Felice Cardone

Type inference with recursive types: syntax and semantics

Abstract In this paper we study type inference systems for λ-calculus with a recursion operator over types. The main syntactical properties, notably the existence of principal type schemes, are proved to hold when recursive types are viewed as finite notations for infinite (regular) type expressions representing their infinite unfoldings. Exploiting the approximation structure of a model for the untyped language of terms, types are interpreted as limits of sequences of their approximations. We show that the interpretation is essentially unique and that two types have equal interpretation if and only if their infinite unfoldings are identical. Finally, a completeness theorem is proved to hold w.r.t. the specific model we consider for a natural (infinitary) extension of the type inference system.

Relational Semantics for Recursive Types and Bounded Quantification

The language Fun [Cardelli, Wegner, 1985] is a typed polymorphic lambda calculus with record types, quantification over subtypes of a given type and inheritance. In this paper it is extended with recursive types, and the consistency of the resulting language is proved by constructing an interpretation of its types as partial equivalence relations of a special kind, terms being interpreted as equivalence classes, modulo such relations, of elements of a model of the underlying language of untyped terms.

Combining type disciplines

We present a type inference system for pure λ-calculus which includes, in addition to arrow types, also universal and existential type quantifiers, intersection and union types, and type recursion. The interest of this system lies in the fact that it offers a possibility to study in a unified framework a wide range of type constructors. We investigate the main syntactical properties of the system, including an analysis of the preservation of types under parallel reduction strategies, leading to a form of the subject-reduction property. We describe a model for this system where types are special subsets of a D∞ model for λ-calculus, without imposing any formal contractiveness constraint on types of the kind considered for a closely related system by MacQueen, Plotkin and Sethi (1986).

Decidability Properties of Recursive Types

In this paper we study decision problems and invertibility for two notions of equivalence of recursive types. In particular, for recursive types presented by means of a recursion operator μ, we describe an algorithm showing that the natural equivalence generated by finitely many steps of folding and unfolding of μ-types is decidable. For recursive types presented by finite systems of recursive equations, we give a thoroughly coinductive characterization of the equivalence induced by their interpretation as infinite (regular) trees, from which the decidability of this equivalence follows. A formal proof of the former result, to our knowledge, has never appeared in the literature. The latter result, on the contrary, is known but we present here a new proof obtained as an application of general coalgebraic facts to the theory of recursive types. From these results invertibility is easily proved for both equivalences.

Recursive types for Fun

Abstract The language Fun [13] is a typed polymorphic lambda calculus with a notion of subtyping and quantifiers ranging over subtypes of a given type. In this paper we show that it is consistent to allow recursive type definitions in Fun, by constructing an interpretation of types as partial equivalence relations of a special kind, terms being interpreted as equivalence classes, modulo such relations, of elements of a model for an underlying untyped language.

The geometry and algebra of commitment

We propose a formal description, by means of graphical and categorical structures, of mechanisms for handling the dynamics of rights and obligations familiar in jurisprudence.We argue that the formal study of commitment in this setting can contribute new insights to the analysis of a large variety of communicative situations relevant to formal pragmatics.

A coinductive completeness proof for the equivalence of recursive types

We prove that the equivalence of recursive types induced by the equality of their infinite unfoldings coincides with the equality of their interpretations as closures over the -model P.

An Algebraic Approach to the Interpretation of Recursive Types

We present an algebraic framework for the interpretation of type systems including type recursion, where types are interpreted as partial equivalence relations over a Scott domain. We use the notion of iterative algebra, introduced by J. Tiuryn [26] as a counterpart to the categorical notion of iterative algebraic theory by C.C. Elgot [15]. We show that a suitable collection of partial equivalence relations is closed under type constructors and forms an iterative algebra. The existence of type interpretations follows from the initiality, in the class of iterative algebras, of the algebra of regular infinite trees obtained by infinitely unfolding recursive types.

Computers and the Mechanics of Communication

Computers have become an integral part of a vast range of coordination patterns among human activities which go far beyond mere calculation. The conceptual relevance of this new field of application of computers has been advocated by Carl Adam Petri (1926–2010) and Anatol W. Holt (1927–2010), two computer scientists best known for their contributions to the subject of Petri nets, a graphical formalism for describing the causal dependence of events in systems distributed in space. We outline some fundamental, mainly epistemological aspects of their vision of the computer as a “communication machine.”

Continuity in Semantic Theories of Programming

Continuity is perhaps the most familiar characterization of the finitary character of the operations performed in computation. We sketch the historical and conceptual development of this notion by interpreting it as a unifying theme across three main varieties of semantical theories of programming: denotational, axiomatic and event-based. Our exploration spans the development of this notion from its origins in recursion theory to the forms it takes in the context of the more recent event-based analyses of sequential and concurrent computations, touching upon the relations of continuity with non-determinism.