TY - GEN

T1 - Principal type-schemes of BCI-lambda-terms

AU - Hirokawa, Sachio

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1991.

PY - 1991

Y1 - 1991

N2 - A BCI-λ-term is a λ-term in which each variable occurs exactly once. It represents a proof figure for implicational formula provable in linear logic. A principal type-scheme is a most general type to the term with respect to substitution. The notion of “relevance relation” is introduced for type-variables in a type. Intuitively an occurrence of a type-variable b is relevant to other occurrence of some type-variable c in a type α, when b is essentially concerned with the deduction of c in α. This relation defines a directed graph G(α) for type-variables in the type. We prove that a type a is a principal type-scheme of BCI-λ-term iff (a), (b) and (c) holds: (a) Each variable occurring in α occurs exactly twice and the occurrences have opposite sign. (b) G(α) is a tree and the right-most type variable in α is its root. (c) For any subtype γ of α, each type variable in γ is relevant to the right-most type variable in γ. A type-schemes of some BCI-λ-term is minimal iff it is not a non-trivial substitution instance of other type-scheme of BCI-λ-term. We prove that the set of BCI-minimal types coincides with the set of principal type-schemes of BCI-λ-terms in βη-normal form.

AB - A BCI-λ-term is a λ-term in which each variable occurs exactly once. It represents a proof figure for implicational formula provable in linear logic. A principal type-scheme is a most general type to the term with respect to substitution. The notion of “relevance relation” is introduced for type-variables in a type. Intuitively an occurrence of a type-variable b is relevant to other occurrence of some type-variable c in a type α, when b is essentially concerned with the deduction of c in α. This relation defines a directed graph G(α) for type-variables in the type. We prove that a type a is a principal type-scheme of BCI-λ-term iff (a), (b) and (c) holds: (a) Each variable occurring in α occurs exactly twice and the occurrences have opposite sign. (b) G(α) is a tree and the right-most type variable in α is its root. (c) For any subtype γ of α, each type variable in γ is relevant to the right-most type variable in γ. A type-schemes of some BCI-λ-term is minimal iff it is not a non-trivial substitution instance of other type-scheme of BCI-λ-term. We prove that the set of BCI-minimal types coincides with the set of principal type-schemes of BCI-λ-terms in βη-normal form.

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U2 - 10.1007/3-540-54415-1_68

DO - 10.1007/3-540-54415-1_68

M3 - Conference contribution

AN - SCOPUS:84972503586

SN - 9783540544159

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 633

EP - 650

BT - Theoretical Aspects of Computer Software - International Conference TACS 1991, Proceedings

A2 - Meyer, Albert R.

A2 - Ito, Takayasu

PB - Springer Verlag

T2 - 1st International Conference on Theoretical Aspects of Computer Software, TACS 1991

Y2 - 24 September 1991 through 27 September 1991

ER -