Luca Paolini

Call-by-Value solvability, revisited

In the call-by-value lambda-calculus solvable terms have been characterised by means of call-by-name reductions, which is disappointing and requires complex reasonings. We introduce the value-substitution lambda-calculus, a simple calculus borrowing ideas from Herbelin and Zimmermans call-by-value λ CBV calculus and from Accattoli and Kesners substitution calculus λ sub . In this new setting, we characterise solvable terms as those terms having normal form with respect to a suitable restriction of the rewriting relation.

Parametric parameter passing λ-calculus

A λ-calculus is defined, which is parametric with respect to a set V of input values and subsumes all the different λ-calculi given in the literature, in particular the classical one and the call-by-value λ-calculus of Plotkin. It is proved that it enjoy the confluence property, and a necessary and sufficient condition is given, under which it enjoys the standardization property. Its operational semantics is given through a reduction machine, parametric with respect to both V and a set Vo of output values.

Semantically linear programming languages

We propose a paradigmatic programming language (called SlPCF) which is linear in a semantic sense. SlPCF is not syntactically linear, namely its programs can contain more than one occurrencies of the same variable. We give an interpretation of SlPCF into a model of linear coherence spaces and we show that such semantics is fully abstract with respect to our language. Furthermore, we discuss the independence of new syntactical operators and we address the universality problem.

Call-by-Value Separability and Computability

The aim of this paper is to study the notion of separability in the call-by-value setting.Separability is the key notion used in the Bohm Theorem, proving that syntactically different s?-normal forms are separable in the classical ?-calculus endowed with s-reduction, i.e. in the call-by-name setting. In the case of call-by-value ?-calculus endowed with s? -reduction and ?? -reduction (see Plotkin [7]), it turns out that two syntactically different s?-normal forms are separable too, while the notion of s? -normal form and ?? -normal form is semantically meaningful.An explicit representation of Kleenes recursive functions is presented. The separability result guarantees that the representation makes sense in every consistent theory of call-by-value, i.e. theories in which not all terms are equals.

Linearity and PCF: a semantic insight!

Linearity is a multi-faceted and ubiquitous notion in the analysis and the development of programming language concepts. We study linearity in a denotational perspective by picking out programs that correspond to linear functions between coherence spaces. We introduce a language, named SlPCF*, that increases the higher-order expressivity of a linear core of PCF by means of new operators related to exception handling and parallel evaluation. SlPCF* allows us to program all the finite elements of the model and, consequently, it entails a full abstraction result that makes the reasoning on the equivalence between programs simpler. Denotational linearity provides also crucial information for the operational evaluation of programs. We formalize two evaluation machineries for the language. The first one is an abstract and concise operational semantics designed with the aim of explaining the new operators, and is based on an infinite-branching search of the evaluation space. The second one is more concrete and it prunes such a space, by exploiting the linear assumptions. This can also be regarded as a base for an implementation.

An operational characterization of strong normalization

This paper introduces the Φ-calculus, a new call-by-value version of the λ-calculus, following the spirit of Plotkins λβv-calculus. The Φ-calculus satisfies some interesting properties, in particular that its set of solvable terms coincides with the set of β-strongly normalizing terms in the classical λ-calculus.

The Parametric λ-Calculus

A calculus is a language equipped with some reduction rules. All the calculi we consider in this book share the same language, which is the language of λ-calculus, while they differ each other in their reduction rules. In order to treat them in an uniform way we define a parametric calculus, the λΔ-calculus, which gives rise to different calculi by different instantiations of the parameter Δ. In Part I we study the syntactical properties of the λΔ-calculus, and in particular those of its two most important instances, the call-by-name and the call-by-value λ-calculi. The λΔ-calculus has been introduced first in [85] and further studied in [74]. We use the terminology of [9].

A Class of Reversible Primitive Recursive Functions

Reversible computing is bi-deterministic which means that its execution is both forward and backward deterministic, i.e. next/previous computational step is uniquely determined. Various approaches exist to catch its extensional or intensional aspects and properties. We present a class RPRF of reversible functions which holds at bay intensional aspects and emphasizes the extensional side of the reversible computation by following the style of Dedekind-Robinson Primitive Recursive Functions. The class RPRF is closed by inversion, can only express bijections on integers - not only natural numbers -, and it is expressive enough to simulate Primitive Recursive Functions, of course, in an effective way.

From Featured Transition Systems to Modal Transition Systems with Variability Constraints

We present an automatic technique to transform a subclass of featured transition systems into modal transition systems with additional sets of variability constraints in the specific format accepted by the variability model checker VMC. Both formal models are widely used in the field of software product line engineering and both come with a dedicated model checker. The transformation serves two purposes. First, it contributes to a better understanding of the fundamental differences between the two approaches, basically concerning the way in which variability constraints are represented (in terms of features and actions, respectively). Second, it paves the way to compare the modelling and analysis of product line behaviour in two different settings.

Essential and Relational Models

Intersection type assignment systems can be used as a general framework for building logical models of �-calculus that allow to reason about the denotation of terms in a finitary way. We define essential models (a new class of logical models) through a parametric type assignment system using non idempotent intersection types. Under an interpretation of terms based on typings instead than the usual one based on types, every suitable instance of the parameters induces a �-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of �-models arising from a category of sets and relations. In the general framework of denotational semantics of λ-calculus, logical models are a particular class of λ-models supplying a finitary description of the interpretation of terms, through type assignment systems. Types are built from a set of constants, via two type-constructors: the arrow (→) and the intersection (∧). Terms are interpreted as sets of types, so reasoning about the interpretation of a term in these models can be done via type inference; in fact, in order to prove the equivalence between two terms, it is sufficient to show that they can be assigned the same set oftypes. Although the type inference is undecidable, logical models are concrete tools for reasoning in finitary way on the interpretation of terms, since a (finite) derivation grasps a finite piece of the semantic-interpretations. The relationship between logical models and domain-theoretical models has been widely studied, and it has been proved that some interesting classes of such models can be described in logical form. For instance, filter models supply a logical description of a class of Scott models based on continuous functions, since they can be seen as a restriction of the domain theory in logical form, which goes back to Stone duality (see Abramsky (1991)). A first characterization of a logical model where types represent continuous functions is in Coppo et al. (1984), a sketch of the proof of the correspondence between