Preprint:
Sam Buss, Noah Fleming, and Russell Impagliazzo
TFNP Characterizations of Proof Systems and
Monotone Circuits
In: Proc. Innovations in Theoretical Computer Science (ITCS),
January 2023.
LIPIcs volume 251, 30:1-30, 2023..
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ITCS conference version.
Abstract: Connections between proof complexity and circuit complexity have become major tools for obtaining lower bounds in both areas. These connections — which take the form of interpolation theorems and query-to-communication lifting theorems — translate efficient proofs into small circuits, and vice versa, allowing tools from one area to be applied to the other. Recently, the theory of TFNP has emerged as a unifying framework underlying these connections. For many of the proof systems which admit such a connection there is a TFNP problem which characterizes it: the class of problems which are reducible to this TFNP problem via query-efficient reductions is equivalent to the tautologies that can be efficiently proven in the system. Through this, proof complexity has become a major tool for proving separations in black-box TFNP Similarly, for certain monotone circuit models, the class of functions that it can compute efficiently is equivalent to what can be reduced to a certain TFNP problem in a communication-efficient manner. When a TFNP problem has both a proof and circuit characterization, one can prove an interpolation theorem. Conversely, many lifting theorems can be viewed as relating the communication and query reductions to TFNP problems. This is exciting, as it suggests that TFNP provides a roadmap for the development of further interpolation theorems and lifting theorems. In this paper we begin to develop a more systematic understanding of when these connections to TFNP occur. We give exact conditions under which a proof system or circuit model admits a characterization by a TFNP problem. We show: