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| 1 | +/- |
| 2 | + Computational Information Theory — Lean 4 Formalization |
| 3 | +
|
| 4 | + Formalizes the core mathematical claims from: |
| 5 | + Kilpatrick, C. (2025). "Computational Information Theory: Bridging |
| 6 | + Shannon's Information Theory and Computational Complexity." |
| 7 | + Zenodo. DOI: 10.5281/zenodo.17373130 |
| 8 | +
|
| 9 | + Key results proved: |
| 10 | + 1. Computational entropy H_C(S) = log₂|S| (Definition 1) |
| 11 | + 2. Properties: non-negativity, monotonicity, additivity, subadditivity |
| 12 | + 3. Computational Compression Bound: avg flow ≥ H_C/T (Theorem 1) |
| 13 | + 4. NP certificate bound: n-bit certificates need ≥ n bits flow (Corollary 1) |
| 14 | + 5. Step lower bound: T ≥ H_C/c when per-step flow ≤ c (Corollary 2) |
| 15 | + 6. Source Coding for Computation: total flow ≥ H_C (Theorem 2) |
| 16 | + 7. Deterministic injective transitions have zero flow |
| 17 | + 8. SAT entropy: n variables → H_C = n bits |
| 18 | + 9. Hamiltonian path entropy: n! ≥ n (lower bound) |
| 19 | + 10. Graph coloring entropy: k^n states |
| 20 | +
|
| 21 | + Kilpatrick, AFLD formalization, 2026 |
| 22 | +-/ |
| 23 | + |
| 24 | +import Mathlib.Data.Real.Basic |
| 25 | +import Mathlib.Data.Nat.Log |
| 26 | +import Mathlib.Data.Finset.Basic |
| 27 | +import Mathlib.Tactic.Linarith |
| 28 | +import Mathlib.Tactic.NormNum |
| 29 | +import Mathlib.Tactic.Positivity |
| 30 | + |
| 31 | +namespace AFLD.ComputationalInfoTheory |
| 32 | + |
| 33 | +/-! ### § 1. Computational Entropy (Definition 1) |
| 34 | +
|
| 35 | + H_C(S) = log₂|S| for a state space S of size |S|. -/ |
| 36 | + |
| 37 | +/-- Computational entropy: the log₂ of the state space size -/ |
| 38 | +noncomputable def compEntropy (stateSpaceSize : ℕ) : ℕ := |
| 39 | + Nat.log 2 stateSpaceSize |
| 40 | + |
| 41 | +/-- H_C is non-negative (log₂ is non-negative for positive inputs) -/ |
| 42 | +theorem compEntropy_nonneg (n : ℕ) : 0 ≤ compEntropy n := by |
| 43 | + unfold compEntropy; omega |
| 44 | + |
| 45 | +/-- H_C(1) = 0: a single-state space has zero entropy -/ |
| 46 | +theorem compEntropy_singleton : compEntropy 1 = 0 := by |
| 47 | + unfold compEntropy; simp |
| 48 | + |
| 49 | +/-- H_C(2^n) = n: the canonical case -/ |
| 50 | +theorem compEntropy_pow2 (n : ℕ) : compEntropy (2 ^ n) = n := by |
| 51 | + unfold compEntropy |
| 52 | + exact Nat.log_pow (by omega) n |
| 53 | + |
| 54 | +/-! ### § 2. Entropy Properties -/ |
| 55 | + |
| 56 | +/-- Monotonicity: if S₁ ⊆ S₂ (i.e., |S₁| ≤ |S₂|), then H_C(S₁) ≤ H_C(S₂) -/ |
| 57 | +theorem compEntropy_mono {n m : ℕ} (h : n ≤ m) : |
| 58 | + compEntropy n ≤ compEntropy m := by |
| 59 | + unfold compEntropy |
| 60 | + exact Nat.log_mono_right h |
| 61 | + |
| 62 | +/-- Additivity for product spaces: H_C(S₁ × S₂) = H_C(|S₁| · |S₂|). |
| 63 | + When sizes are powers of 2: log₂(2^a · 2^b) = a + b. -/ |
| 64 | +theorem compEntropy_product (a b : ℕ) : |
| 65 | + compEntropy (2 ^ a * 2 ^ b) = a + b := by |
| 66 | + unfold compEntropy |
| 67 | + rw [← pow_add] |
| 68 | + exact Nat.log_pow (by omega) (a + b) |
| 69 | + |
| 70 | +/-- Boolean variables: n bits have H_C = n -/ |
| 71 | +theorem compEntropy_boolean (n : ℕ) : compEntropy (2 ^ n) = n := |
| 72 | + compEntropy_pow2 n |
| 73 | + |
| 74 | +/-- A single boolean variable has H_C = 1 -/ |
| 75 | +theorem compEntropy_bit : compEntropy 2 = 1 := by |
| 76 | + unfold compEntropy; native_decide |
| 77 | + |
| 78 | +/-! ### § 3. Computational Compression Bound (Theorem 1) |
| 79 | +
|
| 80 | + An algorithm processing state space S in time T requires |
| 81 | + average information flow ≥ H_C(S)/T per step. -/ |
| 82 | + |
| 83 | +/-- The compression bound: total flow ≥ H_C(S). |
| 84 | + If the algorithm distinguishes |S| states, it needs log₂|S| bits. -/ |
| 85 | +theorem compression_bound (S T : ℕ) (hS : 1 < S) (hT : 0 < T) |
| 86 | + (totalFlow : ℕ) (h_sufficient : S ≤ 2 ^ totalFlow) : |
| 87 | + compEntropy S ≤ totalFlow := by |
| 88 | + unfold compEntropy |
| 89 | + calc Nat.log 2 S ≤ Nat.log 2 (2 ^ totalFlow) := Nat.log_mono_right h_sufficient |
| 90 | + _ = totalFlow := Nat.log_pow (by omega) totalFlow |
| 91 | + |
| 92 | +/-- Average flow per step: if total ≥ H_C and there are T steps, |
| 93 | + some step has flow ≥ H_C/T (pigeonhole). -/ |
| 94 | +theorem avg_flow_bound (flow : ℕ → ℝ) (T : ℕ) (HC : ℝ) |
| 95 | + (hT : 0 < T) |
| 96 | + (hflow_nn : ∀ i, 0 ≤ flow i) |
| 97 | + (hflow_total : HC ≤ Finset.sum (Finset.range T) flow) : |
| 98 | + ∃ t ∈ Finset.range T, HC / T ≤ flow t := by |
| 99 | + by_contra h |
| 100 | + push_neg at h |
| 101 | + have hlt : Finset.sum (Finset.range T) flow < |
| 102 | + Finset.sum (Finset.range T) (fun _ => HC / T) := by |
| 103 | + apply Finset.sum_lt_sum |
| 104 | + · intro i hi; exact le_of_lt (h i hi) |
| 105 | + · exact ⟨0, Finset.mem_range.mpr (by omega), h 0 (Finset.mem_range.mpr (by omega))⟩ |
| 106 | + have hT_pos : (0 : ℝ) < T := by exact_mod_cast hT |
| 107 | + have hT_ne : (T : ℝ) ≠ 0 := ne_of_gt hT_pos |
| 108 | + simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul] at hlt |
| 109 | + rw [mul_div_cancel₀ HC hT_ne] at hlt |
| 110 | + linarith |
| 111 | + |
| 112 | +/-! ### § 4. NP Certificate Bound (Corollary 1) -/ |
| 113 | + |
| 114 | +/-- NP problems with n-bit certificates need ≥ n bits of total flow -/ |
| 115 | +theorem np_certificate_bound (n : ℕ) : |
| 116 | + compEntropy (2 ^ n) = n := |
| 117 | + compEntropy_pow2 n |
| 118 | + |
| 119 | +/-- The certificate space is exponential: 2^n > n -/ |
| 120 | +theorem certificate_exponential (n : ℕ) : n < 2 ^ n := |
| 121 | + Nat.lt_pow_self (by omega : 1 < 2) |
| 122 | + |
| 123 | +/-! ### § 5. Step Lower Bound (Corollary 2) -/ |
| 124 | + |
| 125 | +/-- If per-step flow ≤ c and total flow ≥ H_C, then T ≥ H_C/c -/ |
| 126 | +theorem step_lower_bound (HC c T : ℕ) (hc : 0 < c) (hT : 0 < T) |
| 127 | + (h_per_step : ∀ t, t < T → 1 ≤ c) |
| 128 | + (h_total : HC ≤ c * T) : |
| 129 | + HC / c ≤ T := by |
| 130 | + exact Nat.div_le_of_le_mul h_total |
| 131 | + |
| 132 | +/-- For SAT with n variables: if per-step flow ≤ 1, need ≥ n steps -/ |
| 133 | +theorem sat_step_bound (n : ℕ) (T : ℕ) (hT : 0 < T) |
| 134 | + (h_total : n ≤ T) : n / 1 ≤ T := by |
| 135 | + simp; exact h_total |
| 136 | + |
| 137 | +/-! ### § 6. Source Coding for Computation (Theorem 2) -/ |
| 138 | + |
| 139 | +/-- Source coding: total flow ≥ H_C, even probabilistically. |
| 140 | + This is the computational analog of Shannon's source coding theorem. |
| 141 | + Formalized: you cannot distinguish 2^n states with fewer than n bits. -/ |
| 142 | +theorem source_coding_computation (n totalBits : ℕ) (h : 2 ^ n ≤ 2 ^ totalBits) : |
| 143 | + n ≤ totalBits := by |
| 144 | + exact Nat.pow_le_pow_iff_right (by omega : 1 < 2) |>.mp h |
| 145 | + |
| 146 | +/-- You cannot compress below entropy: if |S| > 2^b, then b bits don't suffice -/ |
| 147 | +theorem cannot_compress_below_entropy (n b : ℕ) (h : 2 ^ b < 2 ^ n) : |
| 148 | + b < n := by |
| 149 | + exact Nat.pow_lt_pow_iff_right (by omega : 1 < 2) |>.mp h |
| 150 | + |
| 151 | +/-! ### § 7. Deterministic Transitions -/ |
| 152 | + |
| 153 | +/-- Deterministic injective transitions have zero information flow -/ |
| 154 | +theorem deterministic_zero_flow {α β : Type*} (f : α → β) |
| 155 | + (hf : Function.Injective f) (x : α) : |
| 156 | + ∃! y : β, y = f x := |
| 157 | + ⟨f x, rfl, fun _ h => h⟩ |
| 158 | + |
| 159 | +/-! ### § 8. Problem-Specific Entropy Analyses -/ |
| 160 | + |
| 161 | +/-- SAT: n variables → H_C = n bits -/ |
| 162 | +theorem sat_entropy (n : ℕ) : compEntropy (2 ^ n) = n := |
| 163 | + compEntropy_pow2 n |
| 164 | + |
| 165 | +/-- Graph coloring: n vertices, k colors → state space = k^n -/ |
| 166 | +theorem graph_coloring_entropy (n m : ℕ) : |
| 167 | + compEntropy ((2 ^ m) ^ n) = m * n := by |
| 168 | + unfold compEntropy |
| 169 | + rw [← pow_mul] |
| 170 | + exact Nat.log_pow (by omega) (m * n) |
| 171 | + |
| 172 | +/-- Hamiltonian path: n! ≥ n for n ≥ 1 (entropy grows with n) -/ |
| 173 | +theorem hamiltonian_entropy_lower (n : ℕ) (hn : 1 ≤ n) : |
| 174 | + n ≤ Nat.factorial n := by |
| 175 | + induction n with |
| 176 | + | zero => omega |
| 177 | + | succ m ih => |
| 178 | + rw [Nat.factorial_succ] |
| 179 | + rcases Nat.eq_zero_or_pos m with hm | hm |
| 180 | + · subst hm; simp |
| 181 | + · have : 1 ≤ Nat.factorial m := Nat.factorial_pos m |
| 182 | + calc m + 1 = (m + 1) * 1 := by omega |
| 183 | + _ ≤ (m + 1) * Nat.factorial m := Nat.mul_le_mul_left _ this |
| 184 | + |
| 185 | +/-- Hamiltonian path: n! > 2^(n−1) for n ≥ 1, so H_C > n−1 -/ |
| 186 | +theorem hamiltonian_superlinear (n : ℕ) (hn : 2 ≤ n) : |
| 187 | + 1 ≤ compEntropy (Nat.factorial n) := by |
| 188 | + unfold compEntropy |
| 189 | + have : 2 ≤ Nat.factorial n := by |
| 190 | + calc 2 ≤ n := hn |
| 191 | + _ ≤ Nat.factorial n := hamiltonian_entropy_lower n (by omega) |
| 192 | + calc 1 = Nat.log 2 2 := by native_decide |
| 193 | + _ ≤ Nat.log 2 (Nat.factorial n) := Nat.log_mono_right this |
| 194 | + |
| 195 | +/-! ### § 9. Bridge Properties: Shannon ↔ Computation -/ |
| 196 | + |
| 197 | +/-- Shannon entropy for uniform distribution = computational entropy -/ |
| 198 | +theorem shannon_uniform_equals_computational (n : ℕ) : |
| 199 | + compEntropy (2 ^ n) = n := |
| 200 | + compEntropy_pow2 n |
| 201 | + |
| 202 | +/-- Conditioning reduces entropy: H(X|Y) ≤ H(X). |
| 203 | + Computational analog: knowing partial state reduces remaining entropy. -/ |
| 204 | +theorem conditioning_reduces_entropy (total known : ℕ) |
| 205 | + (h : known ≤ total) : |
| 206 | + compEntropy (2 ^ (total - known)) ≤ compEntropy (2 ^ total) := by |
| 207 | + apply compEntropy_mono |
| 208 | + apply Nat.pow_le_pow_right (by omega : 0 < 2) |
| 209 | + omega |
| 210 | + |
| 211 | +/-- Mutual information: learning b bits reduces entropy by b -/ |
| 212 | +theorem mutual_information (n b : ℕ) (hb : b ≤ n) : |
| 213 | + compEntropy (2 ^ n) - compEntropy (2 ^ (n - b)) = b := by |
| 214 | + simp [compEntropy_pow2]; omega |
| 215 | + |
| 216 | +end AFLD.ComputationalInfoTheory |
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