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

As shown by Tsukada and Ong, simply-typed, normal and η-long resource terms correspond to plays in Hyland-Ong games, quotiented by Melliès' homotopy equivalence. Though inspiring, their proof is indirect, relying on the injectivity of the relational model {w.r.t.} both sides of the correspondence - in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization.
In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step - and our third contribution - is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential λ-calculus.

Lison Blondeau-Patissier, Pierre Clairambault, and Lionel Vaux Auclair. Strategies as Resource Terms, and Their Categorical Semantics. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 13:1-13:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blondeaupatissier_et_al:LIPIcs.FSCD.2023.13, author = {Blondeau-Patissier, Lison and Clairambault, Pierre and Vaux Auclair, Lionel}, title = {{Strategies as Resource Terms, and Their Categorical Semantics}}, booktitle = {8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)}, pages = {13:1--13:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.13}, URN = {urn:nbn:de:0030-drops-179976}, doi = {10.4230/LIPIcs.FSCD.2023.13}, annote = {Keywords: Resource calculus, Game semantics, Categorical semantics} }

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

In asynchronous games, Melliès proved that innocent strategies are positional: their behaviour only depends on the position, not the temporal order used to reach it. This insightful result shaped our understanding of the link between dynamic (i.e. game) and static (i.e. relational) semantics.
In this paper, we investigate the positionality of innocent strategies in the traditional setting of Hyland-Ong-Nickau-Coquand pointer games. We show that though innocent strategies are not positional, total finite innocent strategies still enjoy a key consequence of positionality, namely positional injectivity: they are entirely determined by their positions. Unfortunately, this does not hold in general: we show a counter-example if finiteness and totality are lifted. For finite partial strategies we leave the problem open; we show however the partial result that two strategies with the same positions must have the same P-views of maximal length.

Lison Blondeau-Patissier and Pierre Clairambault. Positional Injectivity for Innocent Strategies. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 17:1-17:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blondeaupatissier_et_al:LIPIcs.FSCD.2021.17, author = {Blondeau-Patissier, Lison and Clairambault, Pierre}, title = {{Positional Injectivity for Innocent Strategies}}, booktitle = {6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)}, pages = {17:1--17:22}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.17}, URN = {urn:nbn:de:0030-drops-142555}, doi = {10.4230/LIPIcs.FSCD.2021.17}, annote = {Keywords: Game Semantics, Innocence, Relational Semantics, Positionality} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We investigate the expressive power of higher-order recursion schemes (HORS) restricted to linear types. Two formalisms are considered: multiplicative additive HORS (MAHORS), which feature both linear function types and products, and multiplicative HORS (MHORS), based on linear function types only.
For MAHORS, we establish an equi-expressivity result with a variant of tree-stack automata. Consequently, we can show that MAHORS are strictly more expressive than first-order HORS, that they are incomparable with second-order HORS, and that the associated branch languages lie at the third level of the collapsible pushdown hierarchy.
In the multiplicative case, we show that MHORS are equivalent to a special kind of pushdown automata. It follows that any MHORS can be translated to an equivalent first-order MHORS in polynomial time. Further, we show that MHORS generate regular trees and can be translated to equivalent order-0 HORS in exponential time. Consequently, MHORS turn out to have the same expressive power as 0-HORS but they can be exponentially more concise.
Our results are obtained through a combination of techniques from game semantics, the geometry of interaction and automata theory.

Pierre Clairambault and Andrzej S. Murawski. On the Expressivity of Linear Recursion Schemes. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{clairambault_et_al:LIPIcs.MFCS.2019.50, author = {Clairambault, Pierre and Murawski, Andrzej S.}, title = {{On the Expressivity of Linear Recursion Schemes}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {50:1--50:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.50}, URN = {urn:nbn:de:0030-drops-109945}, doi = {10.4230/LIPIcs.MFCS.2019.50}, annote = {Keywords: higher-order recursion schemes, linear logic, game semantics, geometry of interaction} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

Herbrand's theorem, widely regarded as a cornerstone of proof theory, exposes some of the constructive content of classical logic. In its simplest form, it reduces the validity of a first-order purely existential formula to that of a finite disjunction. In the general case, it reduces first-order validity to propositional validity, by understanding the structure of the assignment of first-order terms to existential quantifiers, and the causal dependency between quantifiers.
In this paper, we show that Herbrand's theorem in its general form can be elegantly stated and proved as a theorem in the framework of concurrent games, a denotational semantics designed to faithfully represent causality and independence in concurrent systems, thereby exposing the concurrency underlying the computational content of classical proofs. The causal structure of concurrent strategies, paired with annotations by first-order terms, is used to specify the dependency between quantifiers implicit in proofs. Furthermore concurrent strategies can be composed, yielding a compositional proof of Herbrand's theorem, simply by interpreting classical sequent proofs in a well-chosen denotational model.

Aurore Alcolei, Pierre Clairambault, Martin Hyland, and Glynn Winskel. The True Concurrency of Herbrand's Theorem. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 5:1-5:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{alcolei_et_al:LIPIcs.CSL.2018.5, author = {Alcolei, Aurore and Clairambault, Pierre and Hyland, Martin and Winskel, Glynn}, title = {{The True Concurrency of Herbrand's Theorem}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {5:1--5:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.5}, URN = {urn:nbn:de:0030-drops-96723}, doi = {10.4230/LIPIcs.CSL.2018.5}, annote = {Keywords: Herbrand's theorem, Game semantics, True concurrency} }

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**Published in:** LIPIcs, Volume 119, 27th EACSL Annual Conference on Computer Science Logic (CSL 2018)

We compare three models of the probabilistic lambda-calculus: the probabilistic Böhm trees of Leventis, the probabilistic concurrent games of Winskel et al., and the weighted relational model of Ehrhard et al. Probabilistic Böhm trees and probabilistic strategies are shown to be related by a precise correspondence theorem, in the spirit of existing work for the pure lambda-calculus. Using Leventis' theorem (probabilistic Böhm trees characterise observational equivalence), we derive a full abstraction result for the games model. Then, we relate probabilistic strategies to the weighted relational model, using an interpretation-preserving functor from the former to the latter. We obtain that the relational model is also fully abstract.

Pierre Clairambault and Hugo Paquet. Fully Abstract Models of the Probabilistic lambda-calculus. In 27th EACSL Annual Conference on Computer Science Logic (CSL 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 119, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{clairambault_et_al:LIPIcs.CSL.2018.16, author = {Clairambault, Pierre and Paquet, Hugo}, title = {{Fully Abstract Models of the Probabilistic lambda-calculus}}, booktitle = {27th EACSL Annual Conference on Computer Science Logic (CSL 2018)}, pages = {16:1--16:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-088-0}, ISSN = {1868-8969}, year = {2018}, volume = {119}, editor = {Ghica, Dan R. and Jung, Achim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2018.16}, URN = {urn:nbn:de:0030-drops-96835}, doi = {10.4230/LIPIcs.CSL.2018.16}, annote = {Keywords: Game Semantics, Lambda-calculus, Probabilistic programming, Relational model, Full abstraction} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Distributed/concurrent strategies have been introduced as special maps of event structures. As such they factor through their "rigid images," themselves strategies. By concentrating on such "rigid image" strategies we are able to give an elementary account of distributed strategies and their composition, resulting in a category of games and strategies. This is in contrast to the usual development where composition involves the pullback of event structures explicitly and results in a bicategory. It is shown how, in this simpler setting, to extend strategies to probabilistic strategies; and indicated how through probability we can track nondeterministic branching behaviour, that one might otherwise think lost irrevocably in restricting attention to "rigid image" strategies.

Simon Castellan, Pierre Clairambault, and Glynn Winskel. Distributed Strategies Made Easy. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 81:1-81:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{castellan_et_al:LIPIcs.MFCS.2017.81, author = {Castellan, Simon and Clairambault, Pierre and Winskel, Glynn}, title = {{Distributed Strategies Made Easy}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {81:1--81:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.81}, URN = {urn:nbn:de:0030-drops-81315}, doi = {10.4230/LIPIcs.MFCS.2017.81}, annote = {Keywords: Games, Strategies, Event Structures, Probability} }

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**Published in:** LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

Although Plotkin’s parallel-or is inherently deterministic, it has a non-deterministic interpretation in games based on (prime) event structures - in which an event has a unique causal history - because they do not directly support disjunctive causality. General event structures can express disjunctive causality and have a more permissive notion of determinism, but do not support hiding. We show that (structures equivalent to) deterministic general event structures do support hiding, and construct a new category of games based on them with a deterministic interpretation of aPCFpor, an affine variant of PCF extended with parallel-or. We then exploit this deterministic interpretation to give a relaxed notion of determinism (observable determinism) on the plain event structures model. Putting this together with our previously introduced concurrent notions of well-bracketing and innocence, we obtain an intensionally fully abstract model of aPCFpor.

Simon Castellan, Pierre Clairambault, and Glynn Winskel. Observably Deterministic Concurrent Strategies and Intensional Full Abstraction for Parallel-or. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 12:1-12:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{castellan_et_al:LIPIcs.FSCD.2017.12, author = {Castellan, Simon and Clairambault, Pierre and Winskel, Glynn}, title = {{Observably Deterministic Concurrent Strategies and Intensional Full Abstraction for Parallel-or}}, booktitle = {2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)}, pages = {12:1--12:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-047-7}, ISSN = {1868-8969}, year = {2017}, volume = {84}, editor = {Miller, Dale}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.12}, URN = {urn:nbn:de:0030-drops-77219}, doi = {10.4230/LIPIcs.FSCD.2017.12}, annote = {Keywords: Game semantics, parallel-or, concurrent games, event structures, full abstraction} }

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**Published in:** LIPIcs, Volume 59, 27th International Conference on Concurrency Theory (CONCUR 2016)

We investigate relationships between interleaving and causal notions of game semantics for concurrent programming languages, focusing on the existence of canonical compact causal representations of the interleaving game semantics of programs.
We perform our study on an affine variant of Idealized Parallel Algol (IPA), for which we present two games model: and interleaving model (an adaptation of Ghica and Murawski’s fully abstract games model for IPA up to may-testing), and a causal model (a variant of Rideau and Winskel’s games on event structures). Both models are sound and adequate for affine IPA. Then, we relate the two models. First we give a causality-forgetting operation mapping functorially the causal model to the interleaving one. We show that from an interleaving strategy we can reconstruct a causal strategy, from which it follows that the interleaving model is the observational quotient of the causal one. Then, we investigate several reconstructions of causal strategies from interleaving ones, showing finally that there are programs which are inherently causally ambiguous, with several distinct minimal causal representations.

Simon Castellan and Pierre Clairambault. Causality vs. Interleavings in Concurrent Game Semantics. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 32:1-32:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{castellan_et_al:LIPIcs.CONCUR.2016.32, author = {Castellan, Simon and Clairambault, Pierre}, title = {{Causality vs. Interleavings in Concurrent Game Semantics}}, booktitle = {27th International Conference on Concurrency Theory (CONCUR 2016)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-017-0}, ISSN = {1868-8969}, year = {2016}, volume = {59}, editor = {Desharnais, Jos\'{e}e and Jagadeesan, Radha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2016.32}, URN = {urn:nbn:de:0030-drops-61620}, doi = {10.4230/LIPIcs.CONCUR.2016.32}, annote = {Keywords: Game semantics, concurrency, causality, event structures} }

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

We show that a version of Martin-Löf type theory with extensional identity, a unit type N1, Sigma, Pi, and a base type is a free category with families (supporting these type formers) both in a 1- and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We then show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic.

Simon Castellan, Pierre Clairambault, and Peter Dybjer. Undecidability of Equality in the Free Locally Cartesian Closed Category. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 138-152, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)

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@InProceedings{castellan_et_al:LIPIcs.TLCA.2015.138, author = {Castellan, Simon and Clairambault, Pierre and Dybjer, Peter}, title = {{Undecidability of Equality in the Free Locally Cartesian Closed Category}}, booktitle = {13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)}, pages = {138--152}, 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.2015.138}, URN = {urn:nbn:de:0030-drops-51602}, doi = {10.4230/LIPIcs.TLCA.2015.138}, annote = {Keywords: Extensional type theory, locally cartesian closed categories, undecidab- ility} }

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**Published in:** LIPIcs, Volume 24, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)

Higher-order recursive schemes (HORS) are schematic representations of functional programs. They generate possibly infinite ranked labelled trees and, in that respect, are known to be equivalent to a restricted fragment of the lambda-Y-calculus consisting of ground-type terms whose free variables have types of the form o -> ... -> o (with o being a special case).
In this paper, we show that any lambda-Y-term (with no restrictions on term type or the types of free variables) can actually be represented by a HORS. More precisely, for any lambda-Y-term M, there exists a HORS generating a tree that faithfully represents M's (eta-long) Böhm tree. In particular, the HORS captures higher-order binding information contained in the Böhm tree. An analogous result holds for finitary PCF.
As a consequence, we can reduce a variety of problems related to the
lambda-Y-calculus or finitary PCF to problems concerning higher-order
recursive schemes. For instance, Böhm tree equivalence can be reduced
to the equivalence problem for HORS. Our results also enable MSO
model-checking of Böhm trees, despite the general undecidability of
the problem.

Pierre Clairambault and Andrzej S. Murawski. Böhm Trees as Higher-Order Recursive Schemes. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 24, pp. 91-102, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2013)

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@InProceedings{clairambault_et_al:LIPIcs.FSTTCS.2013.91, author = {Clairambault, Pierre and Murawski, Andrzej S.}, title = {{B\"{o}hm Trees as Higher-Order Recursive Schemes}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {91--102}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.91}, URN = {urn:nbn:de:0030-drops-43644}, doi = {10.4230/LIPIcs.FSTTCS.2013.91}, annote = {Keywords: Lambda calculus, B\"{o}hm trees, Recursion Schemes} }

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