### Abstract

It is shown that the infiniteness problem of proof (α) is polynomial-space complete. The set proof (α) is the set of closed λ-terms in β-normal form which has α as their types. The set is identical to the set of normal form proofs of α in the natural deduction system for implicational fragment of intuitionistic logic. A transformation of a type is defined by F(α)=(((6→α)→α)→b)→b and applied as a deduction of the non-emptiness problem to the infiniteness problem. The non-emptiness is identical to the provability of α, which is polynomial-space complete (Statman, 1979). Therefore, the infiniteness problem is polynomial-space hard. To show the polynomial completeness, an algorithm is shown which searches λ-terms of given type α. It is proved that the infiniteness is determined within the depth of 2|α|^{3}, where the size |α| is the total number of occurrences of symbols in α. Thus, the problem is solved in polynomial space. Hence, the infiniteness problem is polynomial-space complete. The bound is obtained by an estimation of the length of an irredundant chain of sequents in type-assignment system in sequent calculus formulation.

Original language | English |
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Pages (from-to) | 331-339 |

Number of pages | 9 |

Journal | Theoretical Computer Science |

Volume | 206 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Oct 6 1998 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

**Infiniteness of proof(α) is polynomial-space complete.** / Hirokawa, Sachio.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 206, no. 1-2, pp. 331-339. https://doi.org/10.1016/S0304-3975(97)00168-0

}

TY - JOUR

T1 - Infiniteness of proof(α) is polynomial-space complete

AU - Hirokawa, Sachio

PY - 1998/10/6

Y1 - 1998/10/6

N2 - It is shown that the infiniteness problem of proof (α) is polynomial-space complete. The set proof (α) is the set of closed λ-terms in β-normal form which has α as their types. The set is identical to the set of normal form proofs of α in the natural deduction system for implicational fragment of intuitionistic logic. A transformation of a type is defined by F(α)=(((6→α)→α)→b)→b and applied as a deduction of the non-emptiness problem to the infiniteness problem. The non-emptiness is identical to the provability of α, which is polynomial-space complete (Statman, 1979). Therefore, the infiniteness problem is polynomial-space hard. To show the polynomial completeness, an algorithm is shown which searches λ-terms of given type α. It is proved that the infiniteness is determined within the depth of 2|α|3, where the size |α| is the total number of occurrences of symbols in α. Thus, the problem is solved in polynomial space. Hence, the infiniteness problem is polynomial-space complete. The bound is obtained by an estimation of the length of an irredundant chain of sequents in type-assignment system in sequent calculus formulation.

AB - It is shown that the infiniteness problem of proof (α) is polynomial-space complete. The set proof (α) is the set of closed λ-terms in β-normal form which has α as their types. The set is identical to the set of normal form proofs of α in the natural deduction system for implicational fragment of intuitionistic logic. A transformation of a type is defined by F(α)=(((6→α)→α)→b)→b and applied as a deduction of the non-emptiness problem to the infiniteness problem. The non-emptiness is identical to the provability of α, which is polynomial-space complete (Statman, 1979). Therefore, the infiniteness problem is polynomial-space hard. To show the polynomial completeness, an algorithm is shown which searches λ-terms of given type α. It is proved that the infiniteness is determined within the depth of 2|α|3, where the size |α| is the total number of occurrences of symbols in α. Thus, the problem is solved in polynomial space. Hence, the infiniteness problem is polynomial-space complete. The bound is obtained by an estimation of the length of an irredundant chain of sequents in type-assignment system in sequent calculus formulation.

UR - http://www.scopus.com/inward/record.url?scp=0345767008&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0345767008&partnerID=8YFLogxK

U2 - 10.1016/S0304-3975(97)00168-0

DO - 10.1016/S0304-3975(97)00168-0

M3 - Article

AN - SCOPUS:0345767008

VL - 206

SP - 331

EP - 339

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - 1-2

ER -