| import OAI.Computability.UniqueGames.Decoding.ActualSeedSamplingLemmas |
| import OAI.Computability.UniqueGames.Machines.MachineSingleOrbitBridgeLemmas |
| import OAI.Computability.UniqueGames.PCP.BinaryParityReduction |
| import OAI.Computability.UniqueGames.Reduction.FinishReduction |
|
|
| namespace OAI |
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|
|
| namespace UniqueGamesTheorem.FromMatrixGap |
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|
| open UniqueGamesTheorem.Foundations Target |
| open UniqueGamesTheorem.Foundations.Complexity |
| open UniqueGamesTheorem.Integration |
| open UniqueGamesTheorem.Reduction |
| open Explicit |
|
|
| noncomputable section |
|
|
| /-- Choose the parity error only after the matrix tuple length is fixed, then |
| compose the actual raw-input, matrix-test, and final graph machines. -/ |
| def reduction {ε δ : ℝ} (P : ParameterSelection.OuterParameters ε δ) |
| (M : Decoder.MatrixGap.Parameters P.pStar) : |
| MachineOutputContract.BinaryGapReduction ε δ := by |
| classical |
| have hk : 0 < M.k := M.k_pos |
| let ξ := Classical.choose (P.exists_parity_error M.k hk) |
| have hξspec := Classical.choose_spec (P.exists_parity_error M.k hk) |
| have hξ : 0 < ξ := hξspec.1 |
| have hξsmall : ξ < 1 / 100 := hξspec.2.1 |
| have hξbudget : ξ < P.epsilon0 / (2 * M.k * P.repetitions) := hξspec.2.2 |
| let R := Classical.choice (Outer.BinaryParityReduction.exists_reduction ξ hξ hξsmall) |
| let matrix := fun input => MachineSingleOrbitBridge.output M.k M.T (R.reduce input) |
| let matrixMachine := MachineSequential.composeBits |
| (f := R.reduce) (g := MachineSingleOrbitBridge.output M.k M.T) |
| R.computation (MachineSingleOrbitBridge.computableInPolyTime M.k M.T) |
| have matrixFinite : MachineFiniteAlphabet.FiniteAlphabet matrixMachine.tm := |
| MachineFiniteAlphabet.composeBits |
| (f := R.reduce) (g := MachineSingleOrbitBridge.output M.k M.T) |
| R.computation (MachineSingleOrbitBridge.computableInPolyTime M.k M.T) |
| R.finiteAlphabet (MachineSingleOrbitBridge.workAlphabetFinite M.k M.T) |
| have hq : 2 ≤ 2 ^ M.s := Nat.le_self_pow (by have h := M.s_pos; omega) 2 |
| refine FinishReduction.of_matrix_reduction P hq M.s_pos |
| (Encoding.alphabetEquiv M.s).symm matrix ?_ matrixMachine matrixFinite |
| ((M.k : ℚ) * ξ + P.pStar / 2) |
| (P.matrix_error_budget M.k hk ξ hξbudget.le) ?_ ?_ |
| · intro input |
| exact M.output_translations (Outer.HastadSource.asSource (R.reduce input)) |
| · intro input yes |
| obtain ⟨assignment, hfailure⟩ := |
| Outer.BinaryParityReduction.failure_completeness R input yes |
| have h := M.completeness (Outer.HastadSource.asSource (R.reduce input)) |
| assignment ξ hfailure |
| simpa only [matrix, MachineSingleOrbitBridge.output, |
| MachineSingleOrbitBridge.SingleAddress.tableOutput, Outer.HastadSource.asSource, |
| Rat.cast_add, Rat.cast_mul, Rat.cast_natCast, |
| Rat.cast_div, Rat.cast_ofNat] using h |
| · intro input no |
| exact M.sound (Outer.HastadSource.asSource (R.reduce input)) |
| (R.distinct input) (Outer.BinaryParityReduction.parity_soundness R input no) |
|
|
| end |
| end UniqueGamesTheorem.FromMatrixGap |
|
|
| end OAI |
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|
