Lazy Λ-calculus Research Articles

Nominal SOS

Plotkinʼs style of Structural Operational Semantics (SOS) has become a de facto standard in giving operational semantics to formalisms and process calculi. In many such formalisms and calculi, the concepts of names, variables and binders are essential ingredients. In this paper, we propose a formal framework for dealing with names in SOS. The framework is based on the Nominal Logic of Gabbay and Pitts and hence is called Nominal SOS. We define nominal bisimilarity, an adaptation of the notion of bisimilarity that is aware of binding. We provide evidence of the expressiveness of the framework by formulating the early π-calculus and Abramskyʼs lazy λ-calculus within Nominal SOS. For both calculi we establish the operational correspondence with the original calculi. Moreover, in the context of the π-calculus, we prove that nominal bisimilarity coincides with Sangiorgiʼs open bisimilarity and in the context of the λ-calculus we prove that nominal bisimilarity coincides with Abramskyʼs applicative bisimilarity.

Solution to the Range Problem for Combinatory Logic

The λ-theory H is obtained from β-conversion by identifying all closed unsolvable terms (or, equivalently, terms without head normal form). The range problem for the theory H asks whether a closed term has always (up to equality in H) either an infinite range or a singleton range (that is, it is a constant function). Here we give a solution to a natural version of this problem, giving a positive answer for the theory H restricted to Combinatory Logic. The method of proof applies also to the Lazy λ-Calculus.

Strong normalization from an unusual point of view

A new complete characterization of β-strong normalization is given, both in the classical and in the lazy λ-calculus, through the notion of potential valuability inside two suitable parametric calculi.

A semantics for lambda calculi with resources

We present the λ-calculus with resources λr, and two variants of it: a deterministic restriction λm and an extension λcr with a convergence testing operator. These calculi provide a control on the substitution process – deadlocks may arise if not enough resources are available to carry out all the substitutions needed to pursue a computation. The design of these calculi was motivated by Milner's encoding of the λ-calculus in the π-calculus. As Boudol and Laneve have shown elsewhere, the discriminating power of λm (given by the contextual observational equivalence) over λ-terms coincides with that induced by Milner's π-encoding, and coincides also with that provided by the lazy algebraic semantics (Lévy–Longo trees). The main contribution of this paper is model-theoretic. We define and solve an appropriate domain equation, and show that the model thus obtained is fully abstract with respect to λcr. The techniques used are in the line of those used by Abramsky for the lazy λ-calculus, the main departure being that the resource-consciousness of our calculi leads us to introduce a non-idempotent form of intersection types.

Termination, deadlock and divergence in the λ-calculus with multiplicities

The λ-calculus with multiplicities is a refinement of the lazy λ-calculus where the argument in an application comes with a multiplicity, which is an upper bound to the number of its uses. This introduces potential deadlocks in the evaluation. We study various observation scenarios for this calculus, depending on the way we observe termination, deadlock and divergence. We relate these observational semantics with the intensional interpretation of λ-terms, by means of Lévy-Longo trees. We show in particular that the inclusion of such trees coincides with the “flat” observational semantics, where deadlocks and convergence are distinguished.

A Fully Abstract Denotational Model for Higher-Order Processes

A higher-order process calculus is defined in which one can describe processes which transmit as messages other processes; it may be viewed as a generalisation of the lazy λ-calculus. We present a denotational model for the language, obtained by generalising the domain equation for Abramsky′s model of the lazy λ-calculus. It is shown to be fully abstract with respect to three different behavioural preorders. The first is based on observing the ability of processes to perform an action in all contexts, the second on testing , and the final one on satisfying certain kinds of modal formulae.

Full Abstraction in the Lazy Lambda Calculus

A theory of lazy λ-calculus is developed as a basis for lazy functional programming languages. This is motivated by a mismatch between the "standard" (i.e., sensible) theory of the classical λ-calculus and the practice of lazy functional programming. Part I sets up the fundamentals of the lazy λ-calculus starting from the pure lazy language λ l . We develop the rudiments of weakly sensible λ-theory and characterize λ l as the maximal such theory. The axiomatic framework of our theory is based on the notion of applicative transition systems which gives rise to lazy λ-models. The subclass of lazy λ-models which are operationally extensional are called lambda transition systems (lts′s). The canonical model of the lazy λ-calculus is the initial solution to the domain equation D ≅ [ D → D ] ⊥ . We set up the domain logic corresponding to the same type expression. In part II, we recast the full abstraction problem in the lazy λ-calculus. The pure lazy language λ l is not equationally fully abstract w.r.t. D . In fact, λ l , λ l ω , (a labelled version of λ l ), and λ l c (λ l augmented with a convergence testing constant) are all not equationally fully abstract w.r.t. D . We illustrate two approaches to full abstraction: • The expansive approach consists in enriching the language by augmenting it with appropriate constants, thereby enabling all finite semantic information to be represented syntactically as program phrases using the machinery of domain logic developed in Part I. We show that λ, which is λ augmented with a parallel convergence testing constant, is conditionally fully abstract . • The restrictive approach regards the language as prescribed and tailors the model to fit by "cutting down" (i.e., "quotienting out" by a bisimulation logical relation) the existing "over-generous" canonical model to an appropriate substructure. We establish a general method to construct inequationally fully abstract lazy λ-models which are retracts of for a class of sufficiently expressive lts′s in which the -projection maps are internally definable. The restrictive approach solves the abstraction problem for λ l c and λ l ω but not λ l . The full abstraction problem for λ l is reduced to a conjecture of the conservativity of λ l ω over λ l . Finally, we study lazy languages with ground data and show that it is wrong to conflate parallel or with parallel convergence testing.

Functions as processes

This paper exhibits accurate encodings of the λ-calculus in the π-calculus. The former is canonical for calculation with functions, while the latter is a recent step (Milner et al. 1989) towards a canonical treatment of concurrent processes. With quite simple encodings, two λ-calculus reduction strategies are simulated very closely; each reduction in λ-calculus is mimicked by a short sequence of reductions in π-calculus. Abramsky's precongruence of applicative bisimulation (Abramsky 1989) over λ-calculus is compared with that induced by the encoding of the lazy λ-calculus into π-calculus; a similar comparison is made for call-by-value λ-calculus.