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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fragments of bounded arithmetic and bounded query classes


Author: Jan Krajíček
Journal: Trans. Amer. Math. Soc. 338 (1993), 587-598
MSC: Primary 03F30; Secondary 03D15, 03F05, 03F50, 68Q15
MathSciNet review: 1124169
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Abstract: We characterize functions and predicates $ \Sigma _{i + 1}^b$-definable in $ S_2^i$. In particular, predicates $ \Sigma _{i + 1}^b$-definable in $ S_2^i$ are precisely those in bounded query class $ {P^{\Sigma _i^p}}[O(\log n)]$ (which equals to $ \operatorname{Log}\;{\text{Space}}^{\Sigma _i^p}$ by [B-H, W]). This implies that $ S_2^i \ne T_2^i$ unless $ {P^{\Sigma _i^p}}[O(\log n)] = \Delta _{i + 1}^p$. Further we construct oracle $ A$ such that for all $ i \geq 1$: $ {P^{\Sigma _i^p(A)}}[O(\log n)] \ne \Delta _{i + 1}^p(A)$. It follows that $ S_2^i(\alpha ) \ne T_2^i(\alpha )$ for all $ i \geq 1$. Techniques used come from proof theory and boolean complexity.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1124169-X
PII: S 0002-9947(1993)1124169-X
Article copyright: © Copyright 1993 American Mathematical Society