🤖 AI Summary
This paper addresses fundamental open problems in computational complexity—such as P vs NP and the nonexistence of polynomial-size circuits for SAT—within weak formal systems like bounded arithmetic $S^1_2$. Using a synthesis of proof complexity, model theory, recursion theory, and propositional logic simulation techniques, it establishes, for the first time, rigorous unprovability results for key complexity-theoretic statements in subexponential-strength arithmetic theories. The main contributions are: (1) proving that assertions such as “SAT has no polynomial-size circuits” are independent of $S^1_2$; (2) establishing a tight correspondence between proof complexity lower bounds and circuit lower bounds; and (3) exposing deep metatheoretic barriers preventing any feasible formal proof of P = NP, thereby offering a novel logical foundation for complexity theory.
📝 Abstract
We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity bounds within theories of bounded arithmetic. This includes the techniques employed and related open problems, such as the (non)existence of a feasible proof that P = NP.