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Light Affine Set Theory: A Naive Set Theory of Polynomial Time
, 2004
"... 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 ju ..."
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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 nontraditional 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.
On the computational complexity of cutelimination in linear logic
 In Proceedings of ICTCS 2003, volume 2841 of LNCS
, 2003
"... Abstract. Given two proofs in a logical system with a confluent cutelimination procedure, the cutelimination problem (CEP) is to decide whether these proofs reduce to the same normal form. This decision problem has been shown to be ptimecomplete for Multiplicative Linear Logic (Mairson 2003). The ..."
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Abstract. Given two proofs in a logical system with a confluent cutelimination procedure, the cutelimination problem (CEP) is to decide whether these proofs reduce to the same normal form. This decision problem has been shown to be ptimecomplete for Multiplicative Linear Logic (Mairson 2003). The latter result depends upon a restricted simulation of weakening and contraction for boolean values in MLL; in this paper, we analyze how and when this technique can be generalized to other MLL formulas, and then consider CEP for other subsystems of Linear Logic. We also show that while additives play the role of nondeterminism in cutelimination, they are not needed to express deterministic ptime computation. As a consequence, affine features are irrelevant to expressing ptime computation. In particular, Multiplicative Light Linear Logic (MLLL) and Multiplicative Soft Linear Logic (MSLL) capture ptime even without additives nor unrestricted weakening. We establish hierarchical results on the cutelimination problem for MLL(ptimecomplete), MALL(coNPcomplete), MSLL(EXPTIMEcomplete), and for MLLL (2EXPTIMEcomplete). 1