lean/ComparatorChallenges/UniqueGamesTheorem.lean

import Mathlib

namespace OAI

namespace UniqueGamesTheorem.Foundations.Target

/-- Finite forward and inverse tables make totality and invertibility explicit.
An eventual external encoding only needs to serialize the forward table. -/
structure PermutationTable (alphabet : Nat) where
  images : Vector (Fin alphabet) alphabet
  inverseImages : Vector (Fin alphabet) alphabet
  leftInverse : ∀ label : Fin alphabet, inverseImages[images[label]] = label
  rightInverse : ∀ label : Fin alphabet, images[inverseImages[label]] = label

structure Constraint (vertices alphabet : Nat) where
  source : Fin vertices
  target : Fin vertices
  permutation : PermutationTable alphabet

def Constraint.satisfied {n q : Nat} (constraint : Constraint n q)
    (labeling : Fin n → Fin q) : Bool :=
  decide (constraint.permutation.images[labeling constraint.source] = labeling constraint.target)

/-- Constraint occurrences are list entries, so repeated parallel edges retain
their multiplicity. No weights are hidden in this representation. -/
structure Instance (alphabet : Nat) where
  vertices : Nat
  constraints : List (Constraint vertices alphabet)
  nonempty : constraints ≠ []

def countSatisfied {n q : Nat} (labeling : Fin n → Fin q) :
    List (Constraint n q) → Nat
  | [] => 0
  | constraint :: rest =>
      (if constraint.satisfied labeling then 1 else 0) + countSatisfied labeling rest

end UniqueGamesTheorem.Foundations.Target

namespace UniqueGamesTheorem.BinaryFormula

open UniqueGamesTheorem.Foundations

structure Literal where
  name : Nat
  positive : Bool
  deriving DecidableEq

def Literal.eval (literal : Literal) (assignment : Nat → Bool) : Bool :=
  if literal.positive then assignment literal.name else !(assignment literal.name)

abbrev Clause := Vector Literal 3

def Clause.eval (clause : Clause) (assignment : Nat → Bool) : Bool :=
  (clause[0].eval assignment || clause[1].eval assignment) || clause[2].eval assignment

/-- An ordinary conjunction of three-slot clauses with sparse variable names.
Repeated names and repeated literals are permitted in every clause. -/
structure Formula where
  clauses : List Clause

def Formula.Satisfiable (F : Formula) : Prop :=
  ∃ assignment : Nat → Bool, ∀ clause ∈ F.clauses, clause.eval assignment = true

end UniqueGamesTheorem.BinaryFormula

namespace UniqueGamesTheorem.BinaryEncoding

open BinaryFormula

def bitsValue : List Bool → Nat
  | [] => 0
  | b :: bits => Nat.bit b (bitsValue bits)

def parseFrame : List Bool → Option (List Bool × List Bool)
  | false :: rest => some ([], rest)
  | true :: b :: rest => do
      let (bits, trailing) ← parseFrame rest
      return (b :: bits, trailing)
  | _ => none

/-- Equality with the canonical digits rejects alternate padded encodings. -/
def parseName (input : List Bool) : Option (Nat × List Bool) := do
  let (digits, rest) ← parseFrame input
  let name := bitsValue digits
  if digits = name.bits then some (name, rest) else none

def parseLiteral : List Bool → Option (Literal × List Bool)
  | sign :: input => do
      let (name, rest) ← parseName input
      return (⟨name, sign⟩, rest)
  | [] => none

def parseClause (input : List Bool) : Option (Clause × List Bool) := do
  let (a, input) ← parseLiteral input
  let (b, input) ← parseLiteral input
  let (c, rest) ← parseLiteral input
  return (#v[a, b, c], rest)

/-- Fuel bounds the number of clause iterations, not the size of any name. -/
def parseClauses : Nat → List Bool → Option (List Clause × List Bool)
  | 0, _ => none
  | _ + 1, false :: rest => some ([], rest)
  | fuel + 1, true :: input => do
      let (clause, input) ← parseClause input
      let (clauses, rest) ← parseClauses fuel input
      return (clause :: clauses, rest)
  | _ + 1, [] => none

def decodeFormula (input : List Bool) : Option Formula := do
  let (clauses, rest) ← parseClauses (input.length + 1) input
  if rest = [] then some ⟨clauses⟩ else none

end UniqueGamesTheorem.BinaryEncoding

namespace UniqueGamesTheorem.BinaryLanguage

open UniqueGamesTheorem.Foundations

def language (input : List Bool) : Prop :=
  ∃ formula, BinaryEncoding.decodeFormula input = some formula ∧ formula.Satisfiable

end UniqueGamesTheorem.BinaryLanguage

namespace UniqueGamesTheorem.Integration.BinaryLinear

abbrev F2 := ZMod 2

abbrev Vector (n : Nat) := Fin n → F2

end UniqueGamesTheorem.Integration.BinaryLinear

namespace UniqueGamesTheorem.Integration.TranslationTarget

open UniqueGamesTheorem.Foundations
open Target

def IsTranslationInstance {q s : Nat} (coordinates : Fin q ≃ BinaryLinear.Vector s)
    (g : Instance q) : Prop :=
  ∀ constraint ∈ g.constraints, ∃ shift : BinaryLinear.Vector s,
    ∀ label : Fin q,
      coordinates (constraint.permutation.images[label]) = coordinates label + shift

end UniqueGamesTheorem.Integration.TranslationTarget

namespace UniqueGamesTheorem.Foundations.Complexity

def encodeWord (n : Nat) : List Bool := List.replicate n true ++ [false]

def encodeWords : List Nat → List Bool
  | [] => []
  | n :: ns => encodeWord n ++ encodeWords ns

end UniqueGamesTheorem.Foundations.Complexity

namespace UniqueGamesTheorem.Foundations.Complexity

open Target

def tableWords {q : Nat} (table : PermutationTable q) : List Nat :=
  table.images.toList.map Fin.val

def constraintWords {n q : Nat} (constraint : Constraint n q) : List Nat :=
  [constraint.source.val, constraint.target.val] ++ tableWords constraint.permutation

def gameWords {q : Nat} (game : Instance q) : List Nat :=
  [game.vertices, q, game.constraints.length] ++ game.constraints.flatMap constraintWords

def gameBits {q : Nat} (game : Instance q) : List Bool := encodeWords (gameWords game)

end UniqueGamesTheorem.Foundations.Complexity

namespace UniqueGamesTheorem.Foundations.Complexity.MachineFiniteAlphabet

open Turing

/-- Every physical tape has a finite alphabet. Together with `M.kFin`, this
also makes the disjoint union of all physical tape alphabets finite. -/
def FiniteAlphabet (M : FinTM2) : Prop := ∀ k, Finite (M.Γ k)

end UniqueGamesTheorem.Foundations.Complexity.MachineFiniteAlphabet

namespace UniqueGamesTheorem.Explicit.MachineOutputContract

open UniqueGamesTheorem.Foundations
open Target
open scoped BigOperators

/-- A canonical orientation from the false side to the true side. Injectivity
is on occurrence indices, so two distinct list entries cannot be parallel even
when their constraint tables are different. -/
structure SimpleBipartite {q : Nat} (game : Instance q) where
  side : Fin game.vertices → Bool
  sourceSide : ∀ i : Fin game.constraints.length,
    side game.constraints[i].source = false
  targetSide : ∀ i : Fin game.constraints.length,
    side game.constraints[i].target = true
  endpoints_injective : Function.Injective (fun i : Fin game.constraints.length =>
    (game.constraints[i].source, game.constraints[i].target))

/-- Every parameter, coordinate identification, program, and polynomial is
fixed before the input quantifier. The identity input encoding measures runtime
in the original raw bit length, including sparse binary variable names. -/
structure BinaryGapReduction (ε δ : ℝ) where
  alphabet : Nat
  alphabetAtLeastTwo : 2 ≤ alphabet
  dimension : Nat
  dimensionPositive : 1 ≤ dimension
  coordinates : Fin alphabet ≃ Integration.BinaryLinear.Vector dimension
  construct : List Bool → Instance alphabet
  simpleBipartite : ∀ input, SimpleBipartite (construct input)
  translations : ∀ input,
    Integration.TranslationTarget.IsTranslationInstance coordinates (construct input)
  computation : Turing.TM2ComputableInPolyTime (id : List Bool → List Bool)
    Complexity.gameBits construct
  finiteAlphabet : Complexity.MachineFiniteAlphabet.FiniteAlphabet computation.tm
  completeness : ∀ input, BinaryLanguage.language input →
    ∃ labeling, 1 - ε ≤
      (countSatisfied labeling (construct input).constraints : ℝ) /
        (construct input).constraints.length
  soundness : ∀ input, ¬BinaryLanguage.language input →
    ∀ labeling, (countSatisfied labeling (construct input).constraints : ℝ) /
      (construct input).constraints.length ≤ δ

end UniqueGamesTheorem.Explicit.MachineOutputContract

namespace UniqueGamesTheorem

/-- For every fixed pair of errors in `(0, 1/2)`, binary 3SAT reduces in
polynomial time to translation Unique Games with completeness at least `1 - ε`
and soundness at most `δ`. The alphabet and machine depend only on the errors. -/
theorem theorem11 (ε δ : ℝ)
    (hε : 0 < ε) (hεhalf : ε < 1 / 2)
    (hδ : 0 < δ) (hδhalf : δ < 1 / 2) :
    Nonempty (Explicit.MachineOutputContract.BinaryGapReduction ε δ) := by
  sorry

end UniqueGamesTheorem

end OAI