14 research outputs found
Realizability for Peano Arithmetic with Winning Conditions in HON Games
International audienceWe build a realizability model for Peano arithmetic based on winning conditions for HON games. First we define a notion of winning strategies on arenas equipped with winning conditions. We prove that the interpretation of a classical proof of a formula is a winning strategy on the arena with winning condition corresponding to the formula. Finally we apply this to Peano arithmetic with relativized quantifications and give the example of witness extraction for Î 0 2 -formulas
Inductive and Functional Types in Ludics
Ludics is a logical framework in which types/formulas are modelled by sets of terms with the same computational behaviour. This paper investigates the representation of inductive data types and functional types in ludics. We study their structure following a game semantics approach. Inductive types are interpreted as least fixed points, and we prove an internal completeness result giving an explicit construction for such fixed points. The interactive properties of the ludics interpretation of inductive and functional types are then studied. In particular, we identify which higher-order functions types fail to satisfy type safety, and we give a computational explanation
Infinets: The parallel syntax for non-wellfounded proof-theory
Logics based on the ”-calculus are used to model induc-tive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. While traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. The present paper investigates a discrepancy found in such proof systems , between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (”MLL â) and study their correctness and sequentialization. Inductive and coinductive reasoning is pervasive in computer science to specify and reason about infinite data as well as reactive properties. Developing appropriate proof systems amenable to automated reasoning over (co)inductive statements is therefore important for designing programs as well as for analyzing computational systems. Various logical settings have been introduced to reason about such inductive and coinductive statements, both at the level of the logical languages modelling (co)induction (such as Martin Löf's inductive predicates or fixed-point logics, also known as ”-calculi) and at the level of the proof-theoretical framework considered (finite proofs with explicit (co)induction rulesĂ la Park [23] or infinite, non-wellfounded proofs with fixed-point unfold-ings) [6-8, 4, 1, 2]. Moreover, such proof systems have been considered over classical logic [6, 8], intuitionistic logic [9], linear-time or branching-time temporal logic [19, 18, 25, 26, 13-15] or linear logic [24, 16, 4, 3, 14]
Infinets: The parallel syntax for non-wellfounded proof-theory
International audienceLogics based on the ”-calculus are used to model inductive and coinductive reasoning and to verify reactive systems. A well-structured proof-theory is needed in order to apply such logics to the study of programming languages with (co)inductive data types and automated (co)inductive theorem proving. While traditional proof system suffers some defects, non-wellfounded (or infinitary) and circular proofs have been recognized as a valuable alternative, and significant progress have been made in this direction in recent years. Such proofs are non-wellfounded sequent derivations together with a global validity condition expressed in terms of progressing threads. The present paper investigates a discrepancy found in such proof systems , between the sequential nature of sequent proofs and the parallel structure of threads: various proof attempts may have the exact threading structure while differing in the order of inference rules applications. The paper introduces infinets, that are proof-nets for non-wellfounded proofs in the setting of multiplicative linear logic with least and greatest fixed-points (”MLL â) and study their correctness and sequentialization
Categorical models of Linear Logic with fixed points of formulas
We develop a denotational semantics of muLL, a version of propositional
Linear Logic with least and greatest fixed points extending David Baelde's
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.Comment: arXiv admin note: text overlap with arXiv:1906.0559
Game semantics for a polymorphic programming language
This article presents a game semantics for higher-rank polymorphism, leading to a new model of the calculus System F, and a programming language which extends it with mutable variables. In contrast to previous game models of polymorphism, it is quite concrete, extending existing categories of games by a simple development of the notion of
question/answer labelling
and the associated
bracketing condition
to represent âcopycat linksâ between positive and negative occurrences of type variables. Some well-known System F encodings of type constructors correspond in our model to simple constructions on games, such as the lifted sum.
We characterize the
generic
types of our model (those for which instantiation reflects denotational equivalence), and show how to construct an interpretation in which all types are generic. We show how mutable variables (Ă la Scheme) may be interpreted in our model, allowing the definition of polymorphic objects with local state. By proving definability of finitary elements in this model using a decomposition argument, we establish a full abstraction result.
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Imperative programs as proofs via game semantics
Game semantics extends the Curry-Howard isomorphism to a three-way
correspondence: proofs, programs, strategies. But the universe of strategies
goes beyond intuitionistic logics and lambda calculus, to capture stateful
programs. In this paper we describe a logical counterpart to this extension, in
which proofs denote such strategies. The system is expressive: it contains all
of the connectives of Intuitionistic Linear Logic, and first-order
quantification. Use of Laird's sequoid operator allows proofs with imperative
behaviour to be expressed. Thus, we can embed first-order Intuitionistic Linear
Logic into this system, Polarized Linear Logic, and an imperative total
programming language.
The proof system has a tight connection with a simple game model, where games
are forests of plays. Formulas are modelled as games, and proofs as
history-sensitive winning strategies. We provide a strong full completeness
result with respect to this model: each finitary strategy is the denotation of
a unique analytic (cut-free) proof. Infinite strategies correspond to analytic
proofs that are infinitely deep. Thus, we can normalise proofs, via the
semantics