Witnessing Functions in Bounded Arithmetic and Search Problems

We investigate the possibility to characterize (multi)functions that are b i-deenable with small i (i = 1; 2; 3) in fragments of bounded arithmetic T2 in terms of natural search problems deened over polynomial-time structures. We obtain the following results: 1. A reformulation of known characterizations of (multi)functions that are b 1-and b 2-deenable in the theories S 1 2 and T 1 2. 2. New characterizations of (multi)functions that are b 2-and b 3-deen-able in the theory T 2 2. 3. A new non-conservation result: the theory T 2 2 () is not 8 b 1 ()-conservative over the theory S 2 2 (). To prove that the theory T 2 2 () is not 8 b 1 ()-conservative over the theory S 2 2 (), we present two examples of a b 1 ()-principle separating the two theories: (a) the weak pigeonhole principle WPHP(a 2 ; f;g) formalizing that no function f is a bijection between a 2 and a with the inverse g, (b) the iteration principle Iter(a; R; f) formalizing that no function f deened on a strict partial order (f0; : : : ; ag; R) can have increasing iterates.

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