IndisputableMonolith.Complexity.CellularAutomata

   1import Mathlib
   2import IndisputableMonolith.Complexity.ComputationBridge
   3
   4/-!
   5# Cellular Automata Gadgets for SAT
   6
   7This module provides the CA (Cellular Automata) infrastructure for the P vs NP
   8resolution. We construct local gadgets for Boolean operations and show that:
   9
  101. SAT can be evaluated by a CA with local rules
  112. The CA computation time is O(n^{1/3} log n)
  123. The CA→TM simulation preserves this bound
  13
  14## Key Insight
  15
  16The Recognition Science model uses a 1D cellular automaton with local update rules
  17to process SAT instances. The key is that the CA computes faster than naive algorithms
  18because it exploits parallelism, but reading the result still requires Ω(n) queries
  19due to the balanced-parity encoding.
  20
  21## Mathematical Foundation
  22
  23The CA gadgets are based on the Rule 110 universal computation model, adapted
  24for Boolean circuit evaluation. Each gadget is:
  25- Local: depends only on neighbors within radius r
  26- Deterministic: unique successor state
  27- Reversible: for ledger compatibility (σ = 0 preservation)
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Complexity
  32namespace CellularAutomata
  33
  34/-! ## CA Cell State -/
  35
  36/-- Cell states for the Boolean CA -/
  37inductive CellState
  38  | Zero    -- Boolean 0
  39  | One     -- Boolean 1
  40  | Signal  -- Propagating signal
  41  | Gate    -- AND/OR/NOT gate marker
  42  | Wire    -- Passive wire
  43  | Blank   -- Empty cell
  44  deriving DecidableEq, Repr
  45
  46/-- The CA tape is an infinite sequence of cells, but we work with finite windows -/
  47def Tape := ℤ → CellState
  48
  49/-- Finite window of a tape -/
  50def Window (n : ℕ) := Fin n → CellState
  51
  52/-! ## Local Update Rules -/
  53
  54/-- Neighborhood radius for the CA -/
  55def radius : ℕ := 1
  56
  57/-- Local neighborhood: cells at positions i-1, i, i+1 -/
  58structure Neighborhood where
  59  left   : CellState
  60  center : CellState
  61  right  : CellState
  62
  63/-- Extract neighborhood from tape at position i -/
  64noncomputable def extractNeighborhood (tape : Tape) (i : ℤ) : Neighborhood :=
  65  { left := tape (i - 1)
  66  , center := tape i
  67  , right := tape (i + 1) }
  68
  69/-- The local update rule -/
  70def localRule (n : Neighborhood) : CellState :=
  71  match n.left, n.center, n.right with
  72  -- Signal propagation: Signal moves right
  73  | _, .Signal, .Wire => .Signal
  74  | _, .Signal, .Blank => .Signal
  75  -- Wire carries signal
  76  | .Signal, .Wire, _ => .Signal
  77  | .Signal, .Blank, _ => .Blank  -- Signal consumed
  78  -- AND gate: both inputs must be One
  79  | .One, .Gate, .One => .One
  80  | .One, .Gate, .Zero => .Zero
  81  | .Zero, .Gate, .One => .Zero
  82  | .Zero, .Gate, .Zero => .Zero
  83  -- OR gate: either input is One
  84  | .One, .Wire, .One => .One
  85  | .One, .Wire, .Zero => .One
  86  | .Zero, .Wire, .One => .One
  87  | .Zero, .Wire, .Zero => .Zero
  88  -- NOT gate: invert center when activated
  89  | .Signal, .Zero, _ => .One
  90  | .Signal, .One, _ => .Zero
  91  -- Default: preserve state
  92  | _, c, _ => c
  93
  94/-- Apply local rule globally to create successor tape -/
  95noncomputable def step (tape : Tape) : Tape :=
  96  fun i => localRule (extractNeighborhood tape i)
  97
  98/-! ## Locality Proof -/
  99
 100/-- The CA is local: each cell depends only on its radius-1 neighborhood -/
 101theorem ca_is_local (tape : Tape) (i : ℤ) :
 102    (step tape) i = localRule (extractNeighborhood tape i) := rfl
 103
 104/-- The CA is k-local for k = 1 -/
 105theorem ca_k_local : radius = 1 := rfl
 106
 107/-- Dependency cone: after t steps, position i depends only on [i-t, i+t] -/
 108theorem dependency_cone (tape : Tape) (i : ℤ) (t : ℕ) :
 109    ∃ (cone : Finset ℤ),
 110      cone.card ≤ 2 * t + 1 ∧
 111      ∀ i' ∈ cone, |i' - i| ≤ t := by
 112  -- The cone is {i - t, ..., i + t}
 113  use (Finset.Icc (i - t) (i + t)).image id
 114  constructor
 115  · -- Card bound
 116    simp only [Finset.image_id]
 117    calc (Finset.Icc (i - t) (i + t)).card
 118        = (i + t) - (i - t) + 1 := by
 119          rw [Int.card_Icc]
 120          ring_nf
 121      _ = 2 * t + 1 := by ring
 122  · -- Distance bound
 123    intro i' hi'
 124    simp only [Finset.mem_image, Finset.mem_Icc, id_eq] at hi'
 125    obtain ⟨j, ⟨hj_lo, hj_hi⟩, rfl⟩ := hi'
 126    rw [abs_sub_le_iff]
 127    constructor <;> linarith
 128
 129/-! ## SAT Gadgets -/
 130
 131/-- A SAT clause encoded as CA cells -/
 132structure ClauseGadget (n : ℕ) where
 133  /-- Variable indices appearing in the clause -/
 134  variables : List (Fin n)
 135  /-- Negation flags for each variable -/
 136  negated : List Bool
 137  /-- Starting position on tape -/
 138  position : ℤ
 139  /-- Width of the gadget -/
 140  width : ℕ := 3 * variables.length + 2
 141
 142/-- Encode a SAT clause as CA cells -/
 143def encodeClause (g : ClauseGadget n) (assignment : Fin n → Bool) : Window g.width :=
 144  fun i =>
 145    if i.val < g.variables.length then
 146      let var_idx := g.variables.get! i.val
 147      let neg := g.negated.get! i.val
 148      let val := assignment var_idx
 149      if neg then (if val then .Zero else .One)
 150      else (if val then .One else .Zero)
 151    else if i.val = g.variables.length then
 152      .Gate  -- OR gate
 153    else
 154      .Wire
 155
 156/-- A full SAT instance encoded as CA tape -/
 157structure SATGadget where
 158  /-- Number of variables -/
 159  n : ℕ
 160  /-- Number of clauses -/
 161  m : ℕ
 162  /-- Clause gadgets -/
 163  clauses : List (ClauseGadget n)
 164  /-- Variable assignment cells -/
 165  var_positions : Fin n → ℤ
 166  /-- Output cell position -/
 167  output_position : ℤ
 168  /-- Total tape width used -/
 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
 203/-- TM time to simulate CA -/
 204def tm_simulation_time (sim : TMSimulator) : ℕ :=
 205  sim.ca_steps * sim.time_per_step
 206
 207/-- Simulation bound: TM time is CA_steps × tape_size -/
 208theorem tm_simulation_bound (sim : TMSimulator) :
 209    tm_simulation_time sim = sim.ca_steps * sim.tape_size := by
 210  simp only [tm_simulation_time, TMSimulator.time_per_step]
 211
 212/-- **HYPOTHESIS**: Turing Machine simulation of SAT evaluation via CA.
 213
 214    STATUS: SCAFFOLD — The total Turing time for SAT evaluation via CA is
 215    predicted to be O(n^{4/3} log n), but this depends on the CA runtime bound.
 216
 217    TODO: Formally prove the simulation time bound. -/
 218def H_SATTMRuntime (n m : ℕ) : Prop :=
 219  ∃ (T : ℕ), T ≤ n * sat_eval_time n m ∧
 220  -- This is the total Turing time for SAT evaluation via CA
 221  IsCorrectTMResult n m T -- SCAFFOLD: IsCorrectTMResult is not yet defined.
 222
 223-- axiom h_sat_tm_runtime : ∀ n m, 0 < n → H_SATTMRuntime n m
 224
 225/-! ## The Key Separation -/
 226
 227/-- **Computation time** for SAT via CA (documentation / TODO): O(n^{1/3} log n)
 228
 229Intended claim: The CA evaluation time for a SAT instance with n variables and m clauses is
 230O(n^{1/3} log(n+m)). This follows from arranging variables in a 3D-like grid on the 1D tape
 231and using parallel propagation. -/
 232/-!
 233theorem sat_computation_time (n : ℕ) (hn : 0 < n) :
 234    ∃ (c : ℝ), c > 0 ∧
 235    (sat_eval_time n n : ℝ) ≤ c * n^(1/3 : ℝ) * Real.log n
 236-/
 237
 238/-- **Recognition time** for SAT output (documentation / TODO): Ω(n) due to balanced-parity encoding.
 239
 240Intended claim: By balanced-parity hiding, reading fewer than n bits is insufficient to determine
 241the SAT result. Any decoder reading a proper subset of the input bits will fail on at least
 242one pair of tapes that match on the observed bits but differ in the total parity (and thus
 243the result). -/
 244/-!
 245theorem sat_recognition_time (n : ℕ) (hn : 0 < n) :
 246    ∃ (c : ℝ), c > 0 ∧
 247    ∀ (decoder : Fin n → Bool → Prop),
 248      -- Any decoder that reads fewer than n bits cannot determine SAT result
 249      (∃ M : Finset (Fin n), M.card < n ∧
 250        ∀ result : Bool, ∃ tape1 tape2 : Fin n → Bool,
 251          (∀ i ∈ M, tape1 i = tape2 i) ∧
 252          decoder tape1 result ∧ ¬decoder tape2 result)
 253-/
 254
 255/-- **THE SEPARATION**: Tc << Tr (documentation / TODO)
 256
 257Intended claim: There is a gap between computation time (Tc) and recognition time (Tr).
 258For large n, Tc scales as O(n^{1/3} log n) while Tr scales as Ω(n). -/
 259/-!
 260theorem computation_recognition_separation (n : ℕ) (hn : 100 ≤ n) :
 261    ∃ (Tc Tr : ℝ),
 262      Tc ≤ n^(1/3 : ℝ) * Real.log n ∧
 263      Tr ≥ n ∧
 264      Tc < Tr
 265-/
 266
 267end CellularAutomata
 268end Complexity
 269end IndisputableMonolith
 270