A combinatory account of internal structure

Abstract

Traditional combinatory logic uses combinators S and K to represent all Turing-computable functions on natural numbers, but there are Turing-computable functions on the combinators themselves that cannot be so represented, because they access internal structure in ways that S and K cannot. Much of this expressive power is captured by adding a factorisation combinator F. The resulting SF-calculus is structure complete, in that it supports all pattern-matching functions whose patterns are in normal form, including a function that decides structural equality of arbitrary normal forms. A general characterisation of the structure complete, confluent combinatory calculi is given along with some examples. These are able to represent all their Turing-computable functions whose domain is limited to normal forms. The combinator F can be typed using an existential type to represent internal type information.

On the Expressiveness of Symmetric Communication

“…More generally the core approach of compounding proves sufficient to represent many complex data structures and even (in practice) type information. This has been discussed and formalised in different settings [19,20,10] and in many works related to pattern calculus, S F-logic, and CPC.…”

Towards typing for small-step direct reflection

“…They start from the pure factorisation calculus [6], augmented with some usual combinators from the SK combinatory calculus, as well as two new combinators, B and E, respectively for blocking computation and for deciding equality of operators. They then proceed to add syntactic sugar for the identity combinator, λ-abstraction, let and let rec.…”

Pattern Matching and Bisimulation

“…The exploration of intensionality in the concurrent setting is inspired by the increased expressive power that the intensional S F-calculus has over λ-calculus [23]. Since intensionality, as captured by pattern matching, is more expressive in sequential computation, it is natural to explore the expressiveness of intensionality, as captured by pattern unification, in concurrent computation.…”