On the effect of the IO-substitution on the Parikh image of semilinear AFLs

Back in the 80s, 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 mildly-context sensitive formalisms, 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-ubstitution was created so as to exhibit mildly context-sensitive classes of languages which are not semilinear, although they verify the constant-growth property. In the present article, we extend the notion of semilinearity, and characterise the Parikh image of the IO-MCFLs (i.e. languages which belong to the closure of MCFLs under IO-subsitution) as universally-linear. Based on this result and on the work of Fischer on macro-grammars, we then show IO-MCFLs are not closed under inverse homomorphism, which proves that the family of IO-MCFLs is not an abstract family of languages.