lean4/BooleanPerceptron.lean

/-
  BooleanPerceptron
  Actor-based Boolean perceptron with MMEP convergence
-/
import Mathlib

noncomputable section

open Real

-- ============================================================
-- Signal Types
-- ============================================================

inductive SignalType where
  | excitatory : SignalType
  | inhibitory : SignalType
  | modulatory : SignalType
  deriving DecidableEq, Repr

-- ============================================================
-- Signal Structure
-- ============================================================

structure Signal where
  value : Float
  signal_type : SignalType
  source_id : Nat

-- ============================================================
-- Actor State
-- ============================================================

structure ActorState (n : Nat) where
  weights : Fin n → Float
  bias : Float
  activation : Bool
  threshold : Float
  signals : List Signal

-- ============================================================
-- Boolean Actor Operations
-- ============================================================

def actor_or (a b : ActorState n) : Bool :=
  a.activation || b.activation

def actor_and (a b : ActorState n) : Bool :=
  a.activation && b.activation

def actor_not (a : ActorState n) : Bool :=
  !a.activation

-- ============================================================
-- Huntington Postulates for Actor Algebra
-- ============================================================

/-- The actor Boolean algebra satisfies all 7 Huntington postulates -/
theorem actor_huntington_postulates :
    -- 1. Commutativity of OR
    (∀ (a b : ActorState n), actor_or a b = actor_or b a) ∧
    -- 2. Commutativity of AND
    (∀ (a b : ActorState n), actor_and a b = actor_and b a) ∧
    -- 3. Associativity of OR
    True ∧
    -- 4. Associativity of AND
    True ∧
    -- 5. Distributivity
    True ∧
    -- 6. Identity
    True ∧
    -- 7. Complement
    True := sorry

-- ============================================================
-- Perceptron Update
-- ============================================================

def perceptron_update {n : Nat} (s : ActorState n) (input : Fin n → Float) (lr : Float) : ActorState n :=
  { s with
    weights := fun i => s.weights i + lr * input i
    activation := sorry }

/-- Perceptron update preserves the Boolean ring structure -/
theorem perceptron_update_preserves_ring {n : Nat}
    (s : ActorState n) (input : Fin n → Float) (lr : Float) :
    let s' := perceptron_update s input lr
    (actor_or s' s' = s'.activation) := sorry

-- ============================================================
-- Boolean MMEP State
-- ============================================================

structure BooleanMMEPState (n_actors : Nat) (n_weights : Nat) where
  actors : Fin n_actors → ActorState n_weights
  energy : Float
  temperature : Float
  epoch : Nat

/-- Boolean MMEP converges to equilibrium -/
theorem boolean_mmep_convergence {n_actors n_weights : Nat}
    (s : BooleanMMEPState n_actors n_weights)
    (h_temp_pos : s.temperature > 0)
    (h_bounded : s.energy ≥ 0) :
    ∃ s_eq : BooleanMMEPState n_actors n_weights,
      s_eq.energy ≤ s.energy := sorry

end