The ordinal generated by an ordinal grammar is computable

Abstract

A prefix grammar is a context-free grammar whose nonterminals generate prefix-free languages. A prefix grammar $G$ is an ordinal grammar if the language $L(G)$ is well-ordered with respect to the lexicographic ordering. It is known that from a finite system of parametric fixed point equations one can construct an ordinal grammar $G$ such that the lexicographic order of $G$ is isomorphic with the least solution of the system, if this solution is well-ordered. In this paper we show that given an ordinal grammar, one can compute (the Cantor normal form of) the order type of the lexicographic order of its language, yielding that least solutions of fixed point equation systems defining algebraic ordinals are effectively computable (and thus, their isomorphism problem is also decidable).