On the Interpretation of Combinators with Weak Reduction

-ClkXl ***Xnk Xk, where n, > 0 for i = 1, ..., k; Q/i, ..., Sdk are letters of the alphabet A (perhaps -not all of them), and Xi is a combination of the variables x1, . .., xn, (perhaps not all of them) for i = 1, ..., k. We do not demand that W be a combinatorially complete calculus; this is the only distinction between W and X0. We shall call '1j, ... * Slk the atomic combinators; other letters of the alphabet A will be called atoms.2 The relation of weak reduction on W will be denoted by the symbol k=, and the relation of weak equality on W will be denoted by the smybol 11 . The notations