Normal Proofs and Their Grammar

We present grammatical (or equational) descriptions of the set of normal inhabitants fM j ` M : A; M in -normal form g of a given type A under a given basis , both for the standard simple type system (in partial discharge convention) and for the system in total discharge convention (or Prawitz-style natural deduction system). It is shown that in the latter system we can describe the set by a ( nite) context-free grammar, but for the standard system it is not necessarily the case because we may need an in nite supply of fresh (bound) variables to describe the set. In both cases, however, our grammars re ect the structure of normal inhabitants in such a way that, when non-terminals are ignored, a derivation tree of the grammars yielding a -term M can be identi ed with the Bohm tree of M . We give some applications of the grammatical descriptions. Among others, we give simple algorithms for the emptyness/ niteness problem of the set of normal inhabitants of a given type (both for the standard and nonstandard systems).