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

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### 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 331-339 9 Theoretical Computer Science 206 1-2 https://doi.org/10.1016/S0304-3975(97)00168-0 Published - Oct 6 1998

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Polynomials
Polynomial
Normal Form
Natural Deduction
Sequent Calculus
Intuitionistic Logic
Deduction
Term
Completeness
Fragment
Assignment
Closed
Formulation

### All Science Journal Classification (ASJC) codes

• Theoretical Computer Science
• Computer Science(all)

### Cite this

In: Theoretical Computer Science, Vol. 206, No. 1-2, 06.10.1998, p. 331-339.

Research output: Contribution to journalArticle

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