Normal-order reduction grammars

Maciej Bendkowski

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arXiv · 1603.01758

Normal-order reduction grammars

Maciej Bendkowski

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Abstract

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 $S K$-combinators reducing in $n$ normal-order reduction steps. Finally, we investigate the size of generated grammars, giving a primitive recursive upper bound.

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Maciej Bendkowski. 2016-03-05. Normal-order reduction grammars. https://arxiv.org/abs/1603.01758

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