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import OAI.Computability.VertexCover.PCP.QueryIncidence
import OAI.Computability.VertexCover.PCP.AlphabetReduction
import OAI.Computability.VertexCover.PCP.UniformRestriction
namespace OAI
noncomputable section
namespace UniqueGames.Foundations.PCP.AlphabetGraph
open UniqueGames.Foundations.Hastad
open scoped BigOperators
abbrev Address (V E A : Type*) := (V × Cube A) ⊕ (E × Cube (A × A))
abbrev LocalEvent (A : Type*) := Fin 4 × AlphabetReduction.InputCoordinate A ×
Cube (A × A) × Cube (A × A) × Cube (A × A) × Cube (A × A) × Cube (A × A)
abbrev Event (E A : Type*) := E × LocalEvent A
variable {V E A : Type*} [Fintype A] [DecidableEq A] [Nonempty A]
abbrev legal (G : ConstraintGraph V E A) (e : E) (p : A × A) : Prop :=
G.accepts e p.1 p.2 = true
instance legalDecidable (G : ConstraintGraph V E A) (e : E) :
DecidablePred (legal G e) :=
fun p => inferInstanceAs (Decidable (G.accepts e p.1 p.2 = true))
def restrictTape (G : ConstraintGraph V E A) (e : E) (f : Cube (A × A)) :
Cube (AlphabetReduction.LegalPair (G.accepts e)) := UniformRestriction.restrict (legal G e) f
def extendTape (G : ConstraintGraph V E A) (e : E)
(f : Cube (AlphabetReduction.LegalPair (G.accepts e))) : Cube (A × A) :=
UniformRestriction.extend (legal G e) f
def globalizeQuery (G : ConstraintGraph V E A) (e : E) :
AlphabetReduction.InputCoordinate A ⊕ Cube (AlphabetReduction.LegalPair (G.accepts e)) → Address V E A
| .inl (side, tape) => .inl ((if side then G.head e else G.tail e), tape)
| .inr tape => .inr (e, extendTape G e tape)
def verifier (G : ConstraintGraph V E A) : QueryIncidence.Verifier (Event E A) (Address V E A) 6 where
query := fun (e, kind, k, f, g, r₀, r₁, r₂) slot =>
globalizeQuery G e (AssignmentTester.eventQueries (AlphabetReduction.pairEncoding (G.accepts e)) kind k
(restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
(restrictTape G e r₁) (restrictTape G e r₂) slot)
accepts := fun (_, kind, _, _, _, _, _, _) b => AssignmentTester.eventAccepts kind b
def vertexBlock (assignment : Address V E A → Bool) (v : V) : Cube A → Bool :=
fun tape => assignment (.inl (v, tape))
def edgeOracle (G : ConstraintGraph V E A) (assignment : Address V E A → Bool) (e : E) :
Cube (AlphabetReduction.LegalPair (G.accepts e)) → Bool :=
fun tape => assignment (.inr (e, extendTape G e tape))
omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem oracleAnswer_globalize (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) (e : E)
(x : AlphabetReduction.InputCoordinate A ⊕ Cube (AlphabetReduction.LegalPair (G.accepts e))) :
assignment (globalizeQuery G e x) =
AssignmentTester.oracleAnswer
(CodeComposition.pairWord (vertexBlock assignment (G.tail e))
(vertexBlock assignment (G.head e))) (edgeOracle G assignment e) x := by
rcases x with ⟨side, tape⟩ | tape
· cases side <;> rfl
· rfl
omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem response_eq_local (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) (e : E) (kind : Fin 4)
(k : AlphabetReduction.InputCoordinate A) (f g r₀ r₁ r₂ : Cube (A × A)) :
QueryIncidence.response (verifier G) assignment (e, kind, k, f, g, r₀, r₁, r₂) =
fun slot => AssignmentTester.oracleAnswer
(CodeComposition.pairWord (vertexBlock assignment (G.tail e))
(vertexBlock assignment (G.head e))) (edgeOracle G assignment e)
(AssignmentTester.eventQueries (AlphabetReduction.pairEncoding (G.accepts e)) kind k
(restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
(restrictTape G e r₁) (restrictTape G e r₂) slot) := by
funext slot
exact oracleAnswer_globalize G assignment e _
def eventReject (G : ConstraintGraph V E A) (assignment : Address V E A → Bool)
(event : Event E A) : ℝ :=
AssignmentTester.truth (!((verifier G).accepts event (QueryIncidence.response (verifier G) assignment event)))
omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem eventReject_eq_local (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) (e : E) (kind : Fin 4)
(k : AlphabetReduction.InputCoordinate A) (f g r₀ r₁ r₂ : Cube (A × A)) :
eventReject G assignment (e, kind, k, f, g, r₀, r₁, r₂) =
AssignmentTester.eventReject (AlphabetReduction.pairEncoding (G.accepts e))
(CodeComposition.pairWord (vertexBlock assignment (G.tail e))
(vertexBlock assignment (G.head e))) (edgeOracle G assignment e) kind k
(restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
(restrictTape G e r₁) (restrictTape G e r₂) := by
unfold eventReject
rw [response_eq_local]
rfl
theorem expect_prod {I J : Type*} [Fintype I] [Fintype J] (f : I × J → ℝ) :
(𝔼 z : I × J, f z) = 𝔼 i : I, 𝔼 j : J, f (i, j) := by
simpa only [Finset.univ_product_univ] using
Finset.expect_product (Finset.univ : Finset I) (Finset.univ : Finset J) f
def localAverage (G : ConstraintGraph V E A) (assignment : Address V E A → Bool)
(e : E) : ℝ := 𝔼 sample : LocalEvent A, eventReject G assignment (e, sample)
theorem expect_restrict_five {B : Type*} [Fintype B] [DecidableEq B]
(p : B → Prop) [DecidablePred p]
(h : ({b // p b} → Bool) → ({b // p b} → Bool) → ({b // p b} → Bool) →
({b // p b} → Bool) → ({b // p b} → Bool) → ℝ) :
(𝔼 f : B → Bool, 𝔼 g : B → Bool, 𝔼 r₀ : B → Bool,
𝔼 r₁ : B → Bool, 𝔼 r₂ : B → Bool,
h (UniformRestriction.restrict p f) (UniformRestriction.restrict p g)
(UniformRestriction.restrict p r₀) (UniformRestriction.restrict p r₁)
(UniformRestriction.restrict p r₂)) =
𝔼 f : {b // p b} → Bool, 𝔼 g : {b // p b} → Bool,
𝔼 r₀ : {b // p b} → Bool, 𝔼 r₁ : {b // p b} → Bool,
𝔼 r₂ : {b // p b} → Bool, h f g r₀ r₁ r₂ := by
rw [UniformRestriction.expect_restrict p
(fun f => 𝔼 g : B → Bool, 𝔼 r₀ : B → Bool, 𝔼 r₁ : B → Bool,
𝔼 r₂ : B → Bool, h f (UniformRestriction.restrict p g)
(UniformRestriction.restrict p r₀) (UniformRestriction.restrict p r₁)
(UniformRestriction.restrict p r₂))]
apply Finset.expect_congr rfl
intro f _
rw [UniformRestriction.expect_restrict p
(fun g => 𝔼 r₀ : B → Bool, 𝔼 r₁ : B → Bool, 𝔼 r₂ : B → Bool,
h f g (UniformRestriction.restrict p r₀) (UniformRestriction.restrict p r₁)
(UniformRestriction.restrict p r₂))]
apply Finset.expect_congr rfl
intro g _
rw [UniformRestriction.expect_restrict p
(fun r₀ => 𝔼 r₁ : B → Bool, 𝔼 r₂ : B → Bool,
h f g r₀ (UniformRestriction.restrict p r₁) (UniformRestriction.restrict p r₂))]
apply Finset.expect_congr rfl
intro r₀ _
rw [UniformRestriction.expect_restrict p
(fun r₁ => 𝔼 r₂ : B → Bool, h f g r₀ r₁ (UniformRestriction.restrict p r₂))]
apply Finset.expect_congr rfl
intro r₁ _
exact UniformRestriction.expect_restrict p (h f g r₀ r₁)
omit [Nonempty A] in
theorem localAverage_eq_tester (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) (e : E) :
localAverage G assignment e = AlphabetReduction.edgeTesterReject (G.accepts e)
(vertexBlock assignment (G.tail e)) (vertexBlock assignment (G.head e))
(edgeOracle G assignment e) := by
unfold localAverage AlphabetReduction.edgeTesterReject AssignmentTester.explicitTesterReject
simp only [LocalEvent, expect_prod, eventReject_eq_local, restrictTape]
apply Finset.expect_congr rfl
intro kind _
apply Finset.expect_congr rfl
intro side _
apply Finset.expect_congr rfl
intro tape _
exact expect_restrict_five (legal G e)
(AssignmentTester.eventReject (AlphabetReduction.pairEncoding (G.accepts e))
(CodeComposition.pairWord (vertexBlock assignment (G.tail e))
(vertexBlock assignment (G.head e))) (edgeOracle G assignment e) kind (side, tape))
def average [Fintype E] (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) : ℝ :=
𝔼 event : Event E A, eventReject G assignment event
omit [Nonempty A] in
theorem average_eq_testers [Fintype E] (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) :
average G assignment = 𝔼 e : E, AlphabetReduction.edgeTesterReject (G.accepts e)
(vertexBlock assignment (G.tail e)) (vertexBlock assignment (G.head e))
(edgeOracle G assignment e) := by
unfold average
rw [expect_prod]
exact Finset.expect_congr rfl (fun e _ => localAverage_eq_tester G assignment e)
def decodedLabeling (assignment : Address V E A → Bool) : V → A :=
fun v => CodeComposition.nearest (vertexBlock assignment v)
theorem local_gap (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) (e : E) :
AssignmentTester.truth (!(G.edgeSatisfied (decodedLabeling assignment) e)) / 2048 ≤
localAverage G assignment e := by
cases h : G.edgeSatisfied (decodedLabeling assignment) e with
| false =>
rw [localAverage_eq_tester]
simpa [AssignmentTester.truth] using
AlphabetReduction.rejected_edge_tester_bound (G.accepts e)
(vertexBlock assignment (G.tail e)) (vertexBlock assignment (G.head e))
(edgeOracle G assignment e) h
| true =>
simp only [Bool.not_true, AssignmentTester.truth, Bool.false_eq_true, ite_false, zero_div]
apply Finset.expect_nonneg
intro sample _
exact AssignmentTester.truth_nonneg _
theorem average_gap [Fintype E] (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) :
(𝔼 e : E, AssignmentTester.truth (!(G.edgeSatisfied (decodedLabeling assignment) e))) / 2048 ≤
average G assignment := by
unfold average
rw [expect_prod, Finset.expect_div]
exact Finset.expect_le_expect (fun e _ => local_gap G assignment e)
theorem expect_rejection_eq_count {I : Type*} [Fintype I] (b : I → Bool) :
(𝔼 i : I, AssignmentTester.truth (!(b i))) =
((Finset.univ.filter (fun i => b i = false)).card : ℝ) / Fintype.card I := by
rw [Fintype.expect_eq_sum_div_card]
congr 1
calc
(∑ i : I, AssignmentTester.truth (!(b i))) = ∑ i : I, if b i = false then (1 : ℝ) else 0 := by
apply Finset.sum_congr rfl
intro i _
cases b i <;> norm_num [AssignmentTester.truth]
_ = _ := Finset.sum_boole _ _
omit [Nonempty A] in
theorem average_eq_count [Fintype E] (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) :
average G assignment = (QueryIncidence.verifierRejectionCount (verifier G) assignment : ℝ) /
Fintype.card (Event E A) :=
expect_rejection_eq_count _
omit [Nonempty A] in
@[simp] theorem card_event [Fintype E] :
Fintype.card (Event E A) = Fintype.card E * Fintype.card (LocalEvent A) :=
Fintype.card_prod _ _
omit [Nonempty A] in
theorem card_localEvent :
Fintype.card (LocalEvent A) =
4 * (2 * 2 ^ Fintype.card A) * (2 ^ (Fintype.card A * Fintype.card A)) ^ 5 := by
simp [LocalEvent, AlphabetReduction.InputCoordinate, Cube, Fintype.card_prod,
pow_succ, Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]
omit [Nonempty A] in
theorem card_address [Fintype V] [Fintype E] :
Fintype.card (Address V E A) =
Fintype.card V * 2 ^ Fintype.card A +
Fintype.card E * 2 ^ (Fintype.card A * Fintype.card A) := by
simp [Address, Cube, Fintype.card_sum, Fintype.card_prod]
theorem rejection_count_bound [Fintype E] (G : ConstraintGraph V E A)
(assignment : Address V E A → Bool) :
Fintype.card (LocalEvent A) * G.rejectionCount (decodedLabeling assignment) ≤
2048 * QueryIncidence.verifierRejectionCount (verifier G) assignment := by
classical
cases isEmpty_or_nonempty E with
| inl empty =>
let := empty
simp [ConstraintGraph.rejectionCount, ConstraintGraph.rejectedDarts]
| inr inhabited =>
let := inhabited
have h := average_gap G assignment
rw [average_eq_count, expect_rejection_eq_count, card_event, Nat.cast_mul,
div_div] at h
have he : (0 : ℝ) < Fintype.card E := by exact_mod_cast Fintype.card_pos
have hl : (0 : ℝ) < Fintype.card (LocalEvent A) := by
exact_mod_cast Fintype.card_pos
have hc := (div_le_div_iff₀ (mul_pos he (by norm_num)) (mul_pos he hl)).mp h
have hc' :
((Fintype.card (LocalEvent A) : ℝ) *
(G.rejectionCount (decodedLabeling assignment) : ℝ)) * Fintype.card E ≤
(2048 * (QueryIncidence.verifierRejectionCount (verifier G) assignment : ℝ)) * Fintype.card E := by
simpa only [ConstraintGraph.rejectionCount, ConstraintGraph.rejectedDarts,
mul_assoc, mul_comm, mul_left_comm] using hc
have result := (mul_le_mul_iff_left₀ he).mp hc'
exact_mod_cast result
def honestAssignment (G : ConstraintGraph V E A) (labeling : V → A) : Address V E A → Bool
| .inl (v, tape) => tape (labeling v)
| .inr (e, tape) => tape (labeling (G.tail e), labeling (G.head e))
omit [Fintype A] [DecidableEq A] [Nonempty A] in
@[simp] theorem honest_vertexBlock (G : ConstraintGraph V E A) (labeling : V → A) (v : V) :
vertexBlock (honestAssignment G labeling) v = CodeComposition.codeword (labeling v) := rfl
omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem honest_edgeOracle (G : ConstraintGraph V E A) (labeling : V → A) (e : E)
(h : G.edgeSatisfied labeling e = true) :
edgeOracle G (honestAssignment G labeling) e =
fun tape => tape ⟨(labeling (G.tail e), labeling (G.head e)), h⟩ := by
funext tape
exact UniformRestriction.extend_apply_of_mem (legal G e) tape
(labeling (G.tail e), labeling (G.head e)) h
omit [Fintype A] [DecidableEq A] [Nonempty A] in
theorem honest_event (G : ConstraintGraph V E A) (labeling : V → A)
(satisfied : ∀ e, G.edgeSatisfied labeling e = true) (event : Event E A) :
(verifier G).accepts event (QueryIncidence.response (verifier G) (honestAssignment G labeling) event) =
true := by
rcases event with ⟨e, kind, k, f, g, r₀, r₁, r₂⟩
rw [response_eq_local, honest_edgeOracle G labeling e (satisfied e)]
simp only [honest_vertexBlock]
exact AssignmentTester.event_perfect_completeness (AlphabetReduction.pairEncoding (G.accepts e))
⟨(labeling (G.tail e), labeling (G.head e)), satisfied e⟩ kind k
(restrictTape G e f) (restrictTape G e g) (restrictTape G e r₀)
(restrictTape G e r₁) (restrictTape G e r₂)
def graph [DecidableEq V] [DecidableEq E] (G : ConstraintGraph V E A) :
ConstraintGraph (QueryIncidence.Vertex (Event E A) (Address V E A))
(QueryIncidence.Dart (Event E A) 6) (QueryIncidence.Label 6) := QueryIncidence.graph (verifier G) (by omega)
omit [DecidableEq A] [Nonempty A] in
theorem perfect_completeness [DecidableEq V] [DecidableEq E] (G : ConstraintGraph V E A)
(satisfied : G.Satisfiable) : (graph G).Satisfiable := by
obtain ⟨labeling, h⟩ := satisfied
exact QueryIncidence.satisfiable_of_verifier_satisfiable (verifier G) (by omega)
⟨honestAssignment G labeling, honest_event G labeling h⟩
theorem gap_transfer [Fintype E] [DecidableEq V] [DecidableEq E]
(G : ConstraintGraph V E A) (a b : Nat)
(source : ∀ labeling : V → A,
a * Fintype.card E ≤ b * G.rejectionCount labeling)
(labeling : QueryIncidence.Vertex (Event E A) (Address V E A) → QueryIncidence.Label 6) :
a * Fintype.card (QueryIncidence.Dart (Event E A) 6) ≤
(b * 12288) * (graph G).rejectionCount labeling := by
have verifierGap (assignment : Address V E A → Bool) :
a * Fintype.card (Event E A) ≤
(b * 2048) * QueryIncidence.verifierRejectionCount (verifier G) assignment := by
rw [card_event]
calc
a * (Fintype.card E * Fintype.card (LocalEvent A)) =
Fintype.card (LocalEvent A) * (a * Fintype.card E) := by ac_rfl
_ ≤ Fintype.card (LocalEvent A) * (b * G.rejectionCount (decodedLabeling assignment)) :=
Nat.mul_le_mul_left _ (source (decodedLabeling assignment))
_ = b * (Fintype.card (LocalEvent A) * G.rejectionCount (decodedLabeling assignment)) := by
ac_rfl
_ ≤ b * (2048 * QueryIncidence.verifierRejectionCount (verifier G) assignment) :=
Nat.mul_le_mul_left _ (rejection_count_bound G assignment)
_ = (b * 2048) * QueryIncidence.verifierRejectionCount (verifier G) assignment := by ac_rfl
have h := QueryIncidence.gap_transfer (verifier G) (by omega) a (b * 2048) verifierGap labeling
have hcoef : b * 2048 * 6 = b * 12288 := by omega
rw [hcoef] at h
exact h
theorem alphabet_card : Fintype.card (QueryIncidence.Label 6) = 64 := QueryIncidence.six_query_alphabet
end UniqueGames.Foundations.PCP.AlphabetGraph
end
end OAI
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