File size: 3,542 Bytes
ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
import OAI.Computability.MaxCut.Machines.MachineUniformRounding
import OAI.Computability.MaxCut.Games.UniformPartition
import OAI.Computability.MaxCut.PCP.Parity
import OAI.Computability.MaxCut.Encoding.PortTableEncoding
import OAI.Computability.MaxCut.Machines.MachineTableIteration
import OAI.Computability.MaxCut.Games.CloneTable
import OAI.Computability.MaxCut.Games.Theorem
import OAI.Computability.MaxCut.Encoding.Bit
import OAI.Computability.MaxCut.Arithmetic.GridComputer
namespace OAI
noncomputable section
namespace OptimalMaxCut.LongCode.InstanceAdapter
open MaxCutGames.Foundations.Target MaxCutGames.Foundations.Complexity
theorem bits_eq_gameBits {q : ℕ} (g : Instance q) : bits g = gameBits g := by
have hw (l : List ℕ) : CounterMachine.Expr.unaryWords l = encodeWords l := by
induction l with
| nil => rfl
| cons n l ih =>
simp only [CounterMachine.Expr.unaryWords,List.flatMap_cons,encodeWords,encodeWord] at *
rw [ih]
unfold bits gameBits
rw [hw]
rfl
theorem exists_encoded_gridComputer (q : ℕ) (t : ℚ) (ht : t ∈ Set.Icc (-1:ℚ) 1) (K : ℕ) :
∃ pc : Turing.TM2ComputableInPolyTime (gameBits : Instance q → List Bool) ScaledGraph.bits
(fun g => gridGraph g t ht K), MachineFiniteAlphabet.FiniteAlphabet pc.tm := by
obtain ⟨pc,hpc⟩ := exists_gridComputer q t ht K
let pc' : Turing.TM2ComputableInPolyTime (gameBits : Instance q → List Bool) ScaledGraph.bits
(fun g => gridGraph g t ht K) := {
tm := pc.tm
inputAlphabet := pc.inputAlphabet
outputAlphabet := pc.outputAlphabet
time := pc.time
outputsFun := fun g => by simpa only [bits_eq_gameBits] using pc.outputsFun g }
exact ⟨pc',hpc⟩
end OptimalMaxCut.LongCode.InstanceAdapter
namespace OptimalMaxCut
open LongCode LongCode.InstanceAdapter
open MaxCutGames.Foundations.Target MaxCutGames.Foundations.Complexity
/-- Optimal Max-Cut approximation hardness: for EVERY fixed real alpha above
the exact Goemans--Williamson constant, an ordinary binary polynomial reduction
prints a full finite SIMPLE UNWEIGHTED graph and a strict YES/NO optimum gap.
The construction uses an analytic inequality, rational long-code construction,
integer rounding, finite-field derandomization and a graph-printer TM. -/
theorem main : MainStatement := by
intro α ha ha1
obtain ⟨t,ε,y,n,δ,K,ht,ht0,ht1,hε,hεhalf,hδ,hδhalf,hy,hn,hgap,hcomplete,hsound⟩ :=
Game.exists_grid_gap_parameters α ha ha1
obtain ⟨ug⟩ := MaxCutGames.theorem11 (ε : ℝ) δ (by exact_mod_cast hε)
hεhalf hδ hδhalf
obtain ⟨pc,hpc⟩ := exists_encoded_gridComputer ug.alphabet t ht K
refine ⟨{
yesBound := y
noBound := n
yes_pos := hy
no_nonneg := hn
gap := hgap
construct := fun input => gridGraph (ug.construct input) t ht K
computation := MachineSequential.composeBits ug.computation pc
finiteAlphabet := MachineFiniteAlphabet.composeBits ug.computation pc ug.finiteAlphabet hpc
completeness := ?_
soundness := ?_ }⟩
· intro input hin
exact hcomplete (ofInstance (ug.construct input)) (vertexEquiv _ _) (anchor _)
(InstanceAdapter.complete _ _ (ug.completeness input hin))
· intro input hin
exact hsound (by have := ug.alphabetAtLeastTwo; omega)
(ofInstance (ug.construct input)) (vertexEquiv _ _) (anchor _)
(InstanceAdapter.sound _ (ug.simpleBipartite input) _ (ug.soundness input hin))
end OptimalMaxCut
example : OptimalMaxCut.MainStatement := OptimalMaxCut.main
end
end OAI
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