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ae00908 | 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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import OAI.Algebra.Universal.GraphCriterion
import Mathlib.Data.Fintype.Powerset
namespace OAI
/-!
# Decidability of fixed-size graph witnesses
Connectivity is characterized by containment in every Boolean equivalence relation containing
the marked edges. On a finite carrier this gives an executable decision procedure, and hence
a decision procedure for the existence of a graph witness at each fixed size. The size of a
witness in `HasGraphWitness` is not bounded by these results.
-/
namespace FiniteCongruence
/-- A Boolean relation is closed if it is an equivalence containing every marked edge. -/
def ClosedMarking {Lat Carrier : Type} [LE Lat] (color : Carrier → Carrier → Lat)
(seeds : Finset (Carrier × Carrier)) (rel : Carrier → Carrier → Bool) : Prop :=
(∀ point, rel point point = true) ∧
(∀ left right, rel left right = true → rel right left = true) ∧
(∀ left middle right, rel left middle = true → rel middle right = true → rel left right = true) ∧
(∀ left right, Marked color seeds left right → rel left right = true)
/-- Connectivity is the intersection of the Boolean equivalence relations containing all
marked edges. This characterization uses classical logic but allows finite enumeration. -/
theorem connected_iff_closedMarkings {Lat Carrier : Type} [LE Lat]
(color : Carrier → Carrier → Lat) (seeds : Finset (Carrier × Carrier)) (left right : Carrier) :
Connected color seeds left right ↔
∀ rel : Carrier → Carrier → Bool, ClosedMarking color seeds rel → rel left right = true := by
constructor
· intro connected rel closed
induction connected with
| rel first second marked => exact closed.2.2.2 first second marked
| refl point => exact closed.1 point
| symm first second _ inductionHyp => exact closed.2.1 first second inductionHyp
| trans first middle last _ _ firstHyp secondHyp =>
exact closed.2.2.1 first middle last firstHyp secondHyp
· classical
intro contained
let rel : Carrier → Carrier → Bool := fun first second =>
decide (Connected color seeds first second)
have closed : ClosedMarking color seeds rel := by
refine ⟨?_, ?_, ?_, ?_⟩
· intro point
exact decide_eq_true (Relation.EqvGen.refl point)
· intro first second related
exact decide_eq_true (Relation.EqvGen.symm first second (of_decide_eq_true related))
· intro first middle last firstRelated secondRelated
exact decide_eq_true (Relation.EqvGen.trans first middle last
(of_decide_eq_true firstRelated) (of_decide_eq_true secondRelated))
· intro first second marked
exact decide_eq_true (Relation.EqvGen.rel first second marked)
exact of_decide_eq_true (contained rel closed)
instance decidableAdmissible {Lat Carrier : Type} [LE Lat] [DecidableLE Lat] [Fintype Carrier]
(color : Carrier → Carrier → Lat) (placement : Carrier → Carrier) :
Decidable (Admissible color placement) :=
inferInstanceAs (Decidable
(∀ left right, color (placement left) (placement right) ≤ color left right))
instance decidableMarked {Lat Carrier : Type} [LE Lat] [DecidableLE Lat]
[Fintype Carrier] [DecidableEq Carrier] (color : Carrier → Carrier → Lat)
(seeds : Finset (Carrier × Carrier)) (left right : Carrier) :
Decidable (Marked color seeds left right) :=
inferInstanceAs (Decidable
(∃ placement : Carrier → Carrier, Admissible color placement ∧
∃ edge ∈ seeds, placement edge.1 = left ∧ placement edge.2 = right))
instance decidableClosedMarking {Lat Carrier : Type} [LE Lat] [DecidableLE Lat]
[Fintype Carrier] [DecidableEq Carrier] (color : Carrier → Carrier → Lat)
(seeds : Finset (Carrier × Carrier)) (rel : Carrier → Carrier → Bool) :
Decidable (ClosedMarking color seeds rel) :=
inferInstanceAs (Decidable
((∀ point, rel point point = true) ∧
(∀ left right, rel left right = true → rel right left = true) ∧
(∀ left middle right, rel left middle = true → rel middle right = true → rel left right = true) ∧
(∀ left right, Marked color seeds left right → rel left right = true)))
instance decidableConnected {Lat Carrier : Type} [LE Lat] [DecidableLE Lat]
[Fintype Carrier] [DecidableEq Carrier] (color : Carrier → Carrier → Lat)
(seeds : Finset (Carrier × Carrier)) (left right : Carrier) :
Decidable (Connected color seeds left right) :=
decidable_of_iff
(∀ rel : Carrier → Carrier → Bool, ClosedMarking color seeds rel → rel left right = true)
(connected_iff_closedMarkings color seeds left right).symm
/-- An executable fixed-size test, with no classical choice in its computational content. -/
def graphWitnessDecidable (Lat Carrier : Type) [Fintype Lat] [Lattice Lat] [OrderBot Lat]
[DecidableEq Lat] [DecidableLE Lat] [Fintype Carrier] [DecidableEq Carrier]
(color : Carrier → Carrier → Lat) : Decidable (IsGraphWitness color) :=
inferInstanceAs (Decidable
((∀ left right, color left right = color right left) ∧
(∀ left right, color left right = ⊥ ↔ left = right) ∧
(∀ left right middle, color left right ≤ color left middle ⊔ color middle right) ∧
(∀ lower upper : Lat, ¬ lower ≤ upper →
∃ left right, color left right ≤ lower ∧ ¬ color left right ≤ upper) ∧
(∀ seeds : Finset (Carrier × Carrier), seeds.Nonempty → ∀ left right,
color left right ≤ seeds.sup (fun edge => color edge.1 edge.2) →
Connected color seeds left right)))
/-- Existence of a colored-graph witness at a fixed candidate size is decidable. -/
def graphSizeDecidable (Lat : Type) [Fintype Lat] [Lattice Lat] [OrderBot Lat]
[DecidableEq Lat] [DecidableLE Lat] (size : ℕ) :
Decidable (0 < size ∧ ∃ color : Fin size → Fin size → Lat, IsGraphWitness color) := by
letI : DecidablePred (fun color : Fin size → Fin size → Lat => IsGraphWitness color) :=
graphWitnessDecidable Lat (Fin size)
infer_instance
end FiniteCongruence
end OAI
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