### Abstract

The set of relevantly balanced formulas is introduced in implicational fragment of BCK-logic. It is shown that any relevantly balanced formula has unique normal form proof. Such formulas are defined by the ‘relevance relation’ between type variables in a formula. The set of balanced formulas (or equivalently one-two-formulas) is included in the relevantly balanced formulas. The uniqueness of normal form proofs is known for balanced formulas as the coherence theorem. Thus the result extends the theorem with respect to implicational formulas. The set of relevantly balanced formulas is characterized as the set of irrelevant substitution instances of principal type-schemes of BCK-λ-terms.

Original language | English |
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Title of host publication | Category Theory and Computer Science, Proceedings |

Editors | David H. Pitt, Pierre-Louis Curien, Samson Abramsky, Axel Poigne, David E. Rydeheard, Andrew M. Pitts |

Publisher | Springer Verlag |

Pages | 106-120 |

Number of pages | 15 |

ISBN (Print) | 9783540544951 |

DOIs | |

Publication status | Published - Jan 1 1991 |

Event | 4th Biennial Summer Conference on Category Theory and Computer Science, 1991 - Paris, France Duration: Sep 3 1991 → Sep 6 1991 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 530 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 4th Biennial Summer Conference on Category Theory and Computer Science, 1991 |
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Country | France |

City | Paris |

Period | 9/3/91 → 9/6/91 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Category Theory and Computer Science, Proceedings*(pp. 106-120). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 530 LNCS). Springer Verlag. https://doi.org/10.1007/BFb0013460

**BCK-formulas having unique proofs.** / Hirokawa, Sachio.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Category Theory and Computer Science, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 530 LNCS, Springer Verlag, pp. 106-120, 4th Biennial Summer Conference on Category Theory and Computer Science, 1991, Paris, France, 9/3/91. https://doi.org/10.1007/BFb0013460

}

TY - GEN

T1 - BCK-formulas having unique proofs

AU - Hirokawa, Sachio

PY - 1991/1/1

Y1 - 1991/1/1

N2 - The set of relevantly balanced formulas is introduced in implicational fragment of BCK-logic. It is shown that any relevantly balanced formula has unique normal form proof. Such formulas are defined by the ‘relevance relation’ between type variables in a formula. The set of balanced formulas (or equivalently one-two-formulas) is included in the relevantly balanced formulas. The uniqueness of normal form proofs is known for balanced formulas as the coherence theorem. Thus the result extends the theorem with respect to implicational formulas. The set of relevantly balanced formulas is characterized as the set of irrelevant substitution instances of principal type-schemes of BCK-λ-terms.

AB - The set of relevantly balanced formulas is introduced in implicational fragment of BCK-logic. It is shown that any relevantly balanced formula has unique normal form proof. Such formulas are defined by the ‘relevance relation’ between type variables in a formula. The set of balanced formulas (or equivalently one-two-formulas) is included in the relevantly balanced formulas. The uniqueness of normal form proofs is known for balanced formulas as the coherence theorem. Thus the result extends the theorem with respect to implicational formulas. The set of relevantly balanced formulas is characterized as the set of irrelevant substitution instances of principal type-schemes of BCK-λ-terms.

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U2 - 10.1007/BFb0013460

DO - 10.1007/BFb0013460

M3 - Conference contribution

AN - SCOPUS:84972504615

SN - 9783540544951

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

SP - 106

EP - 120

BT - Category Theory and Computer Science, Proceedings

A2 - Pitt, David H.

A2 - Curien, Pierre-Louis

A2 - Abramsky, Samson

A2 - Poigne, Axel

A2 - Rydeheard, David E.

A2 - Pitts, Andrew M.

PB - Springer Verlag

ER -