## 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}* *is computable in polynomial time if and only if it is provably total in*
**LAST**.

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Terui, K. Light Affine Set Theory: A Naive Set Theory of Polynomial Time.
*Studia Logica* **77, **9–40 (2004). https://doi.org/10.1023/B:STUD.0000034183.33333.6f

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- naive set theory
- polynomial time
- linear logic
- light logic
- substructural logics