# Logical characterizations of bounded query classes I: Logspace oracle machines

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## Abstract

We consider three sub-logics of the logic (±HP)*[FO_{s}] (introduced in [Ste90]) and show that these sub-logics capture the complexity classes obtained by considering logspace deterministic oracle Turing machines with oracles in NP where the number of oracle calls is unrestricted and constant, respectively; that is, the classes L^{NP} and L^{NP}[O(1)]. We conclude that if certain logics are of the same expressibility then the Polynomial Hierarchy collapses. We also exhibit some new complete problems for the complexity class L^{NP} via projection translations (the first to be discovered: projection translations are extremely weak logical reductions between problems).

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