Abstract
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 α is a principal type-scheme of BCI-λ-term iff (a), (b) and (c) holds:
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(a)
Each variable occurring in α occurs exactly twice and the occurrences have opposite sign.
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(b)
G(α) is a tree and the right-most type variable in α is its root.
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(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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Supported by a Grant-in-Aid for Encouragement of Young Scientists No.02740115 of the Ministry of Education.
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Hirokawa, S. (1991). Principal type-schemes of BCI-lambda-terms. In: Ito, T., Meyer, A.R. (eds) Theoretical Aspects of Computer Software. TACS 1991. Lecture Notes in Computer Science, vol 526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54415-1_68
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DOI: https://doi.org/10.1007/3-540-54415-1_68
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