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Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs in regular resolution while admitting polynomial-size proofs in resolution. Thus, with respect to the usual notion of simulation, regular resolution is separated from resolution. An alternative, and weaker, notion for comparing proof systems is that of an “effective simulation,” which allows the translation of the formula along with the proof when moving between proof systems. We prove that regular resolution is equivalent to resolution under effective simulations. As a corollary, we recover in a black-box fashion a recent result on the hardness of automating regular resolution. 

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: - Every well-behaved 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 well-behaved proof system which proves its own soundness. - Every well-behaved monotone circuit model which admits a universal family of functions is characterized by a TFNP problem. Conversely, every TFNP problem gives rise to a well-behaved 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.