On Generalized Ajdukiewicz and Lambek Calculi and Grammars

Investigations concerning the description of languages in terms of categorial grammar started with examining concatenation as the only syntactic operation. Type reduction systems such as associative Ajdukiewicz or Lambek calculi, which admit concatenation of types in the calculus (and consequently concatenation of words in a language) as the expression-forming operation, were proved defective. This is because the type reduction schemata they produce neither cover common linguistic phenomena of natural languages nor protect one from accepting some ill-formed strings of words. Diierent measures were put forward in order to make syntactic calculi more exible. They included adding new type reduction schemata, cf. Steedman 14], or changing the associative format of formulas of calculi: Lambek 12] introduced the nonassociative calculus, van Benthem 1] considered calculi with permutation, Buszkowski 4] discussed nonassociative calculi with functor markers. Linguistic evidence, however, called for such a shape of syntactic calculi in which there would exist diierent operators within one system. Ajdukiewicz and Lambek calculi founded on an algebra of arbitrary syntactic operations, here called the generalized calculi, were introduced by Buszkowski in 5], see also 2], and independently, with slightly another motivation under the name of multimodal systems, by Moortgat, see the survey 13]. Koo lowska-Gawiejnowicz in 11] has recently proved the completeness of the generalized Lambek calculus with respect to powerset residuated algebras. Categorial grammars based on associative syntactic calculi generate string languages, which can be evaluated from the point of view of the place they occupy in the Chomsky hierarchy. Grammars employing nonassociative calculi or nonassociative calculi with func-tor markers impose additional structure on generated strings, and produce, respectively, phrase languages and phrase languages with functor markers. There are diierent ways to characterize classes of such languages. One can compare classes of languages generated by grammars based on diierent syntactic calculi. It is also possible to characterize a class of languages by means of some degrees of complexity of elements belonging to them. Finally, each structured language can be deprived of (a part of) its structural components like functor markers or brackets, and such classes of partially structured languages can be examined as well. Details concerning generative power of categorial grammars and relevant issues are presented in Buszkowski's survey 6] and in original papers mentioned there. This paper deals with phrase languages generated by generalized Ajdukiewicz and Lambek categorial grammars. The calculi we will be concerned with can be viewed as 1