Structural Origin and the Minimal Syntax of NP-Hardness: Analysis of SAT from Syntactic Generativity and Compositional Collapse

Abstract

The Boolean satisfiability problem (SAT) holds a central place in computational complexity theory as the first shown NP-complete problem. Due to this role, SAT is often used as the benchmark for polynomial-time reductions: if a problem can be reduced to SAT, it is at least as hard as SAT, and hence considered NP-complete. However, the CDF framework offers a structural inversion of this traditional view. Rather than treating SAT as merely a representative of NP-completeness, we investigate whether the syntactic structure of SAT itself -- especially in its 3SAT form -- is the source of semantic explosion and computational intractability observed in NP problems. In other words, SAT is not just the yardstick of NP-completeness, but may be the structural archetype that induces NP-type complexity. This reframing suggests that the P vs NP question is deeply rooted not only in computational resource limits, but in the generative principles of problem syntax, with 3SAT capturing the recursive and non-local constructions that define the boundary between tractable and intractable problems.