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Volume

LIPIcs, Volume 69

21st International Conference on Types for Proofs and Programs (TYPES 2015)

TYPES 2015, May 18-21, 2015, Tallinn, Estonia

Editors: Tarmo Uustalu

Document

Invited Talk

DOI: 10.4230/LIPIcs.FSCD.2024.1

Meaningfulness and Genericity in a Subsuming Framework (Invited Talk)

Authors: Delia Kesner, Victor Arrial, and Giulio Guerrieri

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

This paper studies the notion of meaningfulness for a unifying framework called dBang-calculus, which subsumes both call-by-name (dCBN) and call-by-value (dCBV). We first define meaningfulness in dBang and then characterize it by means of typability and inhabitation in an associated non-idempotent intersection type system previously appearing in the literature. We validate the proposed notion of meaningfulness by showing two properties: (1) consistency of the smallest theory, called ℋ, equating all meaningless terms, and (2) genericity, stating that meaningless subterms have no bearing on the significance of meaningful terms. The theory ℋ is also shown to have a unique consistent and maximal extension ℋ*, which coincides with a well-known notion of observational equivalence. Last but not least, we show that the notions of meaningfulness and genericity in the literature for dCBN and dCBV are subsumed by the corresponding ones proposed here for the dBang-calculus.

Cite as

Delia Kesner, Victor Arrial, and Giulio Guerrieri. Meaningfulness and Genericity in a Subsuming Framework (Invited Talk). In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kesner_et_al:LIPIcs.FSCD.2024.1,
  author =	{Kesner, Delia and Arrial, Victor and Guerrieri, Giulio},
  title =	{{Meaningfulness and Genericity in a Subsuming Framework}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{1:1--1:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.1},
  URN =		{urn:nbn:de:0030-drops-203305},
  doi =		{10.4230/LIPIcs.FSCD.2024.1},
  annote =	{Keywords: Lambda calculus, Solvability, Meaningfulness, Inhabitation, Genericity}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.4

Univalent Enriched Categories and the Enriched Rezk Completion

Authors: Niels van der Weide

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics of programming languages. In this paper, we study univalent enriched categories. We prove that all essentially surjective and fully faithful functors between univalent enriched categories are equivalences, and we show that every enriched category admits a Rezk completion. Finally, we use the Rezk completion for enriched categories to construct univalent enriched Kleisli categories.

Cite as

Niels van der Weide. Univalent Enriched Categories and the Enriched Rezk Completion. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanderweide:LIPIcs.FSCD.2024.4,
  author =	{van der Weide, Niels},
  title =	{{Univalent Enriched Categories and the Enriched Rezk Completion}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.4},
  URN =		{urn:nbn:de:0030-drops-203337},
  doi =		{10.4230/LIPIcs.FSCD.2024.4},
  annote =	{Keywords: enriched categories, univalent categories, homotopy type theory, univalent foundations, Rezk completion}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.22

Automating Boundary Filling in Cubical Agda

Authors: Maximilian Doré, Evan Cavallo, and Anders Mörtberg

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy Type Theory allows the user to reason about higher structures, such as topological spaces, using higher inductive types (HITs) and univalence. Cubical Agda is an extension of Agda with computational support for HITs and univalence. A difficulty when working in Cubical Agda is dealing with the complex combinatorics of higher structures, an infinite-dimensional generalisation of equational reasoning. To solve these higher-dimensional equations consists in constructing cubes with specified boundaries. We develop a simplified cubical language in which we isolate and study two automation problems: contortion solving, where we attempt to "contort" a cube to fit a given boundary, and the more general Kan solving, where we search for solutions that involve pasting multiple cubes together. Both problems are difficult in the general case - Kan solving is even undecidable - so we focus on heuristics that perform well on practical examples. We provide a solver for the contortion problem using a reformulation of contortions in terms of poset maps, while we solve Kan problems using constraint satisfaction programming. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve goals presented by Cubical Agda. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks - providing the ground for further study of proof automation in cubical type theories.

Cite as

Maximilian Doré, Evan Cavallo, and Anders Mörtberg. Automating Boundary Filling in Cubical Agda. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dore_et_al:LIPIcs.FSCD.2024.22,
  author =	{Dor\'{e}, Maximilian and Cavallo, Evan and M\"{o}rtberg, Anders},
  title =	{{Automating Boundary Filling in Cubical Agda}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.22},
  URN =		{urn:nbn:de:0030-drops-203514},
  doi =		{10.4230/LIPIcs.FSCD.2024.22},
  annote =	{Keywords: Cubical Agda, Automated Reasoning, Constraint Satisfaction Programming}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.23

Mirroring Call-By-Need, or Values Acting Silly

Authors: Beniamino Accattoli and Adrienne Lancelot

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Call-by-need evaluation for the λ-calculus can be seen as merging the best of call-by-name and call-by-value, namely the wise erasing behaviour of the former and the wise duplicating behaviour of the latter. To better understand how duplication and erasure can be combined, we design a degenerated calculus, dubbed call-by-silly, that is symmetric to call-by-need in that it merges the worst of call-by-name and call-by-value, namely silly duplications by-name and silly erasures by-value. We validate the design of the call-by-silly calculus via rewriting properties and multi types. In particular, we mirror the main theorem about call-by-need - that is, its operational equivalence with call-by-name - showing that call-by-silly and call-by-value induce the same contextual equivalence. This fact shows the blindness with respect to efficiency of call-by-value contextual equivalence. We also define a call-by-silly strategy and measure its length via tight multi types. Lastly, we prove that the call-by-silly strategy computes evaluation sequences of maximal length in the calculus.

Cite as

Beniamino Accattoli and Adrienne Lancelot. Mirroring Call-By-Need, or Values Acting Silly. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 23:1-23:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{accattoli_et_al:LIPIcs.FSCD.2024.23,
  author =	{Accattoli, Beniamino and Lancelot, Adrienne},
  title =	{{Mirroring Call-By-Need, or Values Acting Silly}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{23:1--23:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.23},
  URN =		{urn:nbn:de:0030-drops-203527},
  doi =		{10.4230/LIPIcs.FSCD.2024.23},
  annote =	{Keywords: Lambda calculus, intersection types, call-by-value, call-by-need}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.25

Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories

Authors: Ralph Matthes, Kobe Wullaert, and Benedikt Ahrens

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

We describe a generic construction of non-wellfounded syntax involving variable binding and its monadic substitution operation. Our construction of the syntax and its substitution takes place in category theory, notably by using monoidal categories and strong functors between them. A language is specified by a multi-sorted binding signature, say Σ. First, we provide sufficient criteria for Σ to generate a language of possibly infinite terms, through ω-continuity. Second, we construct a monadic substitution operation for the language generated by Σ. A cornerstone in this construction is a mild generalization of the notion of heterogeneous substitution systems developed by Matthes and Uustalu; such a system encapsulates the necessary corecursion scheme for implementing substitution. The results are formalized in the Coq proof assistant, through the UniMath library of univalent mathematics.

Cite as

Ralph Matthes, Kobe Wullaert, and Benedikt Ahrens. Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{matthes_et_al:LIPIcs.FSCD.2024.25,
  author =	{Matthes, Ralph and Wullaert, Kobe and Ahrens, Benedikt},
  title =	{{Substitution for Non-Wellfounded Syntax with Binders Through Monoidal Categories}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{25:1--25:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.25},
  URN =		{urn:nbn:de:0030-drops-203540},
  doi =		{10.4230/LIPIcs.FSCD.2024.25},
  annote =	{Keywords: Non-wellfounded syntax, Substitution, Monoidal categories, Actegories, Tensorial strength, Proof assistant Coq, UniMath library}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2021.22

Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor

Authors: Niccolò Veltri

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

Abstract

The finite powerset functor is a construct frequently employed for the specification of nondeterministic transition systems as coalgebras. The final coalgebra of the finite powerset functor, whose elements characterize the dynamical behavior of transition systems, is a well-understood object which enjoys many equivalent presentations in set-theoretic foundations based on classical logic. In this paper, we discuss various constructions of the final coalgebra of the finite powerset functor in constructive type theory, and we formalize our results in the Cubical Agda proof assistant. Using setoids, the final coalgebra of the finite powerset functor can be defined from the final coalgebra of the list functor. Using types instead of setoids, as it is common in homotopy type theory, one can specify the finite powerset datatype as a higher inductive type and define its final coalgebra as a coinductive type. Another construction is obtained by quotienting the final coalgebra of the list functor, but the proof of finality requires the assumption of the axiom of choice. We conclude the paper with an analysis of a classical construction by James Worrell, and show that its adaptation to our constructive setting requires the presence of classical axioms such as countable choice and the lesser limited principle of omniscience.

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Niccolò Veltri. Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{veltri:LIPIcs.FSCD.2021.22,
  author =	{Veltri, Niccol\`{o}},
  title =	{{Type-Theoretic Constructions of the Final Coalgebra of the Finite Powerset Functor}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{22:1--22:18},
  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.2021.22},
  URN =		{urn:nbn:de:0030-drops-142601},
  doi =		{10.4230/LIPIcs.FSCD.2021.22},
  annote =	{Keywords: type theory, finite powerset, final coalgebra, Cubical Agda}
}

Document

DOI: 10.4230/LIPIcs.CALCO.2019.14

Decomposing Comonad Morphisms

Authors: Danel Ahman and Tarmo Uustalu

Published in: LIPIcs, Volume 139, 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)

Abstract

The analysis of set comonads whose underlying functor is a container functor in terms of directed containers makes it a simple observation that any morphism between two such comonads factors through a third one by two comonad morphisms, whereof the first is identity on shapes and the second is identity on positions in every shape. This observation turns out to generalize into a much more involved result about comonad morphisms to comonads whose underlying functor preserves Cartesian natural transformations to itself on any category with finite limits. The bijection between comonad coalgebras and comonad morphisms from costate comonads thus also yields a decomposition of comonad coalgebras.

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Danel Ahman and Tarmo Uustalu. Decomposing Comonad Morphisms. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ahman_et_al:LIPIcs.CALCO.2019.14,
  author =	{Ahman, Danel and Uustalu, Tarmo},
  title =	{{Decomposing Comonad Morphisms}},
  booktitle =	{8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-120-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{139},
  editor =	{Roggenbach, Markus and Sokolova, Ana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2019.14},
  URN =		{urn:nbn:de:0030-drops-114427},
  doi =		{10.4230/LIPIcs.CALCO.2019.14},
  annote =	{Keywords: container functors (polynomial functors), container comonads, comonad morphisms and comonad coalgebras, cofunctors, lenses}
}

Document

DOI: 10.4230/LIPIcs.CONCUR.2019.40

Reordering Derivatives of Trace Closures of Regular Languages

Authors: Hendrik Maarand and Tarmo Uustalu

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)

Abstract

We provide syntactic derivative-like operations, defined by recursion on regular expressions, in the styles of both Brzozowski and Antimirov, for trace closures of regular languages. Just as the Brzozowski and Antimirov derivative operations for regular languages, these syntactic reordering derivative operations yield deterministic and nondeterministic automata respectively. But trace closures of regular languages are in general not regular, hence these automata cannot generally be finite. Still, as we show, for star-connected expressions, the Antimirov and Brzozowski automata, suitably quotiented, are finite. We also define a refined version of the Antimirov reordering derivative operation where parts-of-derivatives (states of the automaton) are nonempty lists of regular expressions rather than single regular expressions. We define the uniform scattering rank of a language and show that, for a regexp whose language has finite uniform scattering rank, the truncation of the (generally infinite) refined Antimirov automaton, obtained by removing long states, is finite without any quotienting, but still accepts the trace closure. We also show that star-connected languages have finite uniform scattering rank.

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Hendrik Maarand and Tarmo Uustalu. Reordering Derivatives of Trace Closures of Regular Languages. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{maarand_et_al:LIPIcs.CONCUR.2019.40,
  author =	{Maarand, Hendrik and Uustalu, Tarmo},
  title =	{{Reordering Derivatives of Trace Closures of Regular Languages}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{40:1--40:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.40},
  URN =		{urn:nbn:de:0030-drops-109426},
  doi =		{10.4230/LIPIcs.CONCUR.2019.40},
  annote =	{Keywords: Mazurkiewicz traces, trace closure, regular languages, finite automata, language derivatives, scattering rank, star-connected expressions}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2019.18

Modal Embeddings and Calling Paradigms

Authors: José Espírito Santo, Luís Pinto, and Tarmo Uustalu

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

Abstract

We study the computational interpretation of the two standard modal embeddings, usually named after Girard and Gödel, of intuitionistic logic into IS4. As source system we take either the call-by-name (cbn) or the call-by-value (cbv) lambda-calculus with simple types. The target system can be taken to be the, arguably, simplest fragment of IS4, here recast as a very simple lambda-calculus equipped with an indeterminate lax monoidal comonad. A slight refinement of the target and of the embeddings shows that: the target is a calculus indifferent to the calling paradigms cbn/cbv, obeying a new paradigm that we baptize call-by-box (cbb), and enjoying standardization; and that Girard’s (resp. Gödel’s) embbedding is a translation of cbn (resp. cbv) lambda-calculus into this calculus, using a compilation technique we call protecting-by-a-box, enjoying the preservation and reflection properties known for cps translations - but in a stronger form that allows the extraction of standardization for cbn or cbv as consequence of standardization for cbb. The modal target and embeddings achieve thus an unification of call-by-name and call-by-value as call-by-box.

Cite as

José Espírito Santo, Luís Pinto, and Tarmo Uustalu. Modal Embeddings and Calling Paradigms. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{espiritosanto_et_al:LIPIcs.FSCD.2019.18,
  author =	{Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s and Uustalu, Tarmo},
  title =	{{Modal Embeddings and Calling Paradigms}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{18:1--18:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.18},
  URN =		{urn:nbn:de:0030-drops-105256},
  doi =		{10.4230/LIPIcs.FSCD.2019.18},
  annote =	{Keywords: intuitionistic S4, call-by-name, call-by-value, comonadic lambda-calculus, standardization, indifference property}
}

Document

Complete Volume

DOI: 10.4230/LIPIcs.TYPES.2015

LIPIcs, Volume 69, TYPES'15, Complete Volume

Authors: Tarmo Uustalu

Published in: LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018)

Abstract

LIPIcs, Volume 69, TYPES'15, Complete Volume

Cite as

21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Proceedings{uustalu:LIPIcs.TYPES.2015,
  title =	{{LIPIcs, Volume 69, TYPES'15, Complete Volume}},
  booktitle =	{21st International Conference on Types for Proofs and Programs (TYPES 2015)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-030-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{69},
  editor =	{Uustalu, Tarmo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015},
  URN =		{urn:nbn:de:0030-drops-86220},
  doi =		{10.4230/LIPIcs.TYPES.2015},
  annote =	{Keywords: Mathematical Logic and Formal Languages: Mathematical Logic - Lambda calculus and related systems}
}

Document

Front Matter

DOI: 10.4230/LIPIcs.TYPES.2015.0

Front Matter, Table of Contents, Preface, External Reviewers

Authors: Tarmo Uustalu

Published in: LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018)

Abstract

Front Matter, Table of Contents, Preface, External Reviewers

Cite as

21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{uustalu:LIPIcs.TYPES.2015.0,
  author =	{Uustalu, Tarmo},
  title =	{{Front Matter, Table of Contents, Preface, External Reviewers}},
  booktitle =	{21st International Conference on Types for Proofs and Programs (TYPES 2015)},
  pages =	{0:i--0:xii},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-030-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{69},
  editor =	{Uustalu, Tarmo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.0},
  URN =		{urn:nbn:de:0030-drops-84704},
  doi =		{10.4230/LIPIcs.TYPES.2015.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, External Reviewers}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2015.1

A Type Theory for Probabilistic and Bayesian Reasoning

Authors: Robin Adams and Bart Jacobs

Published in: LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018)

Abstract

This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes our probabilistic type theory from quantum type theory, namely the bijective correspondence between predicates and side-effect free actions (called instrument, or assert, maps). The paper shows how suitable computation rules can be derived from this predicate-action correspondence, and uses these rules for calculating conditional probabilities in two well-known examples of Bayesian reasoning in (graphical) models. Our type theory may thus form the basis for a mechanisation of Bayesian inference.

Cite as

Robin Adams and Bart Jacobs. A Type Theory for Probabilistic and Bayesian Reasoning. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 1:1-1:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{adams_et_al:LIPIcs.TYPES.2015.1,
  author =	{Adams, Robin and Jacobs, Bart},
  title =	{{A Type Theory for Probabilistic and Bayesian Reasoning}},
  booktitle =	{21st International Conference on Types for Proofs and Programs (TYPES 2015)},
  pages =	{1:1--1:34},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-030-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{69},
  editor =	{Uustalu, Tarmo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.1},
  URN =		{urn:nbn:de:0030-drops-84714},
  doi =		{10.4230/LIPIcs.TYPES.2015.1},
  annote =	{Keywords: Probabilistic programming, probabilistic algorithm, type theory, effect module, Bayesian reasoning}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2015.2

Heterogeneous Substitution Systems Revisited

Authors: Benedikt Ahrens and Ralph Matthes

Published in: LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018)

Abstract

Matthes and Uustalu (TCS 327(1--2):155--174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is made up from least or greatest fixed points. We extend this work in two directions: we continue the analysis by creating more categorical structure, in particular by organizing substitution systems into a category and studying its properties, and we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.

Cite as

Benedikt Ahrens and Ralph Matthes. Heterogeneous Substitution Systems Revisited. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 2:1-2:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ahrens_et_al:LIPIcs.TYPES.2015.2,
  author =	{Ahrens, Benedikt and Matthes, Ralph},
  title =	{{Heterogeneous Substitution Systems Revisited}},
  booktitle =	{21st International Conference on Types for Proofs and Programs (TYPES 2015)},
  pages =	{2:1--2:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-030-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{69},
  editor =	{Uustalu, Tarmo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.2},
  URN =		{urn:nbn:de:0030-drops-84724},
  doi =		{10.4230/LIPIcs.TYPES.2015.2},
  annote =	{Keywords: formalization of category theory, nested datatypes, Mendler-style recursion schemes, representation of substitution in languages with variable binding}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2015.3

Towards a Cubical Type Theory without an Interval

Authors: Thorsten Altenkirch and Ambrus Kaposi

Published in: LIPIcs, Volume 69, 21st International Conference on Types for Proofs and Programs (TYPES 2015) (2018)

Abstract

Following the cubical set model of type theory which validates the univalence axiom, cubical type theories have been developed that interpret the identity type using an interval pretype. These theories start from a geometric view of equality. A proof of equality is encoded as a term in a context extended by the interval pretype. Our goal is to develop a cubical theory where the identity type is defined recursively over the type structure, and the geometry arises from these definitions. In this theory, cubes are present explicitly, e.g., a line is a telescope with 3 elements: two endpoints and the connecting equality. This is in line with Bernardy and Moulin's earlier work on internal parametricity. In this paper we present a naive syntax for internal parametricity and by replacing the parametric interpretation of the universe, we extend it to univalence. However, we do not know how to compute in this theory. As a second step, we present a version of the theory for parametricity with named dimensions which has an operational semantics. Extending this syntax to univalence is left as further work.

Cite as

Thorsten Altenkirch and Ambrus Kaposi. Towards a Cubical Type Theory without an Interval. In 21st International Conference on Types for Proofs and Programs (TYPES 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 69, pp. 3:1-3:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{altenkirch_et_al:LIPIcs.TYPES.2015.3,
  author =	{Altenkirch, Thorsten and Kaposi, Ambrus},
  title =	{{Towards a Cubical Type Theory without an Interval}},
  booktitle =	{21st International Conference on Types for Proofs and Programs (TYPES 2015)},
  pages =	{3:1--3:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-030-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{69},
  editor =	{Uustalu, Tarmo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2015.3},
  URN =		{urn:nbn:de:0030-drops-84739},
  doi =		{10.4230/LIPIcs.TYPES.2015.3},
  annote =	{Keywords: homotopy type theory, parametricity, univalence}
}

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