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Algebraic theories, monads, and arities [article]

Charles Grellois
2011 arXiv   pre-print
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
more » ... 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.
arXiv:1110.3294v1 fatcat:7xhovcib7fcenaybwuekxuloam

Intuitionistic Non-Normal Modal Logics: A general framework [article]

Tiziano Dalmonte, Charles Grellois, Nicola Olivetti
2019 arXiv   pre-print
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
more » ... 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.
arXiv:1901.09812v1 fatcat:gyubfeko4fhrzijrktaodrxzlm

Probabilistic Termination by Monadic Affine Sized Typing (Long Version) [article]

Ugo Dal Lago, Charles Grellois
2017 arXiv   pre-print
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
more » ... 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.
arXiv:1701.04089v1 fatcat:itbrvj52bbh3zl5fl72vvdg3ue

Probabilistic Termination by Monadic Affine Sized Typing [chapter]

Ugo Dal Lago, Charles Grellois
2017 Lecture Notes in Computer Science  
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
more » ... 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.
doi:10.1007/978-3-662-54434-1_15 fatcat:z4gjpholmnedvgehfdu4m4542u

An infinitary model of linear logic [article]

Charles Grellois, Paul-André Melliès
2015 arXiv   pre-print
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
more » ... 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.
arXiv:1411.4380v3 fatcat:lxj4n52rqbardkoji5747wv3c4

Finitary semantics of linear logic and higher-order model-checking [article]

Charles Grellois, Paul-André Melliès
2015 arXiv   pre-print
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.
more » ... 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.
arXiv:1502.05147v2 fatcat:lseygm2skzcorilqj7s7o55n4m

On the Termination Problem for Probabilistic Higher-Order Recursive Programs

Naoki Kobayashi, Ugo Dal Lago, Charles Grellois
2019 Logical Methods in Computer Science  
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

Naoki Kobayashi, Ugo Dal Lago, Charles Grellois
2019 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
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

Charles Grellois, Paul-André Melliès
2015 Electronic Proceedings in Theoretical Computer Science  
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
more » ... 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.
doi:10.4204/eptcs.177.4 fatcat:76shpi2nn5ckphb4myagjd54b4

Relational semantics of linear logic and higher-order model-checking [article]

Charles Grellois, Paul-André Melliès
2015 arXiv   pre-print
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
more » ... 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.
arXiv:1501.04789v3 fatcat:4yd3ryhkpzgzfhpy52fctlmglq

On the Termination Problem for Probabilistic Higher-Order Recursive Programs [article]

Naoki Kobayashi, Ugo Dal Lago, Charles Grellois
2020 arXiv   pre-print
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

Charles Grellois, Paul-André Melliès
licensed under Creative Commons License CC-BY 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)   unpublished
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
more » ... 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)
fatcat:hkmigkdrhjaqzh7upjifnemtpq

Page 290 of Journal of Classical and Sacred Philology Vol. 1, Issue [page]

1854 Journal of Classical and Sacred Philology  
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  
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

Alejandro Aguirre, Gilles Barthe, Justin Hsu, Alexandra Silva, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
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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