Bounded Arithmetic and Lower Bounds in Boolean Complexity

We study the question of provability of lower bounds on the com plexity of explicitly given Boolean functions in weak fragments of Peano Arithmetic To that end we analyze what is the right frag ment capturing the kind of techniques existing in Boolean complexity at present We give both formal and informal arguments support ing the claim that a conceivable answer is V which in view of RSUV isomorphism is equivalent to S although some major re sults about the complexity of Boolean functions can be proved in presumably weaker subsystems like U As a by product of this analysis we give a more constructive version of the proof of H astad Switching Lemma which probably is interesting in its own right We also present in a uniform way theories which do not involve second order quanti ers and show that they prove the same b theorems as V k U k k Another application of this technique is that the schemes of b replacement b IND and b limited iterated comprehension all of which are given by Boolean combina tions of b formulae together prove all B b consequences of the full b IND scheme