On Protocols for Monotone Feasible Interpolation

Abstract

Dag-like communication protocols, a generalization of the classical tree-like communication protocols, are useful objects in the realm of proof complexity (most importantly for monotone feasible interpolation) and circuit complexity. We consider three kinds of protocols in this article ( d is the degree of a protocol): — IEQ- d -dags: feasible sets of these protocols are described by inequality which means that the feasible sets are combinatorial triangles; these protocols are also called triangle-dags in the literature, — EQ- d -dags: feasible sets are described by equality, and — c -IEQ- d -dags: feasible sets are described by a conjunction of c inequalities. Garg, Göös, Kamath, and Sokolov (Theory of Computing, 2020) mentioned all these protocols, and they noted that EQ- d -dags are a special case of c -IEQ- d -dags. The exact relationship between these types of protocols is unclear. As our main contribution, we prove the following statement: EQ-2-dags can efficiently simulate c -IEQ- d -dags when c and d are constants. This implies that EQ-2-dags are at least as strong as IEQ- d -dags and that EQ-2-dags have the same strength as c -IEQ- d -dags for c ≥ 2 (because 2-IEQ-2-dags can trivially simulate EQ-2-dags). Hrubeš and Pudlák (Information Processing Letters, 2018) proved that IEQ- d -dags over the monotone Karchmer-Wigderson relation are equivalent to monotone real circuits which implies that we have exponential lower bounds for these protocols. Lower bounds for EQ-2-dags would directly imply lower bounds for the proof system R(LIN).