Paula Severi

Perpetual Reductions in λ-Calculus

This paper surveys a part of the theory ofs-reduction in?-calculus which might aptly be calledperpetual reductions. The theory is concerned withperpetual reduction strategies, i.e., reduction strategies that compute infinite reduction paths from?-terms (when possible), and withperpetual redexes, i.e., redexes whose contraction in?-terms preserves the possibility (when present) of infinite reduction paths. The survey not only recasts classical theorems in a unified setting, but also offers new results, proofs, and techniques, as well as a number of applications to problems in?-calculus and type theory.

Pure Type Systems with Definitions

In this paper, an extension of Pure Type Systems (PTSs) with definitions is presented. We prove this extension preserves many of the properties of PTSs. The main result is a proof that for many PTSs, including the Calculus of Constructions, this extension preserves strong normalisation.

Infinitary lambda calculus and discrimination of Berarducci trees

We propose an extension of lambda calculus for which the Berarducci trees equality coincides with observational equivalence, when we observe rootstable or rootactive behavior of terms. In one direction the proof is an adaptation of the classical Bohm out technique. In the other direction the proof is based on confluence for Strongly converging reductions in this extension.

An Extensional Böhm Model

We show the existence of an infinitary confluen t and normalising extension of the finite extensional lambda calculus with beta and eta. Besides infinite beta reductions also infinite eta reductions are possible in this extension, and terms without head normal form can be reduced to bottom. As corollaries we obtain a simple, syntax based construction of an extensional Bohm model of the finite lambda calculus; and a simple, syntax based proof that two lambda terms have the same semantics in this model if and only if theyha ve the same eta-Bohm tree if and only if they are observationally equivalent wrt to beta normal forms. The confluence proof reduces confluence of beta, bottom and eta via infinitary commutation and postponement arguments to confluence of beta and bottom and confluence of eta.We give counterexamples against confluence of similar extensions based on the identification of the terms without weak head normal form and the terms without top normal form (rootactive terms) respectively.

Type inference for pure type systems

Abstract In this paper we define a type inference semi-algorithm for singly sorted pure type systems. For that, we define the notion of pure type systems without the Φ -condition and a mapping from pure type systems without the Φ -condition to pure type systems. This allows us to prove the two main results: first that weak normalisation is preserved by the extension and second the correctness of the type inference semi-algorithm.

Pure type systems with corecursion on streams: from finite to infinitary normalisation

In this paper, we use types for ensuring that programs involving streams are well-behaved. We extend pure type systems with a type constructor for streams, a modal operator next and a fixed point operator for expressing corecursion. This extension is called Pure Type Systems with Corecursion (CoPTS). The typed lambda calculus for reactive programs defined by Krishnaswami and Benton can be obtained as a CoPTS. CoPTSs allow us to study a wide range of typed lambda calculi extended with corecursion using only one framework. In particular, we study this extension for the calculus of constructions which is the underlying formal language of Coq. We use the machinery of infinitary rewriting and formalise the idea of well-behaved programs using the concept of infinitary normalisation. The set of finite and infinite terms is defined as a metric completion. We establish a precise connection between the modal operator (• A) and the metric at a syntactic level by relating a variable of type (• A) with the depth of all its occurrences in a term. This syntactic connection between the modal operator and the depth is the key to the proofs of infinitary weak and strong normalisation.

Studies of a Theory of Specifications with Built-in Program Extraction

We present a Theory of Specifications based on Martin-Löfs type theory, with rules for simultaneously constructing programs and their correctness proofs. The theory contains types for representing specifications whose corresponding notion of implementation is that of a pair formed by a program and a correctness proof. The rules of the theory are such that in implementations the program parts appear mixed together with the proof parts. A confluent and normalizing computational relation performs the task of separating programs from proofs. As a consequence, every implementation computes to a pair composed of a program and a proof of its correctness, and so the program extraction procedure is immediate.

NOMINAL COALGEBRAIC DATA TYPES WITH APPLICATIONS TO LAMBDA CALCULUS

We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.

Continuity and discontinuity in lambda calculus

This paper studies continuity of the normal form and the context operators as functions in the infinitary lambda calculus. We consider the Scott topology on the cpo of the finite and infinite terms with the prefix relation. We prove that the only continuous parametric trees are Bohm and Levy–Longo trees. We also prove a general statement: if the normal form function is continuous then so is the model induced by the normal form; as well as the converse for parametric trees. This allows us to deduce that the only continuous models induced by the parametric trees are the ones of Bohm and Levy–Longo trees. As a first application, we prove that there is an injective embedding from the infinitary lambda calculus of the ∞η-Bohm trees in D∞. As a second application, we study the relation between the Scott topology on the prefix relation and the tree topologies. This allows us to prove that the only parametric tree topologies in which all context operators are continuous and the approximation property holds are the ones of Bohm and Levy–Longo. As a third application, we give an explicit characterisation of the open sets of the Bohm and Levy–Longo tree topologies.

Weakening the Axiom of Overlap in Infinitary Lambda Calculus

In this paper we present a set of necessary and sucient conditions on a set of lambda terms to serve as the set of meaningless terms in an infinitary bottom extension of lambda calculus. So far only a set of sucient conditions was known for choosing a suitable set of meaningless terms to make this construction produce confluent extensions. The conditions covered the three main known examples of sets of meaningless terms. However, the much later construction of many more examples of sets of meaningless terms satisfying the sucient conditions renewed the interest in the necessity question and led us to reconsider the old conditions. The key idea in this paper is an alternative solution for solving the overlap between beta reduction and bottom reduction. This allows us to reformulate the Axiom of Overlap, which now determines together with the other conditions a larger class of sets of meaningless terms. We show that the reformulated conditions are not only sucient but also necessary for obtaining a confluent and normalizing infinitary lambda beta bottom calculus. As an interesting consequence of the necessity proof we obtain for infinitary lambda calculus with beta and bot reduction that confluence implies normalization.