### 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.

Research output: Contribution to journal › Article

*Journal of Logic and Computation*, vol. 3, no. 3, pp. 269-285. https://doi.org/10.1093/logcom/3.3.269

}

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 -