In a controversial paper (De Millo et al. 1979) at the end of the 1970's, R. A. De Millo, R. J. Lipton and A. J. Perlis argued against formal verifications of programs, mostly motivating their position by an analogy with proofs in mathematics, and, in particular, with the impracticality of a strictly formalist approach to this discipline. The recent, impressive achievements in the field of interactive theorem proving provide an interesting ground for a critical revisiting of their theses. We believe that the social nature of proof and program development is uncontroversial and ineluctable, but formal verification is not antithetical to it. Formal verification should strive not only to cope with, but to ease and enhance the collaborative, organic nature of this process, eventually helping us to master the growing complexity of scientific knowledge.

A longstanding open problem is whether there exists a non-syntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective model of λ-calculus, which covers, in particular, all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ or λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. For Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim–Skolem theorem.

The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the sequential topology on the space of continuous real-valued functions on a Polish space need not be regular.

Mathematical modelling is one of the fundamental tools of science and engineering. Very often, models are required to be executable, as a simulation, on a computer. In this paper, we present some contributions to the process-theoretic and logical foundations of discrete-event modelling with resources and processes. We present a process calculus with an explicit representation of resources in which processes and resources co-evolve. The calculus is closely connected to a logic that may be used as a specification language for properties of models. The logic is strong enough to allow requirements that a system has a certain structure: for example, that it is a parallel composite of subsystems. This work consolidates, extends and improves upon aspects of earlier work of ours in this area. An extended example, consisting of a semantics for a simple parallel programming language, indicates a connection with separating logics for concurrency.

We introduce an extension of the pure lambda calculus by endowing the set of terms with the structure of a vector space, or, more generally, of a module, over a fixed set of scalars. Moreover, terms are subject to identities similar to the usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of beta reduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambda calculus. We also provide normalisation results for a simple type system.