Structural Origin and the Minimal Syntax of NP-Hardness: Analysis of SAT from Syntactic Generativity and Compositional Collapse
Yumiko Nishiyama
TL;DR
This work introduces the Construction Defining Functionality (CDF) framework to explain NP-hardness as a structural breakdown within the syntax-semantics-logic triangle. By applying CDF to SAT variants, it shows that 2SAT preserves a ComCDF structure with stable $\alpha_f$, $\beta_f$, and $\gamma_f$, enabling polynomial-time reasoning, while 3SAT and general $k$-SAT exhibit semantic explosion in $\alpha_f$ and non-compositionality ($\gamma_f \neq \beta_f \circ \alpha_f$) with exponential growth in the semantic space. A Growth Theorem and exponential lower bounds formalize the jump from 2SAT to 3SAT, arguing that minimal syntactic changes suffice to generate NP-hardness. Overall, NP-hardness is reinterpreted as a structural instability: resolving it in polynomial time would require redefining the very syntactic rules that generate complexity, effectively altering problem identity.
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.