Normal-order reduction grammars

We present an algorithm which, for given n, generates an unambiguous regular tree grammar defining the set of combinatory logic terms, over the set {S,K} of primitive combinators, requiring exactly n normal-order reduction steps to normalize. As a consequence of Curry and Feys’s standardization theorem, our reduction grammars form a complete syntactic characterization of normalizing combinatory logic terms. Using them, we provide a recursive method of constructing ordinary generating functions counting the number of SK-combinators reducing in n normal-order reduction steps. Finally, we investigate the size of generated grammars, giving a primitive recursive upper bound.