15 Search Results for "Hyland, Martin"


Document
Coinduction in Flow: The Later Modality in Fibrations

Authors: Henning Basold

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


Abstract
This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search. Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results.

Cite as

Henning Basold. Coinduction in Flow: The Later Modality in Fibrations. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 8:1-8:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{basold:LIPIcs.CALCO.2019.8,
  author =	{Basold, Henning},
  title =	{{Coinduction in Flow: The Later Modality in Fibrations}},
  booktitle =	{8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)},
  pages =	{8:1--8:22},
  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.2019.8},
  URN =		{urn:nbn:de:0030-drops-114369},
  doi =		{10.4230/LIPIcs.CALCO.2019.8},
  annote =	{Keywords: Coinduction, Fibrations, Later Modality, Recursive Proofs, Up-to techniques, Probabilistic Logic, Probabilistic Programming}
}
Document
Causal Unfoldings

Authors: Marc de Visme and Glynn Winskel

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


Abstract
In the simplest form of event structure, a prime event structure, an event is associated with a unique causal history, its prime cause. However, it is quite common for an event to have disjunctive causes in that it can be enabled by any one of multiple sets of causes. Sometimes the sets of causes may be mutually exclusive, inconsistent one with another, and sometimes not, in which case they coexist consistently and constitute parallel causes of the event. The established model of general event structures can model parallel causes. On occasion however such a model abstracts too far away from the precise causal histories of events to be directly useful. For example, sometimes one needs to associate probabilities with different, possibly coexisting, causal histories of a common event. Ideally, the causal histories of a general event structure would correspond to the configurations of its causal unfolding to a prime event structure; and the causal unfolding would arise as a right adjoint to the embedding of prime in general event structures. But there is no such adjunction. However, a slight extension of prime event structures remedies this defect and provides a causal unfolding as a universal construction. Prime event structures are extended with an equivalence relation in order to dissociate the two roles, that of an event and its enabling; in effect, prime causes are labelled by a disjunctive event, an equivalence class of its prime causes. With this enrichment a suitable causal unfolding appears as a pseudo right adjoint. The adjunction relies critically on the central and subtle notion of extremal causal realisation as an embodiment of causal history.

Cite as

Marc de Visme and Glynn Winskel. Causal Unfoldings. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{devisme_et_al:LIPIcs.CALCO.2019.9,
  author =	{de Visme, Marc and Winskel, Glynn},
  title =	{{Causal Unfoldings}},
  booktitle =	{8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)},
  pages =	{9:1--9:18},
  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.2019.9},
  URN =		{urn:nbn:de:0030-drops-114376},
  doi =		{10.4230/LIPIcs.CALCO.2019.9},
  annote =	{Keywords: Event Structures, Parallel Causes, Causal Unfolding, Probability}
}
Document
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
Coinductive Resumption Monads: Guarded Iterative and Guarded Elgot

Authors: Paul Blain Levy and Sergey Goncharov

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


Abstract
We introduce a new notion of "guarded Elgot monad", that is a monad equipped with a form of iteration. It requires every guarded morphism to have a specified fixpoint, and classical equational laws of iteration to be satisfied. This notion includes Elgot monads, but also further examples of partial non-unique iteration, emerging in the semantics of processes under infinite trace equivalence. We recall the construction of the "coinductive resumption monad" from a monad and endofunctor, that is used for modelling programs up to bisimilarity. We characterize this construction via a universal property: if the given monad is guarded Elgot, then the coinductive resumption monad is the guarded Elgot monad that freely extends it by the given endofunctor.

Cite as

Paul Blain Levy and Sergey Goncharov. Coinductive Resumption Monads: Guarded Iterative and Guarded Elgot. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{levy_et_al:LIPIcs.CALCO.2019.13,
  author =	{Levy, Paul Blain and Goncharov, Sergey},
  title =	{{Coinductive Resumption Monads: Guarded Iterative and Guarded Elgot}},
  booktitle =	{8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)},
  pages =	{13:1--13:17},
  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.2019.13},
  URN =		{urn:nbn:de:0030-drops-114414},
  doi =		{10.4230/LIPIcs.CALCO.2019.13},
  annote =	{Keywords: Guarded iteration, guarded monads, coalgebraic resumptions}
}
Document
Data Types as Quotients of Polynomial Functors

Authors: Jeremy Avigad, Mario Carneiro, and Simon Hudon

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)


Abstract
A broad class of data types, including arbitrary nestings of inductive types, coinductive types, and quotients, can be represented as quotients of polynomial functors. This provides perspicuous ways of constructing them and reasoning about them in an interactive theorem prover.

Cite as

Jeremy Avigad, Mario Carneiro, and Simon Hudon. Data Types as Quotients of Polynomial Functors. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{avigad_et_al:LIPIcs.ITP.2019.6,
  author =	{Avigad, Jeremy and Carneiro, Mario and Hudon, Simon},
  title =	{{Data Types as Quotients of Polynomial Functors}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.6},
  URN =		{urn:nbn:de:0030-drops-110612},
  doi =		{10.4230/LIPIcs.ITP.2019.6},
  annote =	{Keywords: data types, polynomial functors, inductive types, coinductive types}
}
Document
On the Expressivity of Linear Recursion Schemes

Authors: Pierre Clairambault and Andrzej S. Murawski

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


Abstract
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.

Cite as

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}
}
Document
Bialgebraic Semantics for String Diagrams

Authors: Filippo Bonchi, Robin Piedeleu, Pawel Sobocinski, and Fabio Zanasi

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


Abstract
Turi and Plotkin’s bialgebraic semantics is an abstract approach to specifying the operational semantics of a system, by means of a distributive law between its syntax (encoded as a monad) and its dynamics (an endofunctor). This setup is instrumental in showing that a semantic specification (a coalgebra) satisfies desirable properties: in particular, that it is compositional. In this work, we use the bialgebraic approach to derive well-behaved structural operational semantics of string diagrams, a graphical syntax that is increasingly used in the study of interacting systems across different disciplines. Our analysis relies on representing the two-dimensional operations underlying string diagrams in various categories as a monad, and their bialgebraic semantics in terms of a distributive law for that monad. As a proof of concept, we provide bialgebraic compositional semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our approach reveals a correspondence between two different interpretations of the Frobenius equations on string diagrams and two synchronisation mechanisms for processes, à la Hoare and à la Milner.

Cite as

Filippo Bonchi, Robin Piedeleu, Pawel Sobocinski, and Fabio Zanasi. Bialgebraic Semantics for String Diagrams. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bonchi_et_al:LIPIcs.CONCUR.2019.37,
  author =	{Bonchi, Filippo and Piedeleu, Robin and Sobocinski, Pawel and Zanasi, Fabio},
  title =	{{Bialgebraic Semantics for String Diagrams}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{37:1--37:17},
  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.2019.37},
  URN =		{urn:nbn:de:0030-drops-109398},
  doi =		{10.4230/LIPIcs.CONCUR.2019.37},
  annote =	{Keywords: String Diagram, Structural Operational Semantics, Bialgebraic semantics}
}
Document
The True Concurrency of Herbrand's Theorem

Authors: Aurore Alcolei, Pierre Clairambault, Martin Hyland, and Glynn Winskel

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


Abstract
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.

Cite as

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}
}
Document
An Intensionally Fully-abstract Sheaf Model for pi

Authors: Clovis Eberhart, Tom Hirschowitz, and Thomas Seiller

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


Abstract
Following previous work on CCS, we propose a compositional model for the pi-calculus in which processes are interpreted as sheaves on certain simple sites. We define an analogue of fair testing equivalence in the model and show that our interpretation is intensionally fully abstract for it. That is, the interpretation preserves and reflects fair testing equivalence; and furthermore, any strategy is fair testing equivalent to the interpretation of some process. The central part of our work is the construction of our sites, whose heart is a combinatorial presentation of pi-calculus traces in the spirit of string diagrams. As in previous work, the sheaf condition is analogous to innocence in Hyland-Ong/Nickau games.

Cite as

Clovis Eberhart, Tom Hirschowitz, and Thomas Seiller. An Intensionally Fully-abstract Sheaf Model for pi. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 86-100, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{eberhart_et_al:LIPIcs.CALCO.2015.86,
  author =	{Eberhart, Clovis and Hirschowitz, Tom and Seiller, Thomas},
  title =	{{An Intensionally Fully-abstract Sheaf Model for pi}},
  booktitle =	{6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)},
  pages =	{86--100},
  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.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.86},
  URN =		{urn:nbn:de:0030-drops-55284},
  doi =		{10.4230/LIPIcs.CALCO.2015.86},
  annote =	{Keywords: concurrency, sheaves, causal models, games}
}
Document
Dialectica Categories and Games with Bidding

Authors: Jules Hedges

Published in: LIPIcs, Volume 39, 20th International Conference on Types for Proofs and Programs (TYPES 2014)


Abstract
This paper presents a construction which transforms categorical models of additive-free propositional linear logic, closely based on de Paiva's dialectica categories and Oliva's functional interpretations of classical linear logic. The construction is defined using dependent type theory, which proves to be a useful tool for reasoning about dialectica categories. Abstractly, we have a closure operator on the class of models: it preserves soundness and completeness and has a monad-like structure. When applied to categories of games we obtain 'games with bidding', which are hybrids of dialectica and game models, and we prove completeness theorems for two specific such models.

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Jules Hedges. Dialectica Categories and Games with Bidding. In 20th International Conference on Types for Proofs and Programs (TYPES 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 39, pp. 89-110, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{hedges:LIPIcs.TYPES.2014.89,
  author =	{Hedges, Jules},
  title =	{{Dialectica Categories and Games with Bidding}},
  booktitle =	{20th International Conference on Types for Proofs and Programs (TYPES 2014)},
  pages =	{89--110},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-88-0},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{39},
  editor =	{Herbelin, Hugo and Letouzey, Pierre and Sozeau, Matthieu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2014.89},
  URN =		{urn:nbn:de:0030-drops-54937},
  doi =		{10.4230/LIPIcs.TYPES.2014.89},
  annote =	{Keywords: Linear logic, Dialectica categories, categorical semantics, model theory, game semantics, dependent types, functional interpretations}
}
Document
An Extensional Kleene Realizability Semantics for the Minimalist Foundation

Authors: Maria Emilia Maietti and Samuele Maschio

Published in: LIPIcs, Volume 39, 20th International Conference on Types for Proofs and Programs (TYPES 2014)


Abstract
We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove that both levels of MF are consistent with the formal Church Thesis CT. Since MF consists of two levels, an intensional one, called mtt, and an extensional one, called emtt, linked by an interpretation, it is enough to build a realizability semantics for the intensional level mtt to get one for the extensional one emtt, too. Moreover, both levels consists of type theories based on versions of Martin-Löf's type theory. Our realizability semantics for mtt is a modification of the realizability semantics by Beeson in 1985 for extensional first order Martin-Löf's type theory with one universe. So it is formalized in Feferman's classical arithmetic theory of inductive definitions, called ID1^. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson's one. The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+CT+AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the Axiom of Choice AC on generic types. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-Löf's theory, but it does not validate CT. The semantics we present here seems to be the best approximation of Kleene realizability for the extensional level emtt. Indeed Beeson's semantics is not an option for emtt since AC on generic sets added to it entails the excluded middle.

Cite as

Maria Emilia Maietti and Samuele Maschio. An Extensional Kleene Realizability Semantics for the Minimalist Foundation. In 20th International Conference on Types for Proofs and Programs (TYPES 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 39, pp. 162-186, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{maietti_et_al:LIPIcs.TYPES.2014.162,
  author =	{Maietti, Maria Emilia and Maschio, Samuele},
  title =	{{An Extensional Kleene Realizability Semantics for the Minimalist Foundation}},
  booktitle =	{20th International Conference on Types for Proofs and Programs (TYPES 2014)},
  pages =	{162--186},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-88-0},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{39},
  editor =	{Herbelin, Hugo and Letouzey, Pierre and Sozeau, Matthieu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2014.162},
  URN =		{urn:nbn:de:0030-drops-54966},
  doi =		{10.4230/LIPIcs.TYPES.2014.162},
  annote =	{Keywords: Realizability, Type Theory, formal Church Thesis}
}
Document
Modelling Coeffects in the Relational Semantics of Linear Logic

Authors: Flavien Breuvart and Michele Pagani

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)


Abstract
Various typing system have been recently introduced giving a parametric version of the exponential modality of linear logic. The parameters are taken from a semi-ring, and allow to express coeffects - i.e. specific requirements of a program with respect to the environment (availability of a resource, some prerequisite of the input, etc.). We show that all these systems can be interpreted in the relational category (Rel) of sets and relations. This is possible because of the notion of multiplicity semi-ring and allowing a great variety of exponential comonads in Rel. The interpretation of a particular typing system corresponds then to give a suitable notion of stratification of the exponential comonad associated with the semi-ring parametrising the exponential modality.

Cite as

Flavien Breuvart and Michele Pagani. Modelling Coeffects in the Relational Semantics of Linear Logic. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 567-581, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{breuvart_et_al:LIPIcs.CSL.2015.567,
  author =	{Breuvart, Flavien and Pagani, Michele},
  title =	{{Modelling Coeffects in the Relational Semantics of Linear Logic}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{567--581},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.567},
  URN =		{urn:nbn:de:0030-drops-54384},
  doi =		{10.4230/LIPIcs.CSL.2015.567},
  annote =	{Keywords: relational semantics, bounded linear logic, lambda calculus}
}
Document
Sub-classical Boolean Bunched Logics and the Meaning of Par

Authors: James Brotherston and Jules Villard

Published in: LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)


Abstract
We investigate intermediate logics between the bunched logics Boolean BI and Classical BI, obtained by combining classical propositional logic with various flavours of Hyland and De Paiva's full intuitionistic linear logic. Thus, in addition to the usual multiplicative conjunction (with its adjoint implication and unit), our logics also feature a multiplicative disjunction (with its adjoint co-implication and unit). The multiplicatives behave "sub-classically", in that disjunction and conjunction are related by a weak distribution principle, rather than by De Morgan equivalence. We formulate a Kripke semantics, covering all our sub-classical bunched logics, in which the multiplicatives are naturally read in terms of resource operations. Our main theoretical result is that validity according to this semantics coincides with provability in a corresponding Hilbert-style proof system. Our logical investigation sheds considerable new light on how one can understand the multiplicative disjunction, better known as linear logic's "par", in terms of resource operations. In particular, and in contrast to the earlier Classical BI, the models of our logics include the heap-like memory models of separation logic, in which disjunction can be interpreted as a property of intersection operations over heaps.

Cite as

James Brotherston and Jules Villard. Sub-classical Boolean Bunched Logics and the Meaning of Par. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 325-342, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{brotherston_et_al:LIPIcs.CSL.2015.325,
  author =	{Brotherston, James and Villard, Jules},
  title =	{{Sub-classical Boolean Bunched Logics and the Meaning of Par}},
  booktitle =	{24th EACSL Annual Conference on Computer Science Logic (CSL 2015)},
  pages =	{325--342},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-90-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{41},
  editor =	{Kreutzer, Stephan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.325},
  URN =		{urn:nbn:de:0030-drops-54234},
  doi =		{10.4230/LIPIcs.CSL.2015.325},
  annote =	{Keywords: Bunched logic, linear logic, modal logic, Kripke semantics, model theory}
}
Document
The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation

Authors: Martín Hötzel Escardó and Chuangjie Xu

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


Abstract
If all functions (N -> N) -> N are continuous then 0 = 1. We establish this in intensional (and hence also in extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Sigma type via the Curry-Howard interpretation. But with an intuitionistic notion of anonymous existence, defined as the propositional truncation of Sigma, it is consistent that all such functions are continuous. A model is Johnstone’s topological topos. On the other hand, any of these two intuitionistic conceptions of existence give the same, consistent, notion of uniform continuity for functions (N -> 2) -> N, again valid in the topological topos. It is open whether the consistency of (uniform) continuity extends to homotopy type theory. The theorems of type theory informally proved here are also formally proved in Agda, but the development presented here is self-contained and doesn't show Agda code.

Cite as

Martín Hötzel Escardó and Chuangjie Xu. The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 153-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{hotzelescardo_et_al:LIPIcs.TLCA.2015.153,
  author =	{H\"{o}tzel Escard\'{o}, Mart{\'\i}n and Xu, Chuangjie},
  title =	{{The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{153--164},
  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.2015.153},
  URN =		{urn:nbn:de:0030-drops-51618},
  doi =		{10.4230/LIPIcs.TLCA.2015.153},
  annote =	{Keywords: Dependent type, intensional Martin-L\"{o}f type theory, Curry-Howard interpretation, constructive mathematics, Brouwerian continuity axioms, anonymous exi}
}
Document
Fibrations of Tree Automata

Authors: Colin Riba

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


Abstract
We propose a notion of morphisms between tree automata based on game semantics. Morphisms are winning strategies on a synchronous restriction of the linear implication between acceptance games. This leads to split indexed categories, with substitution based on a suitable notion of synchronous tree function. By restricting to tree functions issued from maps on alphabets, this gives a fibration of tree automata. We then discuss the (fibrewise) monoidal structure issued from the synchronous product of automata. We also discuss how a variant of the usual projection operation on automata leads to an existential quantification in the fibered sense. Our notion of morphism is correct (it respects language inclusion), and in a weaker sense also complete.

Cite as

Colin Riba. Fibrations of Tree Automata. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 302-316, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{riba:LIPIcs.TLCA.2015.302,
  author =	{Riba, Colin},
  title =	{{Fibrations of Tree Automata}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{302--316},
  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.2015.302},
  URN =		{urn:nbn:de:0030-drops-51719},
  doi =		{10.4230/LIPIcs.TLCA.2015.302},
  annote =	{Keywords: Tree automata, Game semantics, Categorical logic.}
}
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