On the meaning of logical completeness

Basaldella, Michele; Terui, Kazushige

doi:10.2168/lmcs-6(4:11)2010

Cited by 3 publications

(15 citation statements)

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“…Computational ludics. By using the approach we present in this paper, Basaldella and Terui [6] have recently extended Terui's computational ludics [39] in order to accommodate exponentials.…”

Section: Resultsmentioning

confidence: 99%

“…We here assume finiteness among the winning conditions. However, recent work by Basaldella and Terui [6] shows an exciting property of interactive types: any material, deterministic and daimon free strategy in a behaviour which is interpretation of a logical formula is finite. We are confident that this result is also valid our setting; we need to check this in detail and we postpone it to a subsequent work.…”

Section: 1mentioning

confidence: 99%

“…On the expressivity of MELLS. In this part, we discuss the relation between MELLS and some other (more standard) polarized variants of MELL (see also [6]).…”

Section: Proofmentioning

confidence: 99%

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Ludics with Repetitions (Exponentials, Interactive Types and Completeness)

Basaldella

Faggian

2009

2009 24th Annual IEEE Symposium on Logic in Computer Science

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Abstract. Ludics is peculiar in the panorama of game semantics: we first have the definition of interaction-composition and then we have semantical types, as a set of strategies which "behave well" and react in the same way to a set of tests. The semantical types which are interpretations of logical formulas enjoy a fundamental property, called internal completeness, which characterizes ludics and sets it apart also from realizability. Internal completeness entails standard full completeness as a consequence.A growing body of work start to explore the potential of this specific interactive approach. However, ludics has some limitations, which are consequence of the fact that in the original formulation, strategies are abstractions of MALL proofs. On one side, no repetitions are allowed. On the other side, the proofs tend to rely on the very specific properties of the MALL proof-like strategies, making it difficult to transfer the approach to semantical types into different settings.In this paper, we provide an extension of ludics which allows repetitions and show that one can still have interactive types and internal completeness. From this, we obtain full completeness w.r.t. a polarized version of MELL. In our extension, we use less properties than in the original formulation, which we believe is of independent interest. We hope this may open the way to applications of ludics approach to larger domains and different settings.

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Section: Resultsmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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Ludics with Repetitions (Exponentials, Interactive Types and Completeness)

Basaldella

Faggian

2009

2009 24th Annual IEEE Symposium on Logic in Computer Science

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“…It is intuitively clear that it holds, since our c-designs reasonably generalize Girard's original designs and lambda terms, both enjoying associativity. A formal proof is given in [2].…”

Section: Associativitymentioning

confidence: 99%

“…[16]; see [15] for an introduction to coinductive techniques). (2) Designs are infinite objects in general, while effective computation requires finitary representation. We therefore introduce a generator producing a design.…”

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confidence: 99%

Computational ludics

Terui

2011

Theoretical Computer Science

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a b s t r a c tWe reformulate the theory of ludics introduced by J.-Y. Girard from a computational point of view. We introduce a handy term syntax for designs, the main objects of ludics. Our syntax also incorporates explicit cuts for attaining computational expressivity. In addition, we consider design generators that allow for finite representation of some infinite designs. A normalization procedure in the style of Krivine's abstract machine directly works on generators, giving rise to an effective means of computation over infinite designs.The acceptance relation between machines and words, a basic concept in computability theory, is well expressed in ludics by the orthogonality relation between designs. Fundamental properties of ludics are then discussed in this concrete context. We prove three characterization results that clarify the computational powers of three classes of designs. (i) Arbitrary designs may capture arbitrary sets of finite data. (ii) When restricted to finitely generated ones, designs exactly capture the recursively enumerable languages. (iii) When further restricted to cut-free ones as in Girard's original definition, designs exactly capture the regular languages.We finally describe a way of defining data sets by means of logical connectives, where the internal completeness theorem plays an essential role.

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Harmony in the Light of Computational Ludics

Naibo

Takahashi

2021

Electron. Proc. Theor. Comput. Sci.

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Prawitz formulated the so-called inversion principle as one of the characteristic features of Gentzen's intuitionistic natural deduction. In the literature on proof-theoretic semantics, this principle is often coupled with another that is called the recovery principle. By adopting the Computational Ludics framework, we reformulate these principles into one and the same condition, which we call the harmony condition. We show that this reformulation allows us to reveal two intuitive ideas standing behind these principles: the idea of "containment" present in the inversion principle, and the idea that the recovery principle is the "converse" of the inversion principle. We also formulate two other conditions in the Computational Ludics framework, and we show that each of them is equivalent to the harmony condition.* The authors thank the anonymous reviewers for their valuable comments on the early version of this paper. Alberto Naibo's work is partially supported by the ANR project PROGRAMme (ANR-17-CE38-0003-01), by the ANR-DFG project FFIUM (ANR-17-FRAL-0003) and by the ANR project GoA (ANR-20-CE27-0004). Yuta Takahashi's work is supported by JSPS KAKENHI Grant Number JP21K12822. Part of the present work has been prepared during Yuta Takahashi's stay at the IHPST as a postdoctoral researcher, with the support of JSPS (Japan Society for the Promotion of Science) Overseas Research Fellowship (July 2019-June 2021).1 Following [19, § 1.3], we say that a derivation in natural deduction is closed if it has no open assumption, otherwise we say that it is open.

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On the meaning of logical completeness

Cited by 3 publications

References 25 publications

Ludics with Repetitions (Exponentials, Interactive Types and Completeness)

Ludics with Repetitions (Exponentials, Interactive Types and Completeness)

Computational ludics

Harmony in the Light of Computational Ludics

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