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 - Mar 15 1996

    All Science Journal Classification (ASJC) codes

    • Theoretical Computer Science
    • Information Systems
    • Computer Science Applications
    • Computational Theory and Mathematics


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