Nonassociative and Commutative Categorial Grammars and their Languages

Categorial grammars adhere to the group of formalisms aiming at a description of natural languages in which the syntactic role of expressions is described by means of types. Types are indices of two forms | primitive or compound. Compound types are built inductively by means of three type-forming operators: left division n , right division / and product. Thus, formally, for a given set Pr of primitive types, the set TP of types (with product) is deened as follows: (i) Pr TP, (ii) if x and y are in TP, then x=y, ynx and x y are in TP. The set of product-free types will be denoted Tp. Primitive types are ascribed to the basic syntactic categories of a language, such as sentences, proper nouns, etc. An expression of a language is assigned a type x=y if it plays the role of a functor looking for its argument on the right-hand side, and producing with an argument of type y, a compound expression of type x. Expressions of the type xny look for their argument on the left-hand side. The type x y is ascribed to those expressions which are the concatenation of two expressions of types x and y respectively. The above ideas can assume the shape of a formal system called the Ajdukiewicz-Bar-Hillel calculus (ABH), cf. Ajdukiewicz 1935, Bar-Hillel 1953. Formulas of ABH are of the shape X ! x, where X 2 TP + and x 2 TP. The only axiom scheme of ABH is (A0) x ! x , where x 2 TP, and the calculus admits the following three rules of inference: