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 → ∞. The answer to this question is equivalent to finding the ‘density’ of inhabited types in the set of all types, or the so-called asymptotic probability of finding an inhabited type in the set of all types. Under the Curry–Howard isomorphism this means finding the density or asymptotic probability of provable intuitionistic propositional formulas in the set of all formulas. For types with one ground type (formulas with one propositional variable), we prove that the limit exists and is equal to 1/2 + √5/10, which is approximately 72.36%. This means that a long random type (formula) has this probability of being inhabited (tautology). We also prove that for every finite number k of ground-type variables, the density of inhabited types is always positive and lies between (4k + 1)/(2k + 1)2 and (3k + 1)/(k + 1)2. Therefore we can easily see that the density is decreasing to 0 with k going to infinity. From the lower and upper bounds presented we can deduce that at least 1/3 of classical tautologies are intuitionistic.

We consider the problem of determining the number of fixed points of a combinator (a closed λ-term). This appears to be a still unsolved problem. We give a partial answer by showing that if there is a fixed point in normal form, then this fixed point is unique or there are infinitely many fixed points.

We study a general algebraic framework that underlies a wide range of computational formalisms that use the notion of names, notably process calculi. The algebraic framework gives a rigorous basis for describing and reasoning about processes semantically, as well as offering new insights into existing constructions. The formal status of the theory is elucidated by introducing its alternative presentation, which is geometric in nature and is based on explicit manipulation of connections among nameless processes. Nameless processes and their relational theory form a coherent universe in their own right, which underlies existing graphical formalisms such as proof nets. We establish the formal equivalence between these two presentations, and illustrate how they can be used complementarily for the precise and effective description of diverse algebras and the dynamics of processes through examples.