Normal Proofs and Their Grammar

Masako Takahashi, Yohji Akama, Sachio Hirokawa

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


We present grammatical (or equational) descriptions of the set of normal inhabitants {M | Γ ⊢ M : A, M in β-normal form} of a given type A under a given basis Γ, both for the standard simple type system (in the partial discharge convention) and for the system in the total discharge convention (or the Prawitz-style natural deduction system). It is shown that in the latter system we can describe the set by a (finite) context-free grammar, but for the standard system this is not necessarily the case because we may need an infinite supply of fresh (bound) variables to describe the set. In both cases, however, our grammars reflect the structure of normal inhabitants in such a way that, when non-terminals are ignored, a derivation tree of the grammars yielding a λ-term M can be identified with Böhm tree of M. We give some applications of the grammatical descriptions. Among others, we give simple algorithms for the emptiness/finiteness problem of the set of normal inhabitants of a given type (both for the standard and non-standard systems).

Original languageEnglish
Pages (from-to)144-153
Number of pages10
JournalInformation and Computation
Issue number2
Publication statusPublished - 1996 Mar 15


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