derivational-coalgebra

# derivational-coalgebra Skill > *Bridge between atemporal derivations (unworld) and temporal observations (coalgebra)* ## The Gap The highest tension in skill space (d=2.131): ``` ATEMPORAL TEMPORAL │ │ unworld temporal-coalgebra dialectica duckdb-temporal-versioning unworlding-involution changelog-generator │ │ └──────────── d = 2.131 ───────────────┘ ``` ## Resolution: Derivational Coalgebra Coalgebraic observation of **derivation streams** rather than time-indexed states: ``` Traditional coalgebra: S → F(S) where F is a functor on temporal states Derivational coalgebra: D → F(D) where D is the derivation poset ``` ### Key Insight Replace `t ∈ Time` with `δ ∈ Derivation`: | Temporal | Derivational | |----------|--------------| | `state(t)` | `state(δ)` | | `t₁ < t₂` | `δ₁ ⊢ δ₂` | | `lim_{t→∞}` | `fixpoint(δ)` | | `∂/∂t` | `∂/∂δ` (sensitivity to derivation step) | ## Core Types ```haskell -- Derivation as a partial order (not linear time) data Derivation a where Base :: a -> Derivation a Step :: Derivation a -> (a -> a) -> Derivation a Join :: Derivation a -> Derivation a -> Derivation a -- parallel -- Coalgebraic observation class DerivationalCoalgebra s where observe :: s -> ObservationF s unfold :: s -> Stream (Derivation s) -- The functor bridges both paradigms data ObservationF s = ObsF { current :: s -- atemporal snapshot , successor :: Derivation s -- derivational continuation , temporal :: Maybe (Time, s) -- optional temporal embedding } ``` ## Implementation ### Python Bridge ```python from dataclasses import dataclass from typing import Generic, TypeVar, Callable, Optional from abc import ABC, abstractmethod S = TypeVar('S') @dataclass class Derivation(Generic[S]): """Derivation step with optional temporal embedding.""" state: S parent: Optional['Derivation[S]'] = None rule: Optional[str] = None # derivation rule name timestamp: Optional[float] = None # temporal embedding (optional) @property def depth(self) -> int: """Derivation depth (atemporal measure).""" if self.parent is None: return 0 return 1 + self.parent.depth def derive(self, f: Callable[[S], S], rule: str = "step") -> 'Derivation[S]': """Create child derivation (atemporal).""" return Derivation( state=f(self.state), parent=self, rule=rule, timestamp=None # no time needed ) def embed_temporal(self, t: float) -> 'Derivation[S]': """Optionally embed into temporal axis.""" return Derivation( state=self.state, parent=self.parent, rule=self.rule, timestamp=t ) class DerivationalCoalgebra(ABC, Generic[S]): """ Bridge between atemporal derivations and temporal observations. Resolves the unworld ↔ temporal-coalgebra tension by: 1. Representing computation as derivation chains (atemporal) 2. Providing coalgebraic observation interface 3. Optionally embedding into temporal axis when needed """ @abstractmethod def observe(self, d: Derivation[S]) -> S: """Observe current state (works both temporally and atemporally).""" pass @abstractmethod def unfold(self, d: Derivation[S]) -> 'Stream[Derivation[S]]': """Coalgebraic unfold: generate derivation stream.""" pass def to_temporal(self, stream: 'Stream[Derivation[S]]', dt: float = 1.0) -> 'Stream[tuple[float, S]]': """Embed derivation stream into temporal axis.""" t = 0.0 for d in stream: yield (t, d.state) t += dt def from_temporal(self, t_stream: 'Stream[tuple[float, S]]') -> 'Stream[Derivation[S]]': """Extract derivational structure from temporal stream.""" prev = None for t, s in t_stream: d = Derivation(state=s, parent=prev, timestamp=t) yield d prev = d ``` ### Gay.jl Color Integration ```python DERIVATIONAL_COLORS = { 'atemporal': '#5713C0', # Stream 4 (pure derivation) 'temporal': '#E6F463', # Stream 2 (time-embedded) 'bridge': '#89DF91', # Stream 3 (translation layer) } def color_derivation(d: Derivation) -> str: """Color based on temporal embedding status.""" if d.timestamp is None: return DERIVATIONAL_COLORS['atemporal'] else: return DERIVATIONAL_COLORS['temporal'] ``` ## Bisimulation for Equivalence Two derivation streams are bisimilar if their observations match: ```python def bisimilar(d1: Derivation[S], d2: Derivation[S], coalg: DerivationalCoalgebra[S]) -> bool: """ Check bisimulation: same observations regardless of whether temporal or atemporal representation. """ obs1 = coalg.observe(d1) obs2 = coalg.observe(d2) if obs1 != obs2: return False # Check continuations (up to finite depth) for next1, next2 in zip(coalg.unfold(d1), coalg.unfold(d2)): if coalg.observe(next1) != coalg.observe(next2): return False return True ``` ## Triangle Inequality Restoration With this bridge skill: ``` d(unworld, derivational-coalgebra) ≈ 1.0 d(derivational-coalgebra, temporal-coalgebra) ≈ 1.0 Therefore: d(unworld, temporal-coalgebra) ≤ 1.0 + 1.0 = 2.0 ✓ (Original: 2.131, now satisfies triangle inequality) ``` ## Use Cases ### 1. Version Control Without Time ```python # Git commits as derivations, not timestamps class GitDerivational(DerivationalCoalgebra[Tree]): def observe(self, d: Derivation[Tree]) -> Tree: return d.state def unfold(self, d: Derivation[Tree]): # Derivation graph, not timeline for parent in d.parents: yield parent ``` ### 2. Proof Assistant States ```python # Narya/Agda proof states as derivations class ProofDerivational(DerivationalCoalgebra[ProofState]): def derive_tactic(self, state: ProofState, tactic: Tactic): return Derivation( state=apply_tactic(state, tactic), parent=state, rule=tactic.name ) ``` ### 3. DuckDB Temporal Queries via Derivation ```python # Bridge DuckDB temporal to derivational def temporal_to_derivational(db: duckdb.Connection): """Convert temporal version table to derivation chain.""" rows = db.execute(""" SELECT * FROM version_history ORDER BY version_id """).fetchall() prev = None for row in rows: d = Derivation( state=row['data'], parent=prev, rule=row['change_type'], timestamp=row['timestamp'] # preserve but not rely on ) yield d prev = d ``` ## Neighbor Skills - **unworld**: Pure atemporal derivations - **temporal-coalgebra**: Pure temporal observations - **duckdb-temporal-versioning**: Practical temporal queries - **proofgeneral-narya**: Proof derivations - **dialectica**: Constructive derivations ## Resources - [Jacobs - Introduction to Coalgebra](https://www.cs.ru.nl/B.Jacobs/CLG/JacsCoalgBook.pdf) - [Rutten - Universal Coalgebra](https://homepages.cwi.nl/~janr/papers/files-of-papers/universal_coalgebra.pdf) - [unworld skill](file:///Users/bob/.claude/skills/unworld/SKILL.md) - [temporal-coalgebra skill](file:///Users/bob/.claude/skills/temporal-coalgebra/SKILL.md) --- ## End-of-Skill Interface ## Commands ```bash # Convert temporal log to derivational python derivational_coalgebra.py --input temporal.log --output derivations.json # Check bisimulation python derivational_coalgebra.py --bisim d1.json d2.json # Embed derivations into time python derivational_coalgebra.py --embed derivations.json --dt 1.0 ``` --- ## Autopoietic Marginalia > **The interaction IS the skill improving itself.** Every use of this skill is an opportunity for worlding: - **MEMORY** (-1): Record what was learned - **REMEMBERING** (0): Connect patterns to other skills - **WORLDING** (+1): Evolve the skill based on use *Add Interaction Exemplars here as the skill is used.*