Marina Lenisa

Distributivity for endofunctors, pointed and co-pointed endofunctors, monads and comonads

We generalise the notion of a distributive law between a monad and a comonad to consider weakened structures such as pointed or co-pointed endofunctors, or endofunctors. We investigate Eilenberg-Moore and Kleisli constructions for each of these possibilities. Then we consider two applications of these weakened notions of distributivity in detail. We characterise Turi and Plotkins model of GSOS as a distributive law of a monad over a co-pointed endofunctor, and we analyse generalised coiteration and coalgebraic coinduction “up-to” in terms of a distributive law of the underlying pointed endofunctor of a monad over an endofunctor.

Category theory for operational semantics

We use the concept of a distributive law of a monad over a copointed endofunctor to define and develop a reformulation and mild generalisation of Turi and Plotkins notion of an abstract operational rule. We make our abstract definition and give a precise analysis of the relationship between it and Turi and Plotkins definition. Following Tuff and Plotkin, our definition, suitably restricted, agrees with the notion of a set of GSOS-rules, allowing one to construct both an operational model and a canonical, internally fully abstract denotational model. Going beyond Turi and Plotkin, we construct what might be seen as large-step operational semantics from small-step operational semantics and we show how our definition allows one to combine distributive laws, in particular accounting for the combination of operational semantics with congruences.

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 introduce set-theoretic generalizations of the coinduction proof principle, in the spirit of Milners 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 the corecursion scheme. We show how to describe coalgebraic F-bisimulations as set-theoretical ones. A list of examples of set-theoretic coinductions which appear not to be easily amenable to coalgebraic terms are discussed.

A Fully Complete PER Model for ML Polymorphic Types

We present a linear realizability technique for building Partial Equivalence Relations (PER) categories over Linear Combinatory Algebras. These PER categories turn out to be linear categories and to form an adjoint model with their co-Kleisli categories. We show that a special linear combinatory algebra of partial involutions, arising from Geometry of Interaction constructions, gives rise to a fully and faithfully complete modelfor ML polymorphic types of system F.

Final semantics for the pi-calculus

In this paper we discuss final semantics for the π-calculus, a process algebra which models systems that can dynamically change the topology of the channels. We show that the final semantics paradigm, originated by Aczel and Rutten for CCS-like languages, can be successfully applied also here. This is achieved by suitably generalizing the standard techniques so as to accommodate the mechanism of name creation and the behaviour of the binding operators peculiar to the λ-calculus. As a preliminary step, we give a higher order presentation of the π-calculus using as metalanguage LF,a logical framework based on typed λ-calculus. Such a presentation highlights the nature of the binding operators and elucidates the role of free and bound channels. The final semantics is defined making use of this higher order presentation, within a category of hypersets.

Generalized Coiteration Schemata

Abstract Coiterative functions can be explained categorically as final coalgebraic morphisms, once coinductive types are viewed as final coalgebras. However, the coiteration schema which arises in this way is too rigid to accommodate directly many interesting classes of circular specifications. In this paper, building on the notion of T-coiteration introduced by the third author and capitalizing on recent work on bialgebras by Turi-Plotkin and Bartels, we introduce and illustrate various generalized coiteration patterns. First we show that, by choosing the appropriate monad T, T-coiteration captures naturally a wide range of coiteration schemata, such as the duals of primitive recursion and course-of-value iteration, and mutual coiteration. Then we show that, in the more structured categorical setting of bialgebras, T-coiteration captures guarded coiterations schemata, i.e. specifications where recursive calls appear guarded by predefined algebraic operations.

Final Semantics for a Higher Order Concurrent Language

We show that adequate semantics can be provided for imperative higher order concurrent languages simply using syntactical final coalgebras. In particular we investigate and compare various behavioural equivalences on higher order processes defined by finality using hypersets and c.m.s. s. Correspondingly, we derive various coinduction and mixed induction-coinduction proof principles for establishing these equivalences.

Some Results on the Full Abstraction Problem for Restricted Lambda Calculi

Issues in the mathematical semantics of two restrictions of the λ-calculus, i.e. λI-calculus and λv-calculus, are discussed. A fully abstract model for the natural evaluation of the former is defined using complete partial orders and strict Scott-continuous functions. A correct, albeit non-fully abstract, model for the SECD evaluation of the latter is denned using Girards coherence spaces and stable functions. These results are used to illustrate the interest of the analysis of the fine structure of mathematical models of programming languages.

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 @b-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 Plotkins call-by-value lambda calculus, to lambda calculi with patterns. But it suggests also a wide spectrum of 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 @b-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 logic, and call-by-value lambda calculus.

Processes and Hyperuniverses

We show how to define domains of processes, which arise in the denotational semantics of concurrent languages, using hypersets, i.e. non-wellfounded sets. In particular we discuss how to solve recursive equations involving set-theoretic operators within hyperuniverses with atoms. Hyperuniverses are transitive sets which carry a uniform topological structure and include as a clopen subset their exponential space (i.e. the set of their closed subsets) with the exponential uniformity. This approach allows to solve many recursive domain equations of processes which cannot be even expressed in standard Zermelo-Fraenkel Set Theory, e.g. when the functors involved have negative occurrences of the argument. Such equations arise in the semantics of concurrrent programs in connection with function spaces and higher order assignment. Finally, we briefly compare our results to those which make use of complete metric spaces, due to de Bakker, America and Rutten.