On combinatory complete sets of proper combinators

Sabine Broda and Luís Damas



A combinatory system (or equivalently the set of its basic combinators) is called combinatorially complete for a functional system, if any member of the latter can be defined by an entity of the former system. In this paper the decision problem of combinatory completeness for finite sets of proper combinators is studied for three subsystems of the pure lambda calculus. Precise characterizations of proper combinator bases for the linear and the affine lambda calculus are given and the respective decision problems are shown to be decidable. Furthermore, it is determined which extensions with proper combinators of bases for the linear lambda calculus are combinatorially complete for the lambda-I-calculus.