| import OAI.Combinatorics.LowSpaceSubsetSum.Word.WordDecisionBranches |
| import OAI.Combinatorics.LowSpaceSubsetSum.Search.DecisionRound |
|
|
| namespace OAI |
|
|
| /-! # Independent fair-word rounds and counter-executed amplification -/ |
|
|
| noncomputable section |
| namespace LowSpaceSubsetSum.WordDecisionRound |
|
|
| variable {n : ℕ} |
|
|
| abbrev Words (I : Instance n) := |
| WordDecisionBranches.CompressionWords I × WordDecisionBranches.RepresentationWords I |
|
|
| def law (I : Instance n) : FiniteLaw (Words I) := |
| (FiniteLaw.uniform : FiniteLaw (WordDecisionBranches.CompressionWords I)).prod |
| (FiniteLaw.uniform : FiniteLaw (WordDecisionBranches.RepresentationWords I)) |
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|
| def combine (c r : (Bool × ℕ) × ℕ) : (Bool × ℕ) × ℕ := |
| ((c.1.1 || r.1.1,c.1.2+r.1.2+3),max c.2 (r.2+8)+16) |
|
|
| def run (I : Instance n) (hn : 180000000 ≤ n) |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) |
| (words : Words I) : (Bool × ℕ) × ℕ := |
| combine (WordDecisionBranches.compression I (by omega) words.1) |
| (WordDecisionBranches.representation I (by omega) hlog words.2) |
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|
| theorem value (I : Instance n) (hn : 180000000 ≤ n) |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) |
| (words : Words I) : |
| (run I hn hlog words).1.1 = |
| ((WordDecisionBranches.compression I (by omega) words.1).1.1 || |
| (WordDecisionBranches.representation I (by omega) hlog words.2).1.1) := rfl |
|
|
| theorem sound (I : Instance n) (hn : 180000000 ≤ n) |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) |
| (words : Words I) (hy : (run I hn hlog words).1.1 = true) : Solvable I := by |
| rw [value,Bool.or_eq_true] at hy |
| rcases hy with hc | hr |
| · exact WordDecisionBranches.compression_sound I (by omega) words.1 hc |
| · exact WordDecisionBranches.representation_sound I (by omega) hlog words.2 hr |
|
|
| theorem probability (I : Instance n) (hn : 180000000 ≤ n) |
| (hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000) |
| (hs : Solvable I) |
| (hden : ((mixerSize n/2 : ℕ) : ℝ)+1 < |
| (2 : ℝ)^((1/1500 : ℝ)*(mixerSize n/2 : ℕ))) |
| (hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) : |
| 1/(inputScale I : ℝ)^110 ≤ |
| (law I).probability (fun words => (run I hn hlog words).1.1 = true) := by |
| let P : WordDecisionBranches.CompressionWords I → Prop := fun words => |
| (WordDecisionBranches.compression I (by omega) words).1.1 = true |
| let Q : WordDecisionBranches.RepresentationWords I → Prop := fun words => |
| (WordDecisionBranches.representation I (by omega) hlog words).1.1 = true |
| by_cases hm : (153/20000 : ℝ)*n ≤ |
| meanDeficiency (CompressionData.mk I).weights Finset.univ (compressionHalfSize n) |
| · have hp := WordDecisionBranches.compression_probability I (by omega) hs hm hscale |
| have hu : (2 : ℝ) ≤ inputScale I := by |
| exact_mod_cast (show 2 ≤ inputScale I by unfold inputScale; omega) |
| have hd : (8 : ℝ) ≤ (inputScale I : ℝ)^110 := |
| (by norm_num : (8 : ℝ) ≤ 2^110).trans (pow_le_pow_left₀ (by norm_num) hu _) |
| apply ((one_div_le_one_div_of_le (by norm_num) hd).trans hp).trans |
| simpa only [law,value,Bool.or_eq_true,P,Q] using |
| FiniteLaw.prod_or_ge_left |
| (FiniteLaw.uniform : FiniteLaw (WordDecisionBranches.CompressionWords I)) |
| (FiniteLaw.uniform : FiniteLaw (WordDecisionBranches.RepresentationWords I)) P Q |
| · have hm' : meanDeficiency (canonicalRepresentationData I).zeroSetup.originalWeights |
| Finset.univ (mixerSize n/2) < (153/20000 : ℝ)*n := lt_of_not_ge hm |
| apply (WordDecisionBranches.representation_probability I hn hlog hs hm' hden hscale).trans |
| simpa only [law,value,Bool.or_eq_true,P,Q] using |
| FiniteLaw.prod_or_ge_right |
| (FiniteLaw.uniform : FiniteLaw (WordDecisionBranches.CompressionWords I)) |
| (FiniteLaw.uniform : FiniteLaw (WordDecisionBranches.RepresentationWords I)) P Q |
|
|
| def repeated (I : Instance n) (hn : 180000000 ≤ n) |
| (hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) |
| (words : Fin (inputScale I^120) → Words I) : (Bool × ℕ) × ℕ := |
| DecisionRepetitionCells.run (run I hn (roundLogBudget I (by omega) hscale)) words |
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|
| theorem repeated_sound (I : Instance n) (hn : 180000000 ≤ n) |
| (hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) |
| (words : Fin (inputScale I^120) → Words I) |
| (hy : (repeated I hn hscale words).1.1 = true) : Solvable I := by |
| rw [repeated,DecisionRepetitionCells.value,DecisionRepetition.value] at hy |
| exact repeatAccept_sound _ _ (sound I hn _) words hy |
|
|
| theorem repeated_probability (I : Instance n) (hn : 180000000 ≤ n) |
| (hscale : Real.logb 2 (inputScale I : ℝ) ≤ (n : ℝ)/1000000000) |
| (hden : ((mixerSize n/2 : ℕ) : ℝ)+1 < |
| (2 : ℝ)^((1/1500 : ℝ)*(mixerSize n/2 : ℕ))) (hs : Solvable I) : |
| (2/3 : ℝ) < ((law I).iid (inputScale I^120)).probability |
| (fun words => (repeated I hn hscale words).1.1 = true) := by |
| simp only [repeated,DecisionRepetitionCells.value,DecisionRepetition.value] |
| exact repetition_acceptance_probability (law I) _ (by unfold inputScale; omega) |
| (probability I hn _ hs hden hscale) |
|
|
| end LowSpaceSubsetSum.WordDecisionRound |
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|
| end |
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|
| end OAI |
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