On Lower Bound Methods for Tree-like Cutting Plane Proofs

In the book Boolean Function Complexity by Stasys Jukna [7], two lower bound techniques for Tree-like Cutting Plane proofs (henceforth, “Tree-CP proofs”) using Karchmer-Widgerson type communication games (henceforth, “KW games”) are presented: The first, applicable to Tree-CP proofs with bounded coefficients, translates Ω(t) deterministic lower bounds on KW games to 2 logn) lower bounds on Tree-CP proof size. The second, applicable to Tree-CP proofs with unbounded coefficients, translates Ω(t) randomized lower bounds on KW games to 2 log 2 n) lower bounds on Tree-CP proof size. The textbook proof in the latter case uses a O(log n)-bit randomized protocol for the GreaterThan function. However in [6], Nisan mentioned using the ideas of [1] to construct a O(log n + log(1/ǫ))-bit randomized protocol for GreaterThan. Nisan did not explicitly give the proof, though later results in his paper assume such a protocol. In this short exposition, we present the full O(log n+ log(1/ǫ))-bit randomized protocol for the GreaterThan function based on the ideas of [1] for “noisy binary search.” As an application, we show how to translate Ω(t) randomized lower bounds on KW games to 2 logn) lower bounds on Tree-CP proof size in the unbounded coefficient case. This equates randomness with coefficient size for the Tree-CP/KW game lower bound method. We believe that, while the O(log n+ log(1/ǫ))-bit randomized protocol for GreaterThan is a “known” result, the explicit connection to Tree-CP proof size lower bounds given here is new.