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 querytocommunication 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 queryefficient 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 blackbox 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 communicationefficient 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:
 Every wellbehaved proof system which can prove its own soundness (a reflection principle) is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a wellbehaved proof system which proves its own soundness.
 Every wellbehaved monotone circuit model which admits a universal family of functions is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a wellbehaved monotone circuit model with a universal problem. As an example, we provide a TFNP characterization of the Polynomial Calculus, answering a question from [Mika Göös et al., 2022], and show that it can prove its own soundness.
BibTeX  Entry
@InProceedings{buss_et_al:LIPIcs.ITCS.2023.30,
author = {Buss, Sam and Fleming, Noah and Impagliazzo, Russell},
title = {{TFNP Characterizations of Proof Systems and Monotone Circuits}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {30:130:40},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772631},
ISSN = {18688969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17533},
URN = {urn:nbn:de:0030drops175332},
doi = {10.4230/LIPIcs.ITCS.2023.30},
annote = {Keywords: Proof Complexity, Circuit Complexity, TFNP}
}
Keywords: 

Proof Complexity, Circuit Complexity, TFNP 
Collection: 

14th Innovations in Theoretical Computer Science Conference (ITCS 2023) 
Issue Date: 

2023 
Date of publication: 

01.02.2023 