Affiliation:
1. Institute of Mathematics, Czech Academy of Sciences and Computer Science Institute, Charles University
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).
Publisher
Association for Computing Machinery (ACM)
Subject
Computational Theory and Mathematics,Theoretical Computer Science