this paper was quite di#erent, stemming from the axiomatics of categories of tangles (although the authors were aware of possible connections to iteration theories. In fact, similar axiomatics in the symmetric case, motivated by flowcharts and "flownomials" had been developed some years earlier by Stefanescu (Stefanescu 2000).) However, the first author realized, following a stimulating discussion with Gordon Plotkin, that traced monoidal categories provided a common denominator for the axiomatics of both the Girard-style and Abramsky-Jagadeesan-style versions of the Geometry of Interaction, at the basic level of the multiplicatives. This insight was presented in (Abramsky 1996), in which Girard-style GoI was dubbed "particle-style", since it concerns information particles or tokens flowing around a network, while the Abramsky-Jagadeesan style GoI was dubbed "wave-style", since it concerns the evolution of a global information state or "wave". Formally, this distinction is based on whether the tensor product (i.e. the symmetric monoidal structure) in the underlying category is interpreted as a coproduct (particle style) or as a product (wave style). This computational distinction between coproduct and product interpretations of the same underlying network geometry turned out to have been partially anticipated, in a rather di#erent context, in a pioneering paper by E. S. Bainbridge (Bainbridge 1976), as observed by Dusko Pavlovic. These two forms of interpretation, and ways of combining them, have also been studied recently in (Stefanescu 2000). He uses the terminology "additive" for coproduct-based (i.e. our "particle-style") and "multiplicative" for product-based (i.e. our "wave-style"); this is not suitable for our purposes, because of the clash with Linear Logic term...

We consider types and typed lambda calculus over a finite number of ground types. We are going to investigate the size of the fraction of inhabited types of the given length n against the number of all types of length n. The plan of this paper is to find the limit of that fraction when n ! 1.

this report, we use a lean version of the usual system of intersection types, whichwe call . Hence, UP is also an appropriate unification problem to characterize typability of -terms in . Quite apart from the new light it sheds on fi-reduction, such an analysis turns out to have several other benefits

Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on simulations of low-level machine models. By contrast, we develop a more structural approach. We show how high-level functional programs can be mapped compositionally (i.e. in a syntax-directed fashion) into a simple kind of automata which are immediately seen to be reversible. The size of the automaton is linear in the size of the functional term. In mathematical terms, we are building a concrete model of functional computation. This construction stems directly from ideas arising in Geometry of Interaction and Linear Logic—but can be understood without any knowledge of these topics. In fact, it serves as an excellent introduction to them. At the same time, an interesting logical delineation between reversible and irreversible forms of computation emerges from our analysis. 1

We present a Counting Algorithm that computes the number of -terms in -normal form that have a given type as a principal type and produces a list of these terms. The design of the algorithm follows the lines of Ben-Yelles' algorithm for counting normal (not necessarily principal) inhabitants of a type . 1 Introduction In [2], Ben-Yelles presented a Counting Algorithm, also described in [3], which given a type computes the number of -terms in -normal form that can receive type in TA . For each type the algorithm decides in a nite number of steps whether the number of closed -normal forms with type is nite or innite, computes this number in the nite case, and lists all relevant terms in both cases. Related to this is the problem of counting the number of -normal forms that have a given type as a principal type. As pointed out in ([3], p. 127), this problem is still open and in this paper we present a Counting Algorithm which solves this case. Analogous to Ben-Y...

It is shown that unifiability of terms in the simply typed lambda calculus with beta and eta rules becomes decidable if there is a bound on the number of bound variables and lambdas in a unifier in eta-expanded beta-normal form.

In [10] it was shown that it is possible to describe the set of normal inhabitants of a given type , in the standard simple type system, using an in nitary extension of the concept of context-free grammar, which allows for an in nite number of non-terminal symbols as well as production rules. The set of normal inhabitants of corresponds then to the set of terms generated by this, possibly in nitary, grammar plus all terms obtained from those by -reduction. In this paper we show that the set of normal inhabitants of a type can in fact be described using a standard ( nite) contextfree grammar, and more interestingly that the sets of normal inhabitants of all types with a same structure are described by context-free grammars which share one unique underlying structure. The de nition of a common scheme for these grammars, which depends uniquely on the given type structure, is based on an alternative representation for types, introduced in [4], which gives us a better insight on the nature of a type's structure and its relation to the structure of the set of its normal inhabitants. 1

We set up a framework for the study of extensionality in the context of higher-order logic programming. For simply typed logic programs we propose a novel declarative semantics, consisting of a model class with a semicomputable initial model, and a notion of extensionality. We show that the initial model of a simply typed logic program, in case the program is extensional, collapses into a simple, set-theoretic representation. Given the undecidability of extensionality in general, we develop a decidable, syntactic criterion which is su#cient for extensionality. Some typical examples of higher-order logic programs are shown to be extensional. 1991 Mathematics Subject Classification: 68N05, 68N17, 68Q60 1991 ACM Computing Classification System: F.3.1, F.4.1 Keywords and Phrases: Logic programming, higher-order logic, simple types. Note: Work carried out under project PNA1.2, Constraint and Integer Programming. To appear in Proceedings ICLP99. 1. Introduction Higher-order logic progra...

We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite. 1

We de ne an alternative representation for types in the simply typed -calculus or equivalently for formulas in the implicational fragment of intuitionist propositional logic, which gives us a better insight on the nature of a types structure and its relation to the structure of the set of its normal inhabitants. Based on this representation of a type ', we de ne the notion of a valid proof tree. Any such valid proof tree represents a nite set of normal inhabitants of ' and every normal inhabitant corresponds to exactly one valid proof tree, constructable with the primitive parts in the formula tree of '. Precise algorithms are given to establish this relation. Finally, we give a simple characterisation of the proof trees that represent principal normal inhabitants of a type '.