On the Effect of the IO-Substitution on the Parikh Image of Semilinear Full AFLs

Back in the 1980's, the class of mildly context-sensitive formalisms was introduced so as to capture the syntax of natural languages. While the languages generated by such formalisms are constrained by the constant-growth property, the most well-known and used ones—like tree-adjoining grammars or multiple context-free grammars—generate languages which verify the stronger property of being semilinear. In (Bourreau et al. 2012), the operation of IO-substitution was created so as to exhibit mildly-context sensitive classes of languages which are not semilinear. In the present article, we extend the notion of semilinearity, and characterize the Parikh image of the languages in IO(L), the closure of a class L of semilinear languages under IO-substitution, as universally-semilinear. Based on this result and on the work of Fischer on macro-grammars, we then show that IO(L) is not closed under inverse homomorphism when L is closed under inverse homomorphism, and encompasses the class of regular languages. This result proves that IO(MCFL) is not a full AFL, where MCFL denotes the class of multiple context-free languages, closing an open question in Bourreau et al. (2012). More importantly, our proof gives an insight into the relation between the non-closure under inverse homomorphism of IO(MCFL) and how IO-substitution breaks semilinearity.