Properties of Set Functors

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Modeling Fresh Names in the π-calculus Using Abstractions

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An Open Logical Framework

The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of \u25a1-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems, which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincar Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP. We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP: strong normalization, confluence and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution and reduction in the arguments. Moreover, we provide a canonical presentation of LFP, based on a suitable extension of the notion of \u3b2\u3b7-long normal form, allowing for smooth formulations of adequacy statements. \ua9 The Author, 2013

A Framework for Defining Logical Frameworks

In this paper, we introduce a General Logical Framework, called GLF, for defining Logical Frameworks, based on dependent types, in the style of the well known Edinburgh Logical Framework LF. The framework GLF features a generalized form of lambda abstraction where beta-reductions fire provided the argument satisfies a logical predicate and may produce an n-ary substitution. The type system keeps track of when reductions have yet to fire. The framework GLF subsumes, by simple instantiation, LF as well as a large class of generalized constrained-based lambda calculi, ranging from well known restricted lambda calculi, such as Plotkin's call-by-value lambda calculus, to lambda calculi with patterns. But it suggests also a wide spectrum of completely new calculi which have intriguing potential as Logical Frameworks. We investigate the metatheoretical properties of the calculus underpinning GLF and illustrate its expressive power. In particular, we focus on two interesting instantiations of GLF. The first is the Pattern Logical Framework (PLF), where applications fire via pattern-matching in the style of Cirstea, Kirchner, and Liquori. The second is the Closed Logical Framework (CLF) which features, besides standard beta-reduction, also a reduction which fires only if the argument is a closed term. For both these instantiations of GLF we discuss standard metaproperties, such as subject reduction, confluence and strong normalization. The GLF framework is particularly suitable, as a metalanguage, for encoding rewriting logics and logical systems, where rules require proof terms to have special syntactic constraints, e.g. logics with rules of proof, in addition to rules of derivations, such as, e.g., modal logics, and call-by-value lambda calculus

Conway games, algebraically and coalgebraically

Using coalgebraic methods, we extend Conway's theory of games to possibly non-terminating, i.e. non-wellfounded games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on non-losing strategies. Hypergames are a fruitful metaphor for non-terminating processes, Conway's sum being similar to shuffling. We develop a theory of hypergames, which extends in a non-trivial way Conway's theory; in particular, we generalize Conway's results on game determinacy and characterization of strategies. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized Grundy-Sprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. We indicate a number of intriguing directions for future work. We briefly compare hypergames with other notions of games used in computer science.Comment: 30 page

On Quantitative Algebraic Higher-Order Theories

International audienceWe explore the possibility of extending Mardare et al.’s quantitative algebras to the structures which naturally emerge from Combinatory Logic and the λ-calculus. First of all, we show that the framework is indeed applicable to those structures, and give soundness and completeness results. Then, we prove some negative results clearly delineating to which extent categories of metric spaces can be models of such theories. We conclude by giving several examples of non-trivial higher-order quantitative algebras

Principal Types as Lambda Nets

We show that there are connections between principal type schemata, cut-free ?-nets, and normal forms of the ?-calculus, and hence there are correspondences between the normalisation algorithms of the above structures, i.e. unification of principal types, cut-elimination of ?-nets, and normalisation of ?-terms. Once the above correspondences have been established, properties of the typing system, such as typability, subject reduction, and inhabitation, can be derived from properties of ?-nets, and vice-versa. We illustrate the above pattern on a specific type assignment system, we study principal types for this system, and we show that they correspond to ?-nets with a non-standard notion of cut-elimination. Properties of the type system are then derived from results on ?-nets

Multigames and strategies, coalgebraically

Coalgebraic games have been recently introduced as a generalization of Conway games and other notions of games arising in different contexts. Using coalgebraic methods, games can be viewed as elements of a final coalgebra for a suitable functor, and operations on games can be analyzed in terms of (generalized) coiteration schemata. Coalgebraic games are sequential in nature, i.e., at each step either the Left (L) or the Right (R) player moves (global polarization); moreover, only a single move can be performed at each step. Recently, in the context of Game Semantics, concurrent games have been introduced, where global polarization is abandoned, and multiple moves are allowed. In this paper, we introduce coalgebraic multigames, which are situated half-way between traditional sequential games and concurrent games: global polarization is still present, however multiple moves are possible at each step, i.e., a team of L/R players moves in parallel. Coalgebraic operations, such as sum and negation, can be naturally defined on multigames. Interestingly, sum on coalgebraic multigames turns out to be related to Conway's selective sum on games, rather than the usual (sequential) disjoint sum. Selective sum has a parallel nature, in that at each step the current player performs a move in at least one component of the sum game, while on disjoint sum the current player performs a move in exactly one component at each step. A presentation of strategies on coalgebraic games is given via a final coalgebra of a pair of mutually recursive functors, and a suitable notion of simulation. A monoidal closed category of coalgebraic multigames in the vein of a Joyal category of Conway games is then built. The relationship between coalgebraic multigames and games is then formalized via an equivalence of the multigame category and a monoidal closed category of coalgebraic games where tensor is selective sum. \ua9 2015

From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems

We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we examine set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation "up-to", and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T -coiterative functions. These are morphisms into final coalgebras, satisfying the T -coiteration scheme, which is a generalization of both the coiteration and the corecursion scheme. We generalize Rutten's transformation from coalgebraic bisimulations to set-theoretic bisimulations, in order to cover also the case of bisimulations "up-to". A list of examples of set-theoretic coinductive specifications which appear not to be easily expressible in coalgebraic terms are discussed. Introduction Coinductive definitions and ..

A Uniform Syntactical Method for Proving Coinduction Principles in λ-calculi

Coinductive characterizations of various observational congruences which arise in the semantics of λ-calculus, when -terms are evaluated according to various reduction strategies, are discussed. We analyze and extend to non-lazy strategies, both deterministic and nondeterministic, Howe's congruence candidate method for proving the coincidence of the applicative (bisimulation) and the contextual equivalences. This purely syntactical method is based itself on a coinductive argument