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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
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