This special double issue of Mathematical Structures in Computer Science is in honour of Roger Hindley and is devoted to the topic of lambda-calculus and logic.

It is a great pleasure for us to greet Roger Hindley on the occasion of his retirement from the University of Wales, Swansea, and his 60th birthday. We have known Roger for many years and we have had the chance to collaborate with him and appreciate his intellectual standard, his remarkable mathematical rigor, and his inexhaustible sense of humour. This has enabled Roger to step back critically even in the face of a difficult mathematical task and help to solve it by a new way of looking at it.

Roger Hindley's dissertation concerned the Church–Rosser Theorem and was a significant contribution to the topic. His subsequent work spanned many aspects of lambda-calculus, covering both its models and applications. To mention just a few, he produced work on axioms for Curry's strong (eta) reduction, comparing lambda and combinatory reductions (and models), models for type assignment, and formulas as types for some nonstandard systems (intersection types, BCK systems, etc.).

Roger Hindley collaborated with Jonathan Seldin on two well-known introductory books on the subject (Bruce Lercher also collaborated as an author on the first of these). More recently, he has published an introduction to type assignment. He was also co-author with H. B. Curry and J. Seldin on Combinatory Logic, vol. II, which is an important research publication on the subject.

Roger has played an important role in the lambda-calculus community over the years as that community has grown; in particular, he has been an active organiser of many conferences on the topic. In fact, his success in disseminating knowledge about the lambda calculus, particularly in the United Kingdom, means that Roger may be considered a ‘Godfather’ of ML and its type system.

(In preparing this special issue of Mathematical Structures in Computer Science, we have been fortunate enough to receive too many excellent papers for one double issue. As a result, additional papers by colleagues who wish to honour Roger will appear in future issues of this journal.)

We use a syntactical notion of Kripke models to obtain interpretations of subsystems of arithmetic in their intuitionistic counterparts. This yields, in particular, a new proof of Buss' result that the Skolem functions of Bounded Arithmetic are polynomial time computable.

We look at two different ways of interpreting logic in the dependent type system λP. The first is by a direct formulas-as-types interpretation à la Howard where the logical derivation rules are mapped to derivation rules in the type system. The second is by viewing λP as a Logical Framework, following Harper et al. (1987) and Harper et al. (1993). The type system is then used as the meta-language in which various logics can be coded.

We give a (brief) overview of known (syntactical) results about λP. Then we discuss two issues in some more detail. The first is the completeness of the formulas-as-types embedding of minimal first-order predicate logic into λP. This is a remarkably complicated issue, a first proof of which appeared in Geuvers (1993), following ideas in Barendsen and Geuvers (1989) and Swaen (1989). The second issue is the minimality of λP as a logical framework. We will show that some of the rules are actually superfluous (even though they contribute nicely to the generality of the presentation of λP).

At the same time we will attempt to provide a gentle introduction to λP and its various aspects and we will try to use little inside knowledge.

Recall that M is easy if it is consistent with every combinator. We say that M is m-easy if there is no proof with < m + 1 steps that M is inconsistent with any combinator. Obviously, if M is easy, it is m-easy for each m. Here we shall show that for infinitely many m there are m but not m + 1 easy terms.

We study the relationship between the encodings of the λ-calculus into π-calculus, the Continuation Passing Style (CPS) transforms, and the compilation of the Higher-Order π-calculus (HOπ) into π-calculus. We factorise the π-calculus encodings of (untyped as well as simply-typed) call-by-name and call-by-value λ-calculus into three steps: a CPS transform, the inclusion of CPS terms into HOπ and the compilation from HOπ to π-calculus. The factorisations are used both to derive the encodings and to prove their correctness.

We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on λ-terms and, more generally, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss, in particular, what we call the cartesian coinduction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of final semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applicative structure can be construed as a strongly extensional coalgebra for the functor ([Pscr ](- × -))[oplus ]([Pscr ](- × -)). In this paper we present two general methods for showing the soundness of this principle. The first applies to approximable applicative structures – many CPO λ-models in the literature and the corresponding quotient models of indexed terms turn out to be approximable applicative structures. The second method is based on Howe's congruence candidates, which applies to many observational equivalences. Structures satisfying cartesian coinduction are precisely those applicative structures that can be modelled using the standard set-theoretic application in non-wellfounded theories of sets, where the Foundation Axiom is replaced by the Axiom X1 of Forti and Honsell.

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.

We give a negative answer to the question of whether every partial combinatory algebra can be completed. The explicit counterexample will be an intricately constructed term model. The construction and the proof that it works depend heavily on syntactic techniques. In particular, it provides a nice example of reasoning with elementary diagrams and descendants. We also include a domain-theoretic proof of the existence of an incompletable partial combinatory algebra.

Natural restrictions on the syntax of the second-order (i.e., polymorphic) lambda calculus are of interest for programming language theory. One of the authors showed in Leivant (1991) that when type abstraction in that calculus is stratified into levels, the definable numeric functions are precisely the super-elementary functions (level [Escr ]4 in the Grzegorczyk Hierarchy). We define here a second-order lambda calculus in which type abstraction is stratified to levels up to ωω, an ordinal that permits highly uniform (and finite) type inference rules. Referring to this system, we show that the numeric functions definable in the calculus using ranks < ω[lscr ] are precisely Grzegorczyk's class [Escr ][lscr ]+3 ([lscr ] [ges ] 1). This generalizes Leivant (1991), where this is proved for [lscr ] = 1. Thus, the numeric functions definable in our calculus are precisely the primitive recursive functions.