting Techniques and Applications (RTA'11) (2011)

In this paper we present a set of necessary and sufficient conditions on a set of lambda terms to serve as the set of meaningless terms in an infinitary bottom extension of lambda calculus.
So far only a set of sufficient conditions was known for choosing a suitable set of meaningless terms to make this construction produce confluent extensions. The conditions covered the three main known examples of sets of meaningless terms. However, the much later construction of many more examples of sets of meaningless terms satisfying the sufficient conditions renewed the interest in the necessity question and led us to reconsider the old conditions.
The key idea in this paper is an alternative solution for solving the overlap between beta reduction and bottom reduction. This allows us to reformulate the Axiom of Overlap, which now determines together
with the other conditions a larger class of sets of meaningless terms. We show that the reformulated conditions are not only sufficient but also necessary for obtaining a confluent and normalizing infinitary lambda beta bottom calculus. As an interesting consequence of the necessity proof we obtain for infinitary lambda calculus with beta and bot reduction that confluence implies normalization.

Paula Severi and Fer-Jan de Vries. Weakening the Axiom of Overlap in Infinitary Lambda Calculus. In 22nd International Conference on Rewriting Techniques and Applications (RTA'11). Leibniz International Proceedings in Informatics (LIPIcs), Volume 10, pp. 313-328, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{severi_et_al:LIPIcs.RTA.2011.313, author = {Severi, Paula and de Vries, Fer-Jan}, title = {{Weakening the Axiom of Overlap in Infinitary Lambda Calculus}}, booktitle = {22nd International Conference on Rewriting Techniques and Applications (RTA'11)}, pages = {313--328}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-30-9}, ISSN = {1868-8969}, year = {2011}, volume = {10}, editor = {Schmidt-Schauss, Manfred}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.RTA.2011.313}, URN = {urn:nbn:de:0030-drops-31312}, doi = {10.4230/LIPIcs.RTA.2011.313}, annote = {Keywords: Infinitary Lambda Calculus, Confluence, Normalization} }

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