sat_eval_time

formal source

 169  total_width : ℕ := n + 3 * m + 10
 170
 171/-- The evaluation time for a SAT instance with n variables and m clauses -/
 172noncomputable def sat_eval_time (n m : ℕ) : ℕ :=
 173  -- Depth of the clause evaluation tree: O(log m) for m clauses
 174  -- Width propagation: O(n^{1/3}) for n variables arranged optimally
 175  -- Total: O(n^{1/3} · log(n·m))
 176  Nat.ceil (Real.rpow n (1/3) * Real.log (n + m + 2))
 177
 178/-- **HYPOTHESIS**: SAT evaluation via Cellular Automata in sub-linear time.
 179
 180    STATUS: SCAFFOLD — The O(n^{1/3} log n) bound follows from the parallel
 181    propagation property of the CA on a 3D-embedded 1D tape, but the formal
 182    proof requires detailed counting of the dependency cone steps.
 183
 184    TODO: Formally prove the O(n^{1/3} log n) bound. -/
 185def H_SATCARuntime (sg : SATGadget) : Prop :=
 186  ∃ (t : ℕ), t ≤ sat_eval_time sg.n sg.m ∧
 187  -- After t steps, the output cell contains the SAT result
 188  IsCorrectSATResult sg t -- SCAFFOLD: IsCorrectSATResult is not yet defined.
 189
 190-- axiom h_sat_ca_runtime : ∀ sg, H_SATCARuntime sg
 191
 192/-! ## CA → TM Simulation -/
 193
 194/-- A Turing Machine simulating the CA -/
 195structure TMSimulator where
 196  /-- Number of CA steps to simulate -/
 197  ca_steps : ℕ
 198  /-- Tape size (number of cells) -/
 199  tape_size : ℕ
 200  /-- TM time per CA step (linear in tape size) -/
 201  time_per_step : ℕ := tape_size
 202