File size: 3,916 Bytes
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import OAI.Computability.MinUncut.Machines.MachineTableIteration
import OAI.Computability.MinUncut.Graphs.GeneratedGraph
import OAI.Computability.MinUncut.Estimates.SelectedSyntax
namespace OAI
section
noncomputable section
namespace MinUncut.Uniform
open MinUncut.Preprocess MinUncut.Inner MinUncut.Outer MinUncut.SourceBridge
open MinUncut.Costed MinUncut.Costed.SourceWords
open MinUncutGames MinUncutGames.Foundations MinUncutGames.Foundations.Complexity
open MinUncutGames.Foundations.Hastad MinUncutGames.Foundations.Hastad.SourceOccurrences
open MinUncutGames.Reduction
def frontend (F : BinaryFormula.Formula) : SourceGeneratorContract.NonemptyFormula :=
MinUncutGames.Outer.HastadSource.gapInput Parameters.baseTable (BinaryOccurrenceRename.renamed F)
def frontendMachine : Turing.TM2ComputableInPolyTime BinaryEncoding.formulaBits
SourceGeneratorContract.inputEncoding frontend :=
MachineSequential.composeBits (f := BinaryOccurrenceRename.renamed)
(g := MinUncutGames.Outer.HastadSource.gapInput Parameters.baseTable) BinaryRenameMachine.computableInPolyTime
(MinUncutGames.Outer.HastadSource.gapMachine Parameters.baseTable)
lemma frontend_finite : MachineFiniteAlphabet.FiniteAlphabet frontendMachine.tm :=
MachineFiniteAlphabet.composeBits _ _ BinaryRenameMachine.computation_finiteAlphabet
(TableIterationFiniteAlphabet.gapMap Parameters.baseTable
(MinUncutGames.Outer.HastadSource.roundMachine Parameters.baseTable)
(MinUncutGames.Outer.HastadSource.roundMachine_finite Parameters.baseTable))
def program (K : ℕ) := completeGraphProgram (selectedParameters K) (chosenEnumeration K)
(chosenRepetition K) (chosenNoise K)
lemma program_computable : Computable (fun K=>(program K).code) := c_selectedGraphCode
def source (K : ℕ) (F : BinaryFormula.Formula) : SourceEncoding.Input :=
(selectedParameters K).sourceReduction.reduce F
lemma source_eq (K : ℕ) (F : BinaryFormula.Formula) :
source K F=sourceInput (frontend F).val (chosenRepetition K) (chosenNoise K)
(SourceNoiseParameter.noiseDenominator_pos (chosenError K)) := rfl
def output (K : ℕ) (F : BinaryFormula.Formula) : MinUncut.Output :=
generatedGraph (selectedParameters K) (source K F) (chosenEnumeration K)
lemma output_yes (K : ℕ) (F : BinaryFormula.Formula) (hF : F.Satisfiable) :
(output K F).opt≤(output K F).threshold := by
obtain ⟨s,hs⟩:=binaryReduction_complete (selectedParameters K).sourceReduction F hF
(selectedParameters K).o.t (selectedParameters K).o.b4 (selectedParameters K).sourceError_budget
exact generatedGraph_yes (selectedParameters K) (source K F) (chosenEnumeration K) s hs
lemma output_no (K : ℕ) (hK : 2≤K) (F : BinaryFormula.Formula) (hF : ¬F.Satisfiable) :
K*(output K F).threshold<(output K F).opt := by
apply generatedGraph_no (selectedParameters K) (source K F) (chosenEnumeration K) hK
(show 8*K≤parameterJ K by unfold parameterJ; omega)
exact binaryReduction_sound (selectedParameters K).sourceError_small
(selectedParameters K).sourceReduction F hF
lemma output_words (K : ℕ) (F : BinaryFormula.Formula) :
∃table junk,completeGraphWords (selectedParameters K) (chosenEnumeration K)
(chosenRepetition K) (chosenNoise K) (formulaWords (frontend F).val)=
(output K F).vertices::((table : List Bool).map Bool.toNat++2::(output K F).threshold::junk) ∧
(output K F).bits=BinaryEncoding.nameBits (output K F).vertices++table++
BinaryEncoding.nameBits (output K F).threshold := by
obtain ⟨junk,hj⟩:=completeGraphWords_spec (selectedParameters K) (chosenEnumeration K)
(frontend F).val (chosenRepetition K) (chosenNoise K)
(SourceNoiseParameter.noiseDenominator_pos (chosenError K)) (frontend F).property
rw [←source_eq] at hj
refine ⟨_,junk,hj,?_⟩
exact GraphRegisters.listOutput_bits _ _ _
end MinUncut.Uniform
end
end
end OAI
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