Core Concepts

If the Implicit Extended Frege proof system (iEF) is not polynomially bounded, then #P is not contained in FP/poly.

Abstract

The paper establishes a conditional connection between proof complexity and circuit complexity. Specifically, it shows that if the Implicit Extended Frege (iEF) proof system is not polynomially bounded, then #P is not contained in FP/poly.
The key insights are:
The proof exploits the formalization inside iEF of the soundness of the sum-check protocol, which allows the authors to connect the proof complexity of iEF to the circuit complexity of #P.
The authors show that if #P is contained in FP/poly, then the #P-powerful prover in the sum-check protocol can be replaced by a polynomial-size circuit, making the protocol a polynomially bounded Merlin-Arthur system.
By derandomizing this Merlin-Arthur system under standard hardness assumptions, the authors obtain a Cook-Reckhow proof system whose lower bounds imply #P is not contained in FP/poly.
To translate this result to the standard iEF system, the authors prove the soundness of the sum-check protocol inside the theory S1^2 + 1-EXP, which corresponds to iEF at the propositional level.
The result provides further evidence that strong proof complexity lower bounds require circuit lower bounds, and it is the first example of a natural proof system that is conditionally Cook-Reckhow and whose lower bounds imply Boolean circuit lower bounds.

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Deeper Inquiries

The conclusion of the main theorem cannot be directly strengthened to NP ⊈ P/poly instead of #P ⊈ FP/poly. The reason for this limitation lies in the nature of the connections established in the proof. The main theorem relies on a conditional connection between proof complexity and circuit complexity, showing that if a certain condition is met, then #P ⊈ FP/poly follows. This condition is crucial for the logical flow of the proof and the implications drawn from it. To strengthen the conclusion to NP ⊈ P/poly, a different set of conditions or a modified proof strategy would be required to establish a direct link between the two complexity classes.

It is theoretically possible to replace iEF in the main theorem by Gentzen's system G or even by Extended Frege by carrying out the formalization inside these weaker systems. However, this replacement would necessitate a thorough analysis and adaptation of the proof techniques to suit the specific characteristics and constraints of Gentzen's system G or Extended Frege. Each proof system has its own set of rules, axioms, and inference mechanisms, which may impact the feasibility and validity of the formalization process. Additionally, the strength and expressiveness of the chosen system play a significant role in determining the effectiveness of the proof and the extent to which the conclusions can be generalized.

There may not be a natural class of formulas over which Extended Frege directly simulates iEF, as the two proof systems have distinct features and capabilities. However, assuming hardness of certain formulas for Extended Frege could potentially imply iEF lower bounds indirectly. By establishing a connection between the hardness of specific formulas in Extended Frege and the resulting implications for iEF, it might be possible to infer lower bounds for iEF based on the properties of Extended Frege. This approach would involve a detailed analysis of the relationships between the two proof systems and the impact of hardness assumptions on their respective proof complexities. Further research and exploration in this direction could provide valuable insights into the interplay between different proof systems and their implications for complexity theory.

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