Games and full abstraction for the lazy λ-calculus

In this paper we present a fully abstract game model for the pure lazy λ-calculus, i.e. the lazy λ-calculus without constants. In order to obtain this result we introduce a new category of games, the monotonic games, whose main characteristic consists in having an order relation on moves.

Lecture Notes in Computer Science, 1999

We study extensional models of the untyped lambda calculus in the setting of the game semantics introduced by Abramsky, Hyland et alii. In particular we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in a standard category of games, induce the same λ-theory. This is H * , the maximal theory induced already by the classical C.P.O. model D∞, introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

1999

Abstract. We study extensional models of the untyped lambda calculus in the setting of the game semantics introduced by Abramsky, Hyland et alii. In particular we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in a standard category of games, induce the same λ-theory. This is H∗, the maximal theory induced already by the classical C.P.O. model D∞, introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

. We study extensional models of the untyped lambda calculus in the setting of game semantics. In particular, we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in the category of games G, introduced by Abramsky, Jagadeesan and Malacaria, induce the same -theory. This is H , the maximal theory induced already by the classical CPO model D1 , introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction. Introduction -theories are congruences over -terms, which extend pure fi-conversion. Their interest lies in the fact that they correspond to the possible operational (obser- vational) semantics of -calculus. Although researchers have mainly focused on only three such operational semantics, namely those given by head reduction, head lazy reduction or call-by-value reduction, the class of -theories is, in effect, unfathomly rich, see e.g. [6...

2015

Into A-calculus we introduce lazy lsts $\tilde{a} $ whose naive meaning is an infinite tit consisting of variables, ($a_{0}, $ $a_{1} $ , a2, $\ldots$). It is shown that there exist maps which form aGalois connection ffom Parigot’s $\mathrm{A}/\mathrm{i}$-calculus to the A-calculus with lazy list. The translations form not only an equational correspondence but also areduction corre-spondence between the two calculi. 1Introduction We introduce lazy lists into A-calculus. The introduction of infinite lists is motived by a study on denotational semantics of type-free Ap-calculus [Pari92, Pari97, BHF99, BHFOI]. Given domains $U\mathrm{x}U\cong U\cong[Uarrow U] $ such as in Lambek-Scott [LS86], we have established acontinuation denotational semantics of type-free $\lambda\mu$-calculus[Fuji02], which formally coincides with the CPS-translation [HS97, SR98, FujiOl] followed by the direct denotational semantics of the A-calculus [ScOt72, StOy77]. See also the literature [HS97, SR98, SeliOl]...

Information Processing Letters, 1988

We provide a precise characterization of the strictness properties of the rantyped lambda-calculus. Using the notion of head evaluation to model lazy evaluatioq we give a definition of strictness appropriate for the lambda-calculus. We establish that strictness is a necessary and sufficient condition for the 'eager' evaluation of function arguments. We describe an algorithm for computing strictness properties of convergent terms. We show that the natural classifkatioz~ of if-then-e&e as strict in its first and, jointly, strict in its second and third arguments is possible only in an applied lambda-calculus that either includes constants or has some form of t;lpe structure.

1994

Abstract: In this paper we define the Lazy Lambda Calculus with constants, which extends Abramsky's pure lazy Lambda Calculus. This calculus forms a model for modern lazy functional programming languages. Such languages usually provide a call-by-value facility which is able to distinguish between the values _| _ and\ x. _| _. We study the operational and denotational semantics of this calculus both with and without a superimposed type inference system.

Theoretical Computer Science, 1989

A denotationaf semantics for the A-calculus is described. The semantics is cotinuationbased, and so reflects the order in which expressions are evaluated. It provides a means by which lazy functional languages can be better understood.

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2017

We present fully abstract encodings of the call-by-name λ-calculus into HOcore, a minimal higher-order process calculus with no name restriction. We consider several equivalences on the λ-calculus side-normal-form bisimilarity, applicative bisimilarity, and contextual equivalence-that we internalize into abstract machines in order to prove full abstraction.

2021

We propose an intersection type system for an imperative λ -calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the category of ω -algebraic lattices; the solution consists of a filter-model generalizing the well known construction for ordinary λ -calculus. Then the type system is obtained out of the term interpretations into the filter-model itself. The so obtained type system satisfies the “type-semantics” property, and it is sound and complete by construction.