Categorial grammars with unique category assignment

A categorial grammar assigns one of several syntactic categories to each symbol of the alphabet, and the category of a string is then deduced from the categories assigned to its symbols using two simple reduction rules. This paper investigates a special class of categorial grammars, in which only one category is assigned to each symbol, thus eliminating ambiguity on the lexical level (in linguistic terms, a unique part of speech is assigned to each word). While unrestricted categorial grammars are equivalent to the context-free grammars, the proposed subclass initially appears weak, as it cannot define even some regular languages. It is proved in the paper that it is actually powerful enough to define a homomorphic encoding of every context-free language, in the sense that for every context-free language $L$ over an alphabet $Σ$ there is a language $L'$ over some alphabet $Ω$ defined by categorial grammar with unique category assignment and a homomorphism $h \colon Σ\to Ω^+$, such that a string $w$ is in $L$ if and only if $h(w)$ is in $L'$. In particular, in Greibach's hardest context-free language theorem, it is sufficient to use a hardest language defined by a categorial grammar with unique category assignment.