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We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance k(n) Connectivity, which asks whether two specified nodes in a graph of size n are connected by a path of length at most k(n). This problem is solvable (by the recursive doubling technique) on circuits of depth O(log k) and size O(kn3). In contrast, we show that solving this problem on formulas of depth log n/(log log n)O(1) requires size nΩ(log k) for all k(n) ≤ log log n. As corollaries: (i) It follows that polynomial-size circuits for Distance k(n) Connectivity require depth Ω(log k) for all k(n) ≤ log log n. This matches the upper bound from recursive doubling and improves a previous Ω(log log k) lower bound of Beame, Impagliazzo and Pitassi [BIP98]. (ii) We get a tight lower bound of sΩ(d) on the size required to simulate size-s depth-d circuits by depth-d formulas for all s(n) = nO(1) and d(n) ≤ log log log n. No lower bound better than sΩ(1) was previously known for any d(n) ≮ O(1). Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size n via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance k(n) Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity.

Formulas vs. circuits for small distance connectivity