SAT requires exhaustive search
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In this paper, we identify the distinction between non-brute-force computation and brute-force computation as the most fundamental problem in computer science. Subsequently, we prove, by the diagonalization method, that constructed self-referential CSPs cannot be solved by non-brute-force computation, which is stronger than P ≠ NP. This constructive method for proving impossibility results is very different (and missing) from existing approaches in computational complexity theory, but aligns with Gödel’s technique for proving logical impossibility. Just as Gödel showed that proving formal unprovability is feasible in mathematics, our results show that proving computational hardness is not hard in mathematics. Specifically, proving lower bounds for many problems, such as 3-SAT, can be challenging because these problems have various effective strategies available to avoid exhaustive search. However, for self-referential examples that are extremely hard, exhaustive search becomes unavoidable, making its necessity easier to prove. Consequently, it renders the separation between non-brute-force computation and brute-force computation much simpler than that between P and NP. Finally, our results are akin to Gödel’s incompleteness theorem, as they reveal the limits of reasoning and highlight the intrinsic distinction between syntax and semantics.
Xu K, Li W. Exact phase transitions in random constraint satisfaction problems. Journal of Artificial Intelligence Research, 2000, 12(1): 93–103
