Bounded Arithmetic and Lower Bounds in Boolean Complexity

We study the question of provability of lower bounds on the complexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic. To that end, we analyze what is the right fragment capturing the kind of techniques existing in Boolean complexity at present. We give both formal and informal arguments supporting the claim that a conceivable answer is V 1 1 (which, in view of RSUV-isomorphism, is equivalent to S 2 1 ), although some major results about the complexity of Boolean functions can be proved in (presumably) weaker subsystems like U 1 1 . As a by-product of this analysis, we give a more constructive version of the proof of Hastad Switching Lemma which probably is interesting in its own right.

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