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Bounded arithmetic for NC, ALogTIME, L and NL. (English) Zbl 0772.03028
The authors define theories of bounded arithmetic whose definable functions or relations are exactly those in certain complexity classes. They define a fragment TNC of bounded first-order arithmetic and prove that functions in the parallel complexity class NC are exactly those functions which are $$\Sigma^ b_ 1$$-definable in TNC. The class NC consists of all functions computable in polylogarithmic time with a polynomial number of processors on a parallel random-access machine. Then the authors define fragments of bounded second-order arithmetic such that suitably definable relations are, respectively, in the complexity classes ALogTIME, NL and L.

##### MSC:
 03F30 First-order arithmetic and fragments 03D15 Complexity of computation (including implicit computational complexity) 03F35 Second- and higher-order arithmetic and fragments
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