Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 269-285 |
| Number of pages | 17 |
| Journal | Journal of Logic and Computation |
| Volume | 3 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - Jun 1 1993 |
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All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Software
- Arts and Humanities (miscellaneous)
- Hardware and Architecture
- Logic
Cite this
The relevance graph of a BCK-formula. / Hirokawa, Sachio.
In: Journal of Logic and Computation, Vol. 3, No. 3, 01.06.1993, p. 269-285.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The relevance graph of a BCK-formula
AU - Hirokawa, Sachio
PY - 1993/6/1
Y1 - 1993/6/1
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.
UR - http://www.scopus.com/inward/record.url?scp=18944404237&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=18944404237&partnerID=8YFLogxK
U2 - 10.1093/logcom/3.3.269
DO - 10.1093/logcom/3.3.269
M3 - Article
AN - SCOPUS:18944404237
VL - 3
SP - 269
EP - 285
JO - Journal of Logic and Computation
JF - Journal of Logic and Computation
SN - 0955-792X
IS - 3
ER -