Formulas versus Circuits for Small Distance Connectivity.
SIAM JOURNAL ON COMPUTING(2018)
摘要
We prove an n(Omega(log k)) lower bound on the AC(0) formula size of DISTANCE k(n) CONNECTIVITY for all k(n) <= log logn and formulas up to depth log n/(log logn)(O(1)). This lower bound strongly separates the power of bounded-depth formulas versus circuits, since DISTANCE k(n) CONNECTIVITY is solvable by polynomial-size AC(0) circuits of depth O(log k). For all d(n) <= log log log n, it follows that polynomial-size depth-d circuits which are a semantic subclass of n(O(d))-size depth-d formulas-are not a semantic subclass of n(o(d))-size formulas of much higher depth log n/(log logn)(O(1)). Our lower bound technique probabilistically associates each gate in an AC(0) formula with an object called a pathset. We show that with high probability these random pathsets satisfy a family of density constraints called smallness, a property akin to low average sensitivity. We then study a complexity measure on small pathsets, which lower bounds the AC(0) formula size of DISTANCE k(n) CONNECTIVITY. The heart of our technique is an n(Omega(log k)) lower bound on this pathset complexity measure.
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关键词
AC(0),formulas,circuits,lower bounds,circuit complexity,st-connectivity,Boolean formulas,bounded depth
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