TY - JOUR

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

AU - Hirokawa, Sachio

N1 - Funding Information:
The author would like to thank N. Suzuki for his suggestiont o simplify the proof of (b)+(a) in Theorem 1. He thanks R. Hindley, Y. Komori, I. Stewart and M. Zaionc for their comments and discussion. He appreciatest he many suggestionsb y the referees. This work was supportedb y Grant-in-Aid for Scientific Research No. 02740115, No. 05680276 and No. 07680364 of the Ministry of Education, Science and Culture, Japan.

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.

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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 -