### 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 a is a principal type-scheme of BCI-λ-term iff (a), (b) and (c) holds: (a) Each variable occurring in α occurs exactly twice and the occurrences have opposite sign. (b) G(α) is a tree and the right-most type variable in α is its root. (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.

Original language | English |
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Title of host publication | Theoretical Aspects of Computer Software - International Conference TACS 1991, Proceedings |

Editors | Albert R. Meyer, Takayasu Ito |

Publisher | Springer Verlag |

Pages | 633-650 |

Number of pages | 18 |

ISBN (Print) | 9783540544159 |

DOIs | |

Publication status | Published - Jan 1 1991 |

Event | 1st International Conference on Theoretical Aspects of Computer Software, TACS 1991 - Sendai, Japan Duration: Sep 24 1991 → Sep 27 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 | 526 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 1st International Conference on Theoretical Aspects of Computer Software, TACS 1991 |
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Country | Japan |

City | Sendai |

Period | 9/24/91 → 9/27/91 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Aspects of Computer Software - International Conference TACS 1991, Proceedings*(pp. 633-650). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 526 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-54415-1_68

**Principal type-schemes of BCI-lambda-terms.** / Hirokawa, Sachio.

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

*Theoretical Aspects of Computer Software - International Conference TACS 1991, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 526 LNCS, Springer Verlag, pp. 633-650, 1st International Conference on Theoretical Aspects of Computer Software, TACS 1991, Sendai, Japan, 9/24/91. https://doi.org/10.1007/3-540-54415-1_68

}

TY - GEN

T1 - Principal type-schemes of BCI-lambda-terms

AU - Hirokawa, Sachio

PY - 1991/1/1

Y1 - 1991/1/1

N2 - 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 a is a principal type-scheme of BCI-λ-term iff (a), (b) and (c) holds: (a) Each variable occurring in α occurs exactly twice and the occurrences have opposite sign. (b) G(α) is a tree and the right-most type variable in α is its root. (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.

AB - 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 a is a principal type-scheme of BCI-λ-term iff (a), (b) and (c) holds: (a) Each variable occurring in α occurs exactly twice and the occurrences have opposite sign. (b) G(α) is a tree and the right-most type variable in α is its root. (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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U2 - 10.1007/3-540-54415-1_68

DO - 10.1007/3-540-54415-1_68

M3 - Conference contribution

AN - SCOPUS:84972503586

SN - 9783540544159

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

SP - 633

EP - 650

BT - Theoretical Aspects of Computer Software - International Conference TACS 1991, Proceedings

A2 - Meyer, Albert R.

A2 - Ito, Takayasu

PB - Springer Verlag

ER -