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Register a new userOn the denotational semantics of Linear Logic with least and greatest fixed points of formulas
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Added by
Thomas Ehrhard
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We develop a denotational semantics of Linear Logic with least and greatest fixed points in coherence spaces (where both fixed points are interpreted in the same way) and in coherence spaces with totality (where they have different interpretations). These constructions can be carried out in many different denotational models of LL (hypercoherences, Scott semantics, finiteness spaces etc). We also present a natural embedding of G{o}del System T in LL with fixed points thus enforcing the expressive power of this system as a programming language featuring both normalization and a huge expressive power in terms of data types.
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We develop a denotational semantics of muLL, a version of propositional Linear Logic with least and greatest fixed points extending David Baeldes propositional muMALL with exponentials. Our general categorical setting is based on the notion of Seely category and on strong functors acting on them. We exhibit two simple instances of this setting. In the first one, which is based on the category of sets and relations, least and greatest fixed points are interpreted in the same way. In the second one, based on a category of sets equipped with a notion of totality (non-uniform totality spaces) and relations preserving them, least and greatest fixed points have distinct interpretations. This latter model shows that muLL enjoys a denotational form of normalization of proofs.
In this paper we provide two new semantics for proofs in the constructive modal logics CK and CD. The first semantics is given by extending the syntax of combinatorial proofs for propositional intuitionistic logic, in which proofs are factorised in a linear fragment (arena net) and a parallel weakening-contraction fragment (skew fibration). In particular we provide an encoding of modal formulas by means of directed graphs (modal arenas), and an encoding of linear proofs as modal arenas equipped with vertex partitions satisfying topological criteria. The second semantics is given by means of winning innocent strategies of a two-player game over modal arenas. This is given by extending the Heijltjes-Hughes-Stra{ss}burger correspondence between intuitionistic combinatorial proofs and winning innocent strategies in a Hyland-Ong arena. Using our first result, we provide a characterisation of winning strategies for games on a modal arena corresponding to proofs with modalities.
The word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as processes which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting
We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have been proposed before are not. It is organised as a model of propositional linear logic in which all the connectives have intuitive probabilistic interpretations. In addition, it has least fixed points for all maps, so it can interpret recursion.
Higher-order recursion schemes are recursive equations defining new operations from given ones called terminals. Every such recursion scheme is proved to have a least interpreted semantics in every Scotts model of lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in lambda-calculus by an endofunctor Hlambda and they explained simultaneous substitution of lambda-terms by proving that the presheaf of lambda-terms is an initial Hlambda-monoid. Here we work with the presheaf of rational infinite lambda-terms and prove that this is an initial iterative Hlambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid.
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