Skip to main content
← All Solvers

# 🧩 Boolean Satisfiability (SAT)

Band 1 — The Mercury transformer. Can a boolean formula be satisfied? The foundational NP-complete problem.

Variables: 15
Clauses: 30
Ready
Generated 3-SAT Formula

### Algorithm: DPLL

Davis–Putnam–Logemann–Loveland algorithm with unit propagation and pure literal elimination. Recursive backtracking with O(2ⁿ) worst case, but unit propagation prunes most branches on random 3-SAT instances. The clause/variable ratio determines difficulty — hardest at ~4.3x.

### Why This Matters

The SAT transformer (Mercury, Band 1) is the most fundamental solver — every other NP problem can be reduced to SAT. The 72-band framework uses SAT as its core decision engine. When the Router identifies a frequency violation, SAT finds the assignment that restores harmony.