Light affine logic is a variant of linear logic with a polynomial cut-elimination procedure. We study the extensional expressive power of light affine logic with respect to a general notion of encoding of functions in the setting of the Curry–Howard correspondence. We consider light affine logic with both fixpoints of formulae and second-order quantifiers, and analyse the properties of polytime soundness and polytime completeness for various fragments of this system. In particular, we show that the implicative propositional fragment is not polytime complete if we place some reasonable conditions on the encodings. Following previous work, we show that second order leads to polytime unsoundness. We then introduce simple constraints on second-order quantification and fixpoints, and prove that the fragments obtained are polytime sound and complete.

We show that the extensional ordering of the sequential functionals of pure type 3, for example, as defined via game semantics (Abramsky et al. 1994; Hyland and Ong 2000), is not cpo-enriched. This shows that this model does not equal Milner's (Milner 1977) fully abstract model for PCF.

The notion of ordinal computability is defined by generalising standard Turing computability on tapes of length $\omega$ to computations on tapes of arbitrary ordinal length. The fundamental theorem on ordinal computability states that a set $x$ of ordinals is ordinal computable from ordinal parameters if and only if $x$ is an element of the constructible universe $\mathbf{L}$. In this paper we present a new proof of this theorem that makes use of a theory SO axiomatising the class of sets of ordinals in a model of set theory. The theory SO and the standard Zermelo–Fraenkel axiom system ZFC can be canonically interpreted in each other. The proof of the fundamental theorem is based on showing that the class of sets that are ordinal computable from ordinal parameters forms a model of SO.

Recent work in constructive mathematics shows that Hilbert's program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform most of the highly abstract proofs of ‘concrete’ statements into elementary proofs. Surprisingly, the arguments we produce are not only elementary but also mathematically clearer, and not necessarily longer. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem. For this we use a general method to transform some logically complex first-order formulae into a geometrical form, which may be interesting in itself.

A useful separation lemma for partial cm-lattices is proved equivalent to PIT, the Prime Ideal Theorem. The relation of various versions of the Lemma to each other and to PIT is also explored.

We give a domain-theoretic analogue of the classical Banach–Alaoglu theorem, showing that the patch topology on the weak$*$ topology is compact. Various theorems follow concerning the stable compactness of spaces of valuations on a topological space. We conclude with reformulations of the patch topology in terms of polar sets or Minkowski functionals, showing, in particular, that the ‘sandwich set’ of linear functionals is compact.

In the paper we introduce and study the uniform regular enumerations for arbitrary recursive ordinals. As an application of the technique, we obtain a uniform generalisation of a theorem of Ash and a characterisation of a class of uniform operators on transfinite sequences of sets of natural numbers.

We investigate natural systems of fundamental sequences for ordinals below the Howard–Bachmann ordinal and study growth rates of the resulting slow growing hierarchies. We consider a specific assignment of fundamental sequences that depends on a non-negative real number $\varepsilon$. We show that the resulting slow growing hierarchy is eventually dominated by a fixed elementary recursive function if $\varepsilon$ is equal to zero. We show further that the resulting slow growing hierarchy exhausts the provably recursive functions of $\sfb{ID}_1$ if $\varepsilon$ is strictly greater than zero. Finally, we show that the resulting fast growing hierarchies exhaust the provably recursive functions of $\sfb{ID}_1$ for all non-negative values of $\varepsilon$. Our result is somewhat surprising since usually the slow growing hierarchy along the Howard–Bachmann ordinal exhausts precisely the provably recursive functions of $\sfb{PA}$. Note that the elementary functions are a very small subclass of the provably recursive functions of $\sfb{PA}$, and the provably recursive functions of $\sfb{PA}$ are a very small subclass of the provably recursive functions of $\sfb{ID}_1$. Thus the jump from $\varepsilon$ equal to zero to $\varepsilon$ greater than zero is one of the biggest jumps in growth rates for subrecursive hierarchies one might think of.

The object-calculus is an imperative and object-based programming language in which every object comes equipped with its own method suite. Consequently, methods need to reside in the store (‘higher-order store’), which complicates the semantics. Abadi and Leino defined a program logic for this language enriching object types by method specifications. We present a new soundness proof for their logic using denotational semantics. It turns out that denotations of store specifications are predicates defined by mixed-variant recursion. A benefit of our approach is that derivability and validity can be kept distinct. Moreover, it reveals which of the limitations of Abadi and Leino's logic are incidental design decisions and which follow inherently from the use of a higher-order store. We discuss the implications for the development of other, more expressive, program logics.

In this article, we characterise all continuous posets that are partially metrisable in their Scott topology. We present conditions for pmetrisability, which are both necessary and sufficient, in terms of measurements, domain-theoretic bases and, in a more general setting, in terms of radially convex metrics. These conditions, together with their refinements and generalisations, set a natural hierarchy on the class of partially metrised posets. We locate the class of countably-based continuous dcpos within this hierarchy.