Journal Article
Light Affine Set Theory: A Naive Set Theory of Polynomial Time
Kazushige Terui
Studia Logica: An International Journal for Symbolic Logic
Vol. 77, No. 1 (Jun., 2004), pp. 9-40
Published
by: Springer
https://www.jstor.org/stable/20016605
Page Count: 32
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Topics: Polynomials, Naive set theory, Natural numbers, Linear logic, Mathematical functions, Logical theorems, Coercion, Turing machines, Syntax
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Abstract
In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL. In this paper, we consider a naive set theory based on Intuitionistic Light Affine Logic (ILAL), a simplification of LLL introduced by [1], and call it Light Affine Set Theory (LAST). The simplicity of LAST allows us to rigorously verify its polytime character. In particular, we prove that a function over $\{0,1\}^{\ast}$ is computable in polynomial time if and only if it is provably total in LAST.
Studia Logica: An International Journal for Symbolic Logic
© 2004 Springer