This special issue of Mathematical Structures in Computer Science is devoted to recent work in subtyping. When subtyping was first proposed, it presented a new vehicle for understanding programming languages, together with challenging theoretical issues. The papers in this special issue include a new approach to the decidability of subtyping, a metatheoretic investigation of transitivity of coercive subtyping for parametrised dependent types, and applications of subtyping to the classic programming language concerns of flow analysis and typing for distributed systems. We believe that the scope of the papers demonstrates convincingly that the theory and practice of subtyping continue to be extended in novel and interesting ways.

We present an algorithm for deciding polarised higher-order subtyping without bounded quantification. Constructors are identified not only modulo β, but also η. We give a direct proof of completeness, without constructing a model or establishing a strong normalisation theorem. Inductive and coinductive types are enriched with a notion of size and the subtyping calculus is extended to account for the inclusions arising between the sized types.

We present a novel approach to computing the context-sensitive flow of values through procedures and data structures. Our approach combines and extends techniques from two seemingly disparate areas: polymorphic subtyping and interprocedural dataflow analysis based on context-free language reachability. The resulting technique offers several advantages over previous approaches: it works directly on higher-order programs; provides demand-driven interprocedural queries; and improves the asymptotic complexity of a known algorithm based on polymorphic subtyping from O(n8) to O(n3) for computing all queries. For intra-procedural flow restricted to equivalence classes, our algorithm yields linear inter-procedural flow queries.

Session types allow high-level specifications of structured patterns of communication, such as client-server protocols, to be expressed as types and verified by static typechecking. In collaboration with Malcolm Hole, we previously introduced a notion of subtyping for session types, which was formulated for an extended pi calculus. Subtyping allows one part of a system, for example, a server, to be refined without invalidating type-correctness of other parts, for example, clients. In this paper we introduce bounded polymorphism, which is based on the same notion of subtyping, in order to support more precise and flexible specifications of protocols; in particular, a choice of type in one message may affect the types of future messages. We formalise the syntax, operational semantics and typing rules of an extended pi calculus, and prove that typechecking guarantees the absence of run-time communication errors. We study algorithms for checking instances of the subtype relation in two versions of our system, which we call Kernel S≤ and Full S≤, and establish that subtyping in Kernel S≤ is decidable, and that subtyping in Full S≤ is undecidable.

In this paper we study subtyping for inductive types in dependent type theories in the framework of coercive subtyping. General structural subtyping rules for parameterised inductive types are formulated based on the notion of inductive schemata. Certain extensional equality rules play an important role in proving some of the crucial properties of the type system with these subtyping rules. In particular, it is shown that the structural subtyping rules are coherent and that transitivity is admissible in the presence of the functorial rules of computational equality.

A longstanding open problem in lambda calculus is whether there exist continuous models of the untyped lambda calculus whose theory is exactly the λβ or the least sensible λ-theory ℋ (which is generated by equating all the unsolvable terms). A related question is whether, given a class of lambda models, there are a minimal λ-theory and a minimal sensible λ-theory represented by it. In this paper, we give a positive answer to this question for the class of graph models à la Plotkin, Scott and Engeler. In particular, we build two graph models whose theories are the set of equations satisfied in, respectively, any graph model and any sensible graph model. We conjecture that the least sensible graph theory, where ‘graph theory’ means ‘λ-theory of a graph model’, is equal to ℋ, while in one of the main results of the paper we show the non-existence of a graph model whose equational theory is exactly the λβ theory.

Another related question is whether, given a class of lambda models, there is a maximal sensible λ-theory represented by it. In the main result of the paper, we characterise the greatest sensible graph theory as the λ-theory ℬ generated by equating λ-terms with the same Böhm tree. This result is a consequence of the main technical theorem of the paper, which says that all the equations between solvable λ-terms that have different Böhm trees fail in every sensible graph model. A further result of the paper is the existence of a continuum of different sensible graph theories strictly included in ℬ.

Given a map f in the category ω-Cpo of ω-complete posets, exponentiability of f in ω-Cpo easily implies exponentiability of f in the category Pos of posets, while the converse is not true. We investigate the extra conditions needed on f exponentiable in Pos to be exponentiable in ω-Cpo by showing the existence of partial products of the two-point ordered set S={0<1} (Theorem 2.8). Using this characterisation and the embedding through the Scott topology of ω-Cpo in the category Top of topological spaces, we compare exponentiability in each setting and find that a morphism in ω-Cpo that is exponentiable in both Top and Pos is exponentiable in ω-Cpo also. Furthermore, we show that the exponentiability in Top and Pos are independent of each other.