### Sabine Broda and Luís Damas

DCC-FC & LIACC, Universidade do Porto
R. do Campo Alegre 823, 4150-180 Porto, Portugal
Phone: 351 22 6078830, Fax: 351 22 6003654
E-mail: {sbb,luis}@ncc.up.pt}
Junho 2000

## Abstract

In 1996 Takahashi et al. showed that it is possible to describe the set of normal inhabitants of a given type $\tau$, in the standard simple type system, using an infinitary extension of the concept of context-free grammar, which allows for an infinite number of non-terminal symbols as well as production rules. The set of normal inhabitants of $\tau$ corresponds then to the set of terms generated by this, possibly infinitary, grammar plus all terms obtained from those by $\eta$-reduction. In this paper we show that the set of normal inhabitants of a type $\tau$ can in fact be described using a standard (finite) context-free grammar, and more interestingly that the sets of normal inhabitants of all types with a same structure are described by context-free grammars which share one unique underlying structure. The definition of a common scheme for these grammars, which depends uniquely on the given type structure, is based on an alternative representation for types, introduced in 2000 (BrodaDamas-Wollic-2000), which gives us a better insight on the nature of a type's structure and its relation to the structure of the set of its normal inhabitants.