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Algebraic theories, monads, and arities
[article]
2011
arXiv
pre-print
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Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structures like n-categories, the computing process being realized using Kan extensions. This Master thesis
arXiv:1110.3294v1
fatcat:7xhovcib7fcenaybwuekxuloam
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... piles the required material in order to understand this question of arities and reconstruction of monads, and tries to give some examples of relevant interest from both semantics and higher category theory. A discussion on the promising field of operads is then provided as appendix.
Intuitionistic Non-Normal Modal Logics: A general framework
[article]
2019
arXiv
pre-print
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We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert
arXiv:1901.09812v1
fatcat:gyubfeko4fhrzijrktaodrxzlm
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... isation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic.
Probabilistic Termination by Monadic Affine Sized Typing (Long Version)
[article]
2017
arXiv
pre-print
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We introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of
arXiv:1701.04089v1
fatcat:itbrvj52bbh3zl5fl72vvdg3ue
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... obabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique.
Probabilistic Termination by Monadic Affine Sized Typing
[chapter]
2017
Lecture Notes in Computer Science
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We introduce a system of monadic affine sized types, which substantially generalise usual sized types, and allows this way to capture probabilistic higher-order programs which terminate almost surely. Going beyond plain, strong normalisation without losing soundness turns out to be a hard task, which cannot be accomplished without a richer, quantitative notion of types, but also without imposing some affinity constraints. The proposed type system is powerful enough to type classic examples of
doi:10.1007/978-3-662-54434-1_15
fatcat:z4gjpholmnedvgehfdu4m4542u
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... obabilistically terminating programs such as random walks. The way typable programs are proved to be almost surely terminating is based on reducibility, but requires a substantial adaptation of the technique.
An infinitary model of linear logic
[article]
2015
arXiv
pre-print
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In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the relational semantics with a notion of color or priority in the sense of parity games. This extension enables us to define a new fixpoint operator Y
arXiv:1411.4380v3
fatcat:lxj4n52rqbardkoji5747wv3c4
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... ining both inductive and coinductive policies. We conclude the paper by sketching the connection between the resulting model of lambda-calculus with recursion and higher-order model-checking.
Finitary semantics of linear logic and higher-order model-checking
[article]
2015
arXiv
pre-print
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In this paper, we explain how the connection between higher-order model-checking and linear logic recently exhibited by the authors leads to a new and conceptually enlightening proof of the selection problem originally established by Carayol and Serre using collapsible pushdown automata. The main idea is to start from an infinitary and colored relational semantics of the lambdaY-calculus already formulated, and to replace it by its finitary counterpart based on finite prime-algebraic lattices.
arXiv:1502.05147v2
fatcat:lseygm2skzcorilqj7s7o55n4m
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... iven a higher-order recursion scheme G, the finiteness of its interpretation in the model enables us to associate to any MSO formula phi a new higher-order recursion scheme G_phi resolving the selection problem.
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
2019
Logical Methods in Computer Science
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Dal Lago and Grellois [DLG17] generalized sized types [HPS96, BFG + 04] to obtain a sound but highly incomplete technique. ...
doi:10.23638/lmcs-16(4:2)2020
fatcat:nq4rufu6mrfwhavixxztkmlrke
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
2019
2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
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Dal Lago and Grellois [17] generalized sized types [36, 4] to obtain a sound but highly incomplete technique. ...
doi:10.1109/lics.2019.8785679
dblp:conf/lics/KobayashiLG19
fatcat:zwimghbbgffxjpcbv274c3vlli
Indexed linear logic and higher-order model checking
2015
Electronic Proceedings in Theoretical Computer Science
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In recent work, Kobayashi observed that the acceptance by an alternating tree automaton A of an infinite tree T generated by a higher-order recursion scheme G may be formulated as the typability of the recursion scheme G in an appropriate intersection type system associated to the automaton A. The purpose of this article is to establish a clean connection between this line of work and Bucciarelli and Ehrhard's indexed linear logic. This is achieved in two steps. First, we recast Kobayashi's
doi:10.4204/eptcs.177.4
fatcat:76shpi2nn5ckphb4myagjd54b4
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... lt in an equivalent infinitary intersection type system where intersection is not idempotent anymore. Then, we show that the resulting type system is a fragment of an infinitary version of Bucciarelli and Ehrhard's indexed linear logic. While this work is very preliminary and does not integrate key ingredients of higher-order model-checking like priorities, it reveals an interesting and promising connection between higher-order model-checking and linear logic.
Relational semantics of linear logic and higher-order model-checking
[article]
2015
arXiv
pre-print
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In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the
arXiv:1501.04789v3
fatcat:4yd3ryhkpzgzfhpy52fctlmglq
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... set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how his analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.
On the Termination Problem for Probabilistic Higher-Order Recursive Programs
[article]
2020
arXiv
pre-print
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Dal Lago and Grellois [DLG17] generalized sized types [HPS96, BFG + 04] to obtain a sound but highly incomplete technique. ...
arXiv:1811.02133v4
fatcat:g6pr4m7sjbht5h5qmx5q6cdd44
Editor: Stephan Kreutzer
licensed under Creative Commons License CC-BY 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)
unpublished
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In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of a higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the
fatcat:hkmigkdrhjaqzh7upjifnemtpq
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... t of initial states of the tree automaton from which the infinite tree generated by the recursion scheme is accepted. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes. 1 Introduction Thanks to the seminal works by Girard and Reynolds on polymorphism and parametricity in the 1970s, it has been recognized that every finite tree t on a given signature Σ can be seen alternatively as a simply-typed λ-term of an appropriate type depending on Σ. This correspondence between trees and λ-terms is even bijective if one considers λ-terms up to βη-equivalence, see for instance Girard [8]. Typically, a finite tree t on the signature Σ = { a : 2 , b : 1 , c : 0 } (1) is the same thing under this Church encoding as a simply-typed λ-term t of type (o → o → o) → (o → o) → o → o (2) modulo βη-equivalence. The idea underlying the correspondence is that every construc-tor a, b, c of the signature Σ should be treated as a variable a : o → o → o b : o → o c : o (3)
Page 290 of Journal of Classical and Sacred Philology Vol. 1, Issue
[page]
1854
Journal of Classical and Sacred Philology
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By Grellois. [Among other antiquities several funeral in- scriptions have been found, and some thousands of coins, ranging from the time of Augustus to that of Gallienus]. ...
By Grellois.—On some statuettes and tombs found at Géromomont, in Luxembourg. By D’Haart.—On the places at which the Roman coins found at or near Metz were struck. By Boulangé. ...
Page 946 of The Lancet Vol. , Issue 2574
[page]
1872
The Lancet
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Grellois, 226— A Treatise on the Causes of Heart Disease; with a Chapter on the reason of its Prevalence in the Army : by Francis W. ...
Treatment of Diseases of the Skin, vith on an Analysis of 11,000 consecutive Cases: by McCall Anderson, 746—The Practice of Su a Manual: by Thomas Bryant, F. — o The Ex of the Emoti n Man and Animals: by Charles ...
Almost Sure Productivity
2018
International Colloquium on Automata, Languages and Programming
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Acknowledgements We thank Benjamin Kaminski, Charles Grellois and Ugo dal Lago for helpful discussions on preliminary versions of this work, and we are grateful to the anonymous reviewers for pointing ...
doi:10.4230/lipics.icalp.2018.113
dblp:conf/icalp/0001BH018
fatcat:avdlpvgusbdmzlvgl3bbderiui
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