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import OAI.Computability.BinPacking.Machines.GraphPackingPreparationFinish
namespace OAI
noncomputable section
namespace BinPackingGap.GraphPackingPreparation
section
open Turing BinPackingGames.Foundations.Complexity
open GraphPackingRegisters GraphPackingProgram PackingMachineBlocks
open FiniteTapeProgram BinaryRegisterProgram
variable (fixed : InventoryData) (K : Nat)
@[simp] private theorem graphSlots_apply (i : Fin 7) : graphSlots fixed K i =
![.inr .input, .inr .scanScratch, .inr .scanTemporary,
.inr .threshold, .inr .endpoints, .inr .vertices, .inr .edges] i := by
simp only [graphSlots]
rfl
@[simp] private theorem thresholdSlots_apply (i : Fin 3) : thresholdSlots fixed K i =
![.inr .threshold, .inr .decodeScratch, slots fixed K (.inl (input fixed K .k))] i := by
simp only [thresholdSlots]
rfl
@[simp] private theorem setup_apply (r : PackingSetupMachine.Reg (parameters fixed K)) :
setup fixed K r = .inl (.inr (.inl r)) := rfl
def parsePhases : List (Program fixed K) :=
[prepareGraph (graphSlots fixed K),
tallyToBinary fixed K .vertices .n,
tallyToBinary fixed K .edges .m,
PackingArithmeticProgram.code (PackingFieldDecodeMachine.code (thresholdSlots fixed K))]
def parsePolynomial (_fixed : InventoryData) (_K : Nat) : Polynomial Nat :=
Polynomial.C 6 * (Polynomial.X + 1) ^ 2 + Polynomial.C 15 * Polynomial.X + Polynomial.C 15
private def fieldTapes (x : GraphReductionInput) : Tape fixed K → List Bool
| .inr .input => graphBits x
| .inr .threshold => BinaryEncoding.natBits x.k
| .inr .endpoints => GraphFieldMachine.fieldBits .endpoints x
| .inr .vertices => List.replicate x.graph.n true
| .inr .edges => List.replicate x.graph.edges.length true
| _ => []
private theorem preparation_frame (x : GraphReductionInput) :
GraphPreparationMachine.preparedTapes (graphSlots fixed K) (initialTapes fixed K x) x =
fieldTapes fixed K x := by
have atSlot (j : Fin 7) :
GraphPreparationMachine.preparedTapes (graphSlots fixed K) (initialTapes fixed K x) x
(graphSlots fixed K j) = fieldTapes fixed K x (graphSlots fixed K j) := by
rw [GraphPreparationMachine.preparedTapes_register]
fin_cases j <;> simp [initialTapes, fieldTapes]
funext tape
cases tape with
| inl r =>
simpa only [initialTapes, fieldTapes] using
GraphPreparationMachine.preparedTapes_other (graphSlots fixed K)
(initialTapes fixed K x) x (.inl r)
(by simp) (by simp)
(by simp) (by simp)
| inr extra =>
by_cases h₃ : extra = .threshold
· subst extra; exact atSlot 3
by_cases h₄ : extra = .endpoints
· subst extra; exact atSlot 4
by_cases h₅ : extra = .vertices
· subst extra; exact atSlot 5
by_cases h₆ : extra = .edges
· subst extra; exact atSlot 6
rw [GraphPreparationMachine.preparedTapes_other (graphSlots fixed K)
(initialTapes fixed K x) x (.inr extra)
(by simpa using h₃) (by simpa using h₄)
(by simpa using h₅) (by simpa using h₆)]
cases extra <;> simp_all [initialTapes, fieldTapes]
private def vertexTapes (x : GraphReductionInput) : Tape fixed K → List Bool :=
Function.update (Function.update (fieldTapes fixed K x) (.inr .vertices) [])
(regTape fixed K (input fixed K .n)) x.graph.n.bits
private def edgeTapes (x : GraphReductionInput) : Tape fixed K → List Bool :=
Function.update (Function.update (vertexTapes fixed K x) (.inr .edges) [])
(regTape fixed K (input fixed K .m)) x.graph.edges.length.bits
private theorem decoded_frame (x : GraphReductionInput) :
PackingFieldDecodeMachine.decodedTapes (thresholdSlots fixed K) (edgeTapes fixed K x)
x.k.bits [] = parsedTapes fixed K x := by
funext tape
rcases tape with (r | scratch) | extra
· change _ = registerTapes (slots fixed K) (baseTapes fixed K x)
(PackingMachineLayout.values (Numerator fixed) PackingItemExpression.denominator
(fun _ => 0) (parsedOther fixed K x)) (slots fixed K (.inl r))
rw [registerTapes_reg]
rcases r with (v | (setupReg | workReg)) | node
· simp [PackingFieldDecodeMachine.decodedTapes, edgeTapes,
vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers,
PackingMachineLayout.values, PackingSetupMachine.input]
· rcases setupReg with i | (o | node)
· cases i <;> simp [PackingFieldDecodeMachine.decodedTapes,
edgeTapes, vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers,
PackingMachineLayout.values, parsedOther, PackingSetupMachine.frame,
PackingSetupMachine.input, PackingSetupExpression.inputs]
change Function.update (β := fun _ => List Bool) _ (thresholdSlots fixed K 2) x.k.bits
(thresholdSlots fixed K 2) = x.k.bits
exact Function.update_self _ _ _
· simp [PackingFieldDecodeMachine.decodedTapes, edgeTapes,
vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers,
PackingMachineLayout.values, parsedOther, PackingSetupMachine.frame,
PackingSetupMachine.input]
· simp [PackingFieldDecodeMachine.decodedTapes, edgeTapes,
vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers,
PackingMachineLayout.values, parsedOther, PackingSetupMachine.frame,
PackingSetupMachine.input]
· simp [PackingFieldDecodeMachine.decodedTapes, edgeTapes,
vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers,
PackingMachineLayout.values, parsedOther, PackingSetupMachine.input]
· simp [PackingFieldDecodeMachine.decodedTapes, edgeTapes,
vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers,
PackingMachineLayout.values, PackingSetupMachine.input]
· change _ = registerTapes (slots fixed K) (baseTapes fixed K x)
(PackingMachineLayout.values (Numerator fixed) PackingItemExpression.denominator
(fun _ => 0) (parsedOther fixed K x)) (slots fixed K (.inr scratch))
rw [registerTapes_scratch]
simp [PackingFieldDecodeMachine.decodedTapes, edgeTapes,
vertexTapes, fieldTapes, regTape, slots, PackingMachineLayout.registers]
· change _ = PackingMachineLayout.tapes (Numerator fixed) PackingItemExpression.denominator
(baseTapes fixed K x) (fun _ => 0) (parsedOther fixed K x) (.inr extra)
rw [PackingMachineLayout.tapes_extra]
cases extra <;> simp [PackingFieldDecodeMachine.decodedTapes,
edgeTapes, vertexTapes, fieldTapes, regTape, slots, PackingMachineLayout.registers,
baseTapes]
private theorem tally_exec (source : Extra) (destination : PackingSetupExpression.Input)
(different : source ≠ .scanScratch) (base : Tape fixed K → List Bool)
(n : Nat) (state : State)
(sourceWord : base (.inr source) = List.replicate n true)
(outputEmpty : base (regTape fixed K (input fixed K destination)) = [])
(scratchEmpty : base (.inr .scanScratch) = []) :
∃ steps ≤ 3 * (n + 1) ^ 2 + 1,
Exec (tallyToBinary fixed K source destination) ⟨state, base⟩ steps
⟨.arithmetic (BinaryAddMachine.clean ()),
Function.update (Function.update base (.inr source) [])
(regTape fixed K (input fixed K destination)) n.bits⟩ := by
have hSO : (Sum.inr source : Tape fixed K) ≠ regTape fixed K (input fixed K destination) := by
simp [regTape, slots, PackingMachineLayout.registers]
have hSC : (Sum.inr source : Tape fixed K) ≠ .inr .scanScratch := by simpa using different
have hOC : regTape fixed K (input fixed K destination) ≠ Sum.inr Extra.scanScratch := by
simp [regTape, slots, PackingMachineLayout.registers]
have run := BinaryTallyMachine.plainTallyFromTapesInTime (.inr source)
(regTape fixed K (input fixed K destination)) (.inr .scanScratch) hSO hSC hOC
id none (BinaryTallyMachine.instruction (.inr source)
(regTape fixed K (input fixed K destination)) (.inr .scanScratch) id none)
(fun _ => rfl) base () n sourceWord outputEmpty scratchEmpty
simpa only [tallyToBinary, arithmeticControl] using
invokeExecInTime arithmeticControl (BinaryAddMachine.clean ()) .scan
(BinaryTallyMachine.instruction (.inr source)
(regTape fixed K (input fixed K destination)) (.inr .scanScratch) id none)
⟨state, base⟩ (BinaryAddMachine.clean ()) _ _ run
theorem parse_exec (x : GraphReductionInput) (state : State) :
∃ steps ≤ (parsePolynomial fixed K).eval (graphBits x).length,
PhaseRuns (parsePhases fixed K) ⟨state, initialTapes fixed K x⟩ steps
⟨.arithmetic (BinaryAddMachine.clean ()), parsedTapes fixed K x⟩ := by
have first := prepareGraph_exec (graphSlots fixed K) ⟨state, initialTapes fixed K x⟩ x
(by rfl) (by rfl) (by rfl)
rw [preparation_frame fixed K x] at first
obtain ⟨a, ha, second⟩ := tally_exec fixed K .vertices .n (by decide)
(fieldTapes fixed K x) x.graph.n (.graph (GraphFieldMachine.clean ()))
rfl rfl rfl
change Exec (tallyToBinary fixed K .vertices .n)
⟨.graph (GraphFieldMachine.clean ()), fieldTapes fixed K x⟩ a
⟨.arithmetic (BinaryAddMachine.clean ()), vertexTapes fixed K x⟩ at second
obtain ⟨b, hb, third⟩ := tally_exec fixed K .edges .m (by decide)
(vertexTapes fixed K x) x.graph.edges.length (.arithmetic (BinaryAddMachine.clean ()))
(by simp [vertexTapes, fieldTapes, regTape, slots, PackingMachineLayout.registers])
(by simp [vertexTapes, fieldTapes, regTape, slots, input,
PackingMachineLayout.registers, PackingSetupMachine.input])
(by simp [vertexTapes, fieldTapes, regTape, slots, PackingMachineLayout.registers])
change Exec (tallyToBinary fixed K .edges .m)
⟨.arithmetic (BinaryAddMachine.clean ()), vertexTapes fixed K x⟩ b
⟨.arithmetic (BinaryAddMachine.clean ()), edgeTapes fixed K x⟩ at third
have decoded := PackingFieldDecodeMachine.decodeNat_exec (thresholdSlots fixed K)
(edgeTapes fixed K x) () x.k []
(by simp [ edgeTapes, vertexTapes, fieldTapes, regTape, slots,
PackingMachineLayout.registers])
(by simp [ edgeTapes, vertexTapes, fieldTapes, regTape, slots,
PackingMachineLayout.registers])
rw [decoded_frame fixed K x] at decoded
have fourth := PackingArithmeticProgram.exec _ _ _ _
(.arithmetic (BinaryAddMachine.clean ())) decoded
refine ⟨(1 + GraphPreparationMachine.steps x) + (a + (b + (1 + (2 * x.k.size + 2)))), ?_, ?_⟩
· have hp := prepareGraph_cost x
have hn := Nat.pow_le_pow_left (Nat.add_le_add_right (graph_vertexCount_le_bits x) 1) 2
have hm := Nat.pow_le_pow_left (Nat.add_le_add_right (graph_edgeCount_le_bits x) 1) 2
have hk : 2 * x.k.size + 2 ≤ (graphBits x).length + 1 := by
rw [PackingFieldDecodeMachine.decodeNat_steps]
simpa only [GraphFieldMachine.thresholdBits] using
Nat.add_le_add_right (GraphFieldMachine.fieldBits_length_le .threshold x) 1
simp only [parsePolynomial, Polynomial.eval_add, Polynomial.eval_mul,
Polynomial.eval_pow, Polynomial.eval_C, Polynomial.eval_X, Polynomial.eval_one]
omega
· exact PhaseRuns.cons first (PhaseRuns.cons second
(PhaseRuns.cons third (PhaseRuns.single fourth)))
end
section
open BinPackingGames.Foundations.Complexity
variable (fixed : InventoryData) (K : Nat)
def powerPolynomial : Polynomial Nat :=
80 * (Polynomial.X + 1) ^ 3 +
2 * BinaryPowerMachine.timePolynomial.comp (2 * GraphPackingWidths.polynomial fixed K) + 4
private theorem tally_bound {n s : Nat} (h : n ≤ s) :
BinaryToTallyMachine.runtimeBound n ≤ 40 * (s + 1) ^ 3 := by
have size := (nat_size_le_self n).trans h
calc
BinaryToTallyMachine.runtimeBound n = 40 * (n + 1) * (n.size + 1) ^ 2 := rfl
_ ≤ 40 * (s + 1) * (s + 1) ^ 2 :=
Nat.mul_le_mul (Nat.mul_le_mul_left 40 (by omega))
(Nat.pow_le_pow_left (by omega) 2)
_ = _ := by ring
theorem powerBudget_le (x : GraphReductionInput) :
powerBudget fixed K x ≤ (powerPolynomial fixed K).eval (graphBits x).length := by
let w := GraphPackingWidths.width fixed K x
have ns : x.graph.n ≤ (graphBits x).length := graph_vertexCount_le_bits x
have ms : x.graph.edges.length ≤ (graphBits x).length := graph_edgeCount_le_bits x
have sw : (graphBits x).length + 1 ≤ w := GraphPackingWidths.input_length_le_width fixed K x
have three : (3 : Nat).size ≤ w :=
(nat_size_le_self 3).trans (GraphPackingWidths.small_constant_width fixed K x)
have radix : (repetitions fixed K x + 1).size ≤ w :=
(nat_size_le_self _).trans (GraphPackingWidths.repetitions_width fixed K x)
have vertex := MachineComposition.natPolynomial_eval_mono BinaryPowerMachine.timePolynomial
(show (3 : Nat).size + x.graph.n ≤ 2 * w by omega)
have edge := MachineComposition.natPolynomial_eval_mono BinaryPowerMachine.timePolynomial
(show (repetitions fixed K x + 1).size + x.graph.edges.length ≤ 2 * w by omega)
have tn := tally_bound ns
have tm := tally_bound ms
simp only [powerPolynomial, Polynomial.eval_add, Polynomial.eval_mul, Polynomial.eval_pow,
Polynomial.eval_ofNat, Polynomial.eval_one, Polynomial.eval_X, Polynomial.eval_comp]
change _ ≤ 80 * ((graphBits x).length + 1) ^ 3 +
2 * BinaryPowerMachine.timePolynomial.eval (2 * w) + 4
unfold powerBudget PackingPowerProgram.budget
omega
def arithmeticPolynomial : Polynomial Nat :=
PackingSetupMachine.timePolynomial (GraphPackingRegisters.parameters fixed K) +
powerPolynomial fixed K + 1
end
open GraphPackingRegisters BinaryRegisterProgram FiniteTapeProgram PackingMachineBlocks
open GraphPackingProgram
variable (fixed : InventoryData) (K : Nat)
def timePolynomial : Polynomial Nat :=
parsePolynomial fixed K + arithmeticPolynomial fixed K + finishPolynomial fixed K + 1
theorem prepare_exec (x : GraphReductionInput) (state : State) :
∃ steps ≤ (timePolynomial fixed K).eval (graphBits x).length,
Exec (prepare fixed K) ⟨state, initialTapes fixed K x⟩ steps
⟨.arithmetic (BinaryAddMachine.clean ()), preparedTapes fixed K x⟩ := by
obtain ⟨a, ha, arun⟩ := parse_exec fixed K x state
obtain ⟨b, hb, brun⟩ := setup_exec fixed K x (.arithmetic (BinaryAddMachine.clean ()))
obtain ⟨c, hc, crun⟩ := powers_exec fixed K x (.arithmetic (BinaryAddMachine.clean ()))
obtain ⟨d, hd, drun⟩ := finish_exec fixed K x (.arithmetic (BinaryAddMachine.clean ()))
have middle := PhaseRuns.cons brun crun
have whole := arun.append (middle.append drun)
have actual := whole.toExec
refine ⟨a + (b + c + d) + 1, ?_, ?_⟩
· have powers := powerBudget_le fixed K x
simp only [timePolynomial, arithmeticPolynomial, Polynomial.eval_add,
Polynomial.eval_one] at ⊢
omega
· simpa only [prepare, parsePhases, powerPhases, List.cons_append, List.nil_append] using actual
end BinPackingGap.GraphPackingPreparation
end
end OAI
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