### Abstract

BCK-λ-terms are the λ-terms in which each variable occurs at most once. The principal type of a λ-term is the most general type of the term. In this paper we prove that if two BCK-λ-terms in β-normal form have the same principal type then they are identical. This solves the following problem (Y. Komori, 1987) in more general form: if two closed βη-normal form BCK-λ-terms are assigned to the same minimal BCK-formula, are they identical? A minimal BCK-formula is the most general formula among BCK-provable formulas with respect to substitutions for type variables. To analyze type assignment, the notion of "connection" is introduced. A connection is a series of occurrences of a type in a type assignment figure. Connected occurrences of a type have the same meaning. The occurrences of the type in distinct connection classes can be rewritten separately; as a result, we have more general type assignment. By "formulae-as-types" correspondence, the result implies the uniqueness of the normal proof figure for principal BCK-formulas. The result is valid for BCI-logic or implicational fragment of linear logic as well.

Original language | English |
---|---|

Pages (from-to) | 253-276 |

Number of pages | 24 |

Journal | Theoretical Computer Science |

Volume | 107 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 18 1993 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*107*(2), 253-276. https://doi.org/10.1016/0304-3975(93)90171-O

**Principal types of BCK-lambda-terms.** / Hirokawa, Sachio.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 107, no. 2, pp. 253-276. https://doi.org/10.1016/0304-3975(93)90171-O

}

TY - JOUR

T1 - Principal types of BCK-lambda-terms

AU - Hirokawa, Sachio

PY - 1993/1/18

Y1 - 1993/1/18

N2 - BCK-λ-terms are the λ-terms in which each variable occurs at most once. The principal type of a λ-term is the most general type of the term. In this paper we prove that if two BCK-λ-terms in β-normal form have the same principal type then they are identical. This solves the following problem (Y. Komori, 1987) in more general form: if two closed βη-normal form BCK-λ-terms are assigned to the same minimal BCK-formula, are they identical? A minimal BCK-formula is the most general formula among BCK-provable formulas with respect to substitutions for type variables. To analyze type assignment, the notion of "connection" is introduced. A connection is a series of occurrences of a type in a type assignment figure. Connected occurrences of a type have the same meaning. The occurrences of the type in distinct connection classes can be rewritten separately; as a result, we have more general type assignment. By "formulae-as-types" correspondence, the result implies the uniqueness of the normal proof figure for principal BCK-formulas. The result is valid for BCI-logic or implicational fragment of linear logic as well.

AB - BCK-λ-terms are the λ-terms in which each variable occurs at most once. The principal type of a λ-term is the most general type of the term. In this paper we prove that if two BCK-λ-terms in β-normal form have the same principal type then they are identical. This solves the following problem (Y. Komori, 1987) in more general form: if two closed βη-normal form BCK-λ-terms are assigned to the same minimal BCK-formula, are they identical? A minimal BCK-formula is the most general formula among BCK-provable formulas with respect to substitutions for type variables. To analyze type assignment, the notion of "connection" is introduced. A connection is a series of occurrences of a type in a type assignment figure. Connected occurrences of a type have the same meaning. The occurrences of the type in distinct connection classes can be rewritten separately; as a result, we have more general type assignment. By "formulae-as-types" correspondence, the result implies the uniqueness of the normal proof figure for principal BCK-formulas. The result is valid for BCI-logic or implicational fragment of linear logic as well.

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UR - http://www.scopus.com/inward/citedby.url?scp=38249005852&partnerID=8YFLogxK

U2 - 10.1016/0304-3975(93)90171-O

DO - 10.1016/0304-3975(93)90171-O

M3 - Article

VL - 107

SP - 253

EP - 276

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 2

ER -