TY - JOUR

T1 - The relevance graph of a BCK-formula

AU - Hirokawa, Sachio

N1 - Funding Information:
This work was partially supported by a Grant-m-Aid No.02740115 of the Ministry of Education.

PY - 1993/6

Y1 - 1993/6

N2 - It is known that the set of BCK-formulas which is provable by the detachment rule of Meredith is identical to the set pts(BCK) of principal type-schemes of BCK-λ-terms. This paper shows a characterization of the set pts(BCK-β) of principal type-schemes of BCK-λ-terms in β-normal form. To characterize the set pts(BCK), a 'relevance relation' is defined between type variables in a type. A type variable b is relevant to a type variable c in a type α iff α contains a negative occurrence of a subtype of the form (... → b) → ... → c. The relevance graph G(α) of the type α is the directed graph induced by this relevance relation. A type variable is said to be positive iff it occurs in a positive position and negative otherwise. It is proved that a type α is in pts(BCK-β) iff α satisfies: (a) every type variable occurs exactly once in a negative position and at most once in a positive position; (b) no negative type variable is relevant to any type variable but itself and the subgraph of G(α) whose nodes are positive type variables of α is a tree whose root is the rightmost type variable in α; (c) each positive type variable in a subtype γ is relevant to the right-most type variable in γ.

AB - It is known that the set of BCK-formulas which is provable by the detachment rule of Meredith is identical to the set pts(BCK) of principal type-schemes of BCK-λ-terms. This paper shows a characterization of the set pts(BCK-β) of principal type-schemes of BCK-λ-terms in β-normal form. To characterize the set pts(BCK), a 'relevance relation' is defined between type variables in a type. A type variable b is relevant to a type variable c in a type α iff α contains a negative occurrence of a subtype of the form (... → b) → ... → c. The relevance graph G(α) of the type α is the directed graph induced by this relevance relation. A type variable is said to be positive iff it occurs in a positive position and negative otherwise. It is proved that a type α is in pts(BCK-β) iff α satisfies: (a) every type variable occurs exactly once in a negative position and at most once in a positive position; (b) no negative type variable is relevant to any type variable but itself and the subgraph of G(α) whose nodes are positive type variables of α is a tree whose root is the rightmost type variable in α; (c) each positive type variable in a subtype γ is relevant to the right-most type variable in γ.

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U2 - 10.1093/logcom/3.3.269

DO - 10.1093/logcom/3.3.269

M3 - Article

AN - SCOPUS:18944404237

SN - 0955-792X

VL - 3

SP - 269

EP - 285

JO - Journal of Logic and Computation

JF - Journal of Logic and Computation

IS - 3

ER -