Combinatory abstraction using B , 6 ’ and friends *

Trigg, P., J.R. Hindley and M.W. Bunder, Combinatory abstraction using B, B’ and friends, Theoretical Computer Science 135 (1994) 4055422. In this paper we characterise precisely the sets of terms whose abstractions can be defined using the following partial bases of combinators: { B, B’, I}, ( B, B’, I, W}, {B, B’, I, K}, { B, T, I}. [ B,T, I, W) and (B, T, I, K}. The reduction axioms for B’ and T are B’X YZ D Y(XZ), TX YZ L> YXZ The first two B’-bases correspond via type-assignment to two interesting implicational logics. T has the re-ordering property of B’ but not its bracketing property, and turns out to be strictly stronger than B’ but strictly weaker than Cl whose reduction axiom is CIX Y D YX. Correspondence to: J.R. Hindley, Mathematics Department, University College Swansea, Singleton Park, Swansea, SA2 8PP, UK. Email: [email protected]. *This paper is a development by Bunder and Hindley of results in the M. Phil, thesis of Trigg, 1141. The authors are very grateful to the referees for some significant improvements to the exposition and content, as well as some references to the literature. Peter Trigg would also like to thank the British Science and Engineering Research Council for financial support throughout the work leading to 1141. 0304-3975/94/$07.00