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Infinitary Completeness in Ludics

Michele Basaldella, Kazushige Terui
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
In order to extend the completeness theorem of the original ludics to the infinitary setting, we modify the notion of orthogonality by defining it via safety rather than termination of the interaction.  ...  Our work arises from studies on recursive types in denotational and operational semantics, but is conceptually simpler, due to the purely logical setting of ludics, the completeness theorem, and use of  ...  PROOF SYSTEM AND FULL COMPLETENESS In this section we introduce an infinitary (coinductive) proof system and show interactive completeness. A.  ... 
doi:10.1109/lics.2010.47 dblp:conf/lics/BasaldellaT10 fatcat:vcnjuqapwvbybgv5zo46sf4gki

The Ackermann Award 2018

Dexter Kozen, Thomas Schwentick, Michael Wagner
2018 Annual Conference for Computer Science Logic  
The Ackermann Award is the EACSL Outstanding Dissertation Award for Logic in Computer Science. It is presented during the annual conference of the EACSL (CSL'xx).  ...  linear logic with fixed points in Ludics, the thesis investigates completeness problems in more expressive logics and develops potential connections with ω-automata.  ...  By identifying new connections between infinitary proofs and automata theory (e.g., non-determinization of alternating parity automata), she has managed to obtain a new constructive completeness argument  ... 
doi:10.4230/lipics.csl.2018.1 dblp:conf/csl/KozenS18 fatcat:enjy7e5ltvbarlumyxu5tzq2zy

An Infinitary Affine Lambda-Calculus Isomorphic to the Full Lambda-Calculus

Damiano Mazza
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
of this space is shown to yield an infinitary affine lambda-calculus, whose quotient under a suitable partial equivalence relation is exactly the full (non-affine) lambda-calculus.  ...  It is well known that the real numbers arise from the metric completion of the rational numbers, with the metric induced by the usual absolute value.  ...  For instance, Theorem 22 is false in KKSV's infinitary calculi, where complete developments do not always exist (the infinite family of redexes in the term I(I(I . . .)) mentioned above has no complete  ... 
doi:10.1109/lics.2012.57 dblp:conf/lics/Mazza12 fatcat:ldqkg5sxyzb75lfgmanlhbwsl4

Infinitary Classical Logic: Recursive Equations and Interactive Semantics

Michele Basaldella
2014 Electronic Proceedings in Theoretical Computer Science  
In this paper, we present an interactive semantics for derivations in an infinitary extension of classical logic.  ...  We show that in our setting every recursive formula equation has a unique solution. As for derivations, we use an infinitary variant of Tait-calculus to derive sequents.  ...  The motivation for introducing our semantics comes from our interest in extending the completeness theorem of ludics [5] to logics which are not necessarily polarized fragments of linear logic.  ... 
doi:10.4204/eptcs.164.4 fatcat:cjkeyarht5ek5imvact2w2so6u

On the Meaning of Focalization [chapter]

Michele Basaldella, Alexis Saurin, Kazushige Terui
2011 Lecture Notes in Computer Science  
In this paper, we use Girard's Ludics to analyze focalization, a fundamental property of linear logic.  ...  In particular, our study of Focalization in Ludics was primarily motivated by the concluding remarks of the third author's paper on Computational Ludics [17] where focalization on data designs was conjectured  ...  Ludics in three pages Syntax.  ... 
doi:10.1007/978-3-642-19211-1_5 fatcat:2jqf6is5t5dzlnbb57hhjbii2q

On the meaning of logical completeness

Michele Basaldella, Kazushige Terui, Pierre-Louis Curien
2010 Logical Methods in Computer Science  
Goedel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs.  ...  Compared with proofs of full completeness in game semantics, ours exhibits a striking similarity with proofs of Goedel's completeness, in that it explicitly constructs a countermodel essentially using  ...  In particular, infinitary designs are included in our syntax, just as in the original ludics [21] .  ... 
doi:10.2168/lmcs-6(4:11)2010 fatcat:hoy3cxrdtnaylnew6zr562bghu

On the Meaning of Logical Completeness [chapter]

Michele Basaldella, Kazushige Terui
2009 Lecture Notes in Computer Science  
Gödel's completeness theorem is concerned with provability, while Girard's theorem in ludics (as well as full completeness theorems in game semantics) are concerned with proofs.  ...  Compared with proofs of full completeness in game semantics, ours exhibits a striking similarity with proofs of Gödel's completeness, in that it explicitly constructs a countermodel essentially using König's  ...  In particular, infinitary designs are included in our syntax, just as in the original ludics [21] .  ... 
doi:10.1007/978-3-642-02273-9_6 fatcat:6k4dx42s6rexhj5v5gsstxlslu

Infinitary λ-Calculi from a Linear Perspective (Long Version) [article]

Ugo Dal Lago
2016 arXiv   pre-print
Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.  ...  We introduce a linear infinitary λ-calculus, called ℓΛ_∞, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted  ...  In all the obtained calculi, however, many of the properties one expects are not true: the Complete Developments Theorem (i.e. the infinitary analogue of the Finite Complete Theorem [4] ) does not hold  ... 
arXiv:1604.08248v1 fatcat:wykju4emo5azna5iywpvds536y

Infinitary Lambda Calculi from a Linear Perspective

Ugo Dal Lago
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.  ...  We introduce a linear infinitary λ-calculus, called Λ∞, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted coinductively  ...  In all the obtained calculi, however, many of the properties one expects are not true: the Complete Developments Theorem (i.e. the infinitary analogue of the Finite Complete Theorem [4] ) does not hold  ... 
doi:10.1145/2933575.2934505 dblp:conf/lics/Lago16 fatcat:om7mpkcnkzf2zcypjjpyukthgm

Infinitary Proof Theory: the Multiplicative Additive Case

David Baelde, Amina Doumane, Alexis Saurin, Marc Herbstritt
2016 Annual Conference for Computer Science Logic  
ludics' designs and proofs in linear logic with fixed points [5].  ...  To the best of our knowledge, while reversibility has since long been a key-ingredient in completeness arguments based on infinitary proof systems, focalization has simply never been studied in such settings  ...  Because infinitary systems are easier to work with than the finitary proof systems (or axiomatizations) based on Kozen-Park (co)induction schemes, they are often found in completeness arguments for such  ... 
doi:10.4230/lipics.csl.2016.42 dblp:conf/csl/BaeldeDS16 fatcat:vpem436twbaarpzosgwkng6kfu

Non-linearity as the Metric Completion of Linearity [chapter]

Damiano Mazza
2013 Lecture Notes in Computer Science  
., lambda-terms in which abstractions bind variables appearing at most once.  ...  We summarize some recent results showing how the lambdacalculus may be obtained by considering the metric completion (with respect to a suitable notion of distance) of a space of affine lambda-terms, i.e  ...  Roughly speaking, we take objects which are very much related to the designs of Girard's ludics [13] , introduce a metric completely analogous to the one given here, and construct the model in the completed  ... 
doi:10.1007/978-3-642-38946-7_3 fatcat:nxjfhz2hmfe6dczq3rwkmn644y

From Focalization of Logic to the Logic of Focalization

Michele Basaldella, Alexis Saurin, Kazushige Terui
2010 Electronical Notes in Theoretical Computer Science  
The aim of the present paper is to give a more abstract account on focalization in the framework of ludics.  ...  Andreoli originally discovered focalization as a concrete proof search strategy in proof theory of linear logic, putting to the foreground the role of polarity in logic.  ...  Logical behaviours are described in Section 6 together with internal completeness. Finally, we complete our study in Section 7 providing a ludics account of focalization.  ... 
doi:10.1016/j.entcs.2010.08.010 fatcat:knwkldyquzhzdeymgefo42q3ne

Computational ludics

Kazushige Terui
2011 Theoretical Computer Science  
The acceptance relation between machines and words, a basic concept in computability theory, is well expressed in ludics by the orthogonality relation between designs.  ...  We finally describe a way of defining data sets by means of logical connectives, where the internal completeness theorem plays an essential role.  ...  The designs of ludics extend the lambda terms in this well-behaved fragment in several ways. First, designs can be infinitary.  ... 
doi:10.1016/j.tcs.2010.12.026 fatcat:gtwfqbwzezeqpmcqlfpbmadmia

Jumbo λ-Calculus [chapter]

Paul Blain Levy
2006 Lecture Notes in Computer Science  
Firstly, we show that the jumbo lambda-calculus provides a "complete" range of connectives, in the sense of including every possible connective that, within the beta-eta theory, possesses a reversible  ...  Secondly, in the presence of effects, we show that there is no decomposition of jumbo connectives into non-jumbo ones that is valid in both call-by-value and call-by-name.  ...  But how many connectives must we include to obtain a "complete" range?  ... 
doi:10.1007/11787006_38 fatcat:jkkpbktghjckvkiepmgzkweeca

Imperative Programs as Proofs via Game Semantics

Martin Churchill, James Laird, Guy McCusker
2011 2011 IEEE 26th Annual Symposium on Logic in Computer Science  
In this paper we describe a logical counterpart to this extension, in which proofs denote such strategies.  ...  We establish a full completeness theorem for our logic, showing that every bounded strategy is the denotation of a proof.  ...  The rest of this section sketches the proof of this full completeness result, and describes an extension to reify unbounded strategies as infinitary core proofs. A.  ... 
doi:10.1109/lics.2011.19 dblp:conf/lics/ChurchillLM11 fatcat:redxq676rjfwnlmqn2glx5zj5a
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