bisimulation-game

Bisimulation Game Skill

"Two systems are bisimilar if they cannot be distinguished by any observation."

Overview

The bisimulation game provides a framework for:

1. Resilient skill dispersal across multiple AI agents
2. GF(3) conservation during state transitions
3. Observational bridge types for version-aware synchronization
4. Self-rewriting capabilities via MCP Tasks protocol

Game Rules

Players

Player	Role	Trit	Color
Attacker	Tries to distinguish systems	-1	Blue
Defender	Maintains equivalence	+1	Red
Arbiter	Verifies conservation	0	Green

Moves

┌─────────────────────────────────────────────────────────────┐
│  Round n:                                                   │
│                                                             │
│  1. Attacker chooses: system S₁ or S₂                       │
│  2. Attacker makes: transition s₁ →ᵃ s₁'                    │
│  3. Defender responds: matching transition s₂ →ᵃ s₂'        │
│  4. Arbiter verifies: GF(3) conservation                    │
│                                                             │
│  If Defender cannot respond → Attacker wins (distinguishable)│
│  If game continues forever → Defender wins (bisimilar)      │
└─────────────────────────────────────────────────────────────┘

Implementation

Hy (DiscoHy) Implementation

;;; bisimulation_game.hy

(import [splitmix_ternary [SplitMixTernary]])

(defclass BisimulationGame []
  (defn __init__ [self system1 system2 seed]
    (setv self.s1 system1
          self.s2 system2
          self.rng (SplitMixTernary seed)
          self.history []))
  
  (defn attacker-move [self choice transition]
    "Attacker chooses system and transition."
    (setv trit (self.rng.next-ternary))
    (.append self.history {:role "attacker" 
                           :choice choice 
                           :transition transition
                           :trit trit})
    trit)
  
  (defn defender-respond [self matching-transition]
    "Defender provides matching transition."
    (setv trit (self.rng.next-ternary))
    (.append self.history {:role "defender"
                           :response matching-transition
                           :trit trit})
    trit)
  
  (defn arbiter-verify [self]
    "Arbiter checks GF(3) conservation."
    (setv recent-trits (lfor m (cut self.history -3 None) (get m "trit")))
    (setv conserved (= (% (sum recent-trits) 3) 0))
    (.append self.history {:role "arbiter" :conserved conserved :trit 0})
    conserved))