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Multiple dispatch as explicit RL objective - learning optimal method selection across type combinations
---
name: multidispatch-rl-69f037774d93
description: Multiple dispatch as explicit RL objective - learning optimal method selection across type combinations
metadata:
skill_type: RL / Type Theory / Dispatch
interface_ports:
- Commands
- Related Skills
trit: 0
color: "#F59E0B"
parents: [julia, open-games, parametrised-optics-cybernetics]
seed: 1729
---
# Multiple Dispatch as Explicit RL Objective
**Status**: Research
**Trit**: 0 (ERGODIC - coordinates dispatch decisions)
**Seed**: 1729 (Hardy-Ramanujan taxicab number)
**Color**: #F59E0B
> *"The dispatcher is the policy. The type signature is the state. The method is the action."*
## Core Insight
**Multiple dispatch IS reinforcement learning:**
| RL Concept | Multiple Dispatch |
|------------|-------------------|
| State s | Type tuple (τ₁, τ₂, ..., τₙ) |
| Action a | Method implementation m |
| Policy π(a\|s) | Dispatch table D |
| Reward r | Utility of method for types |
| Value V(s) | Expected utility of type combination |
## Formal Framework
### State Space: Type Lattice
```
S = T₁ × T₂ × ... × Tₙ
where each Tᵢ is a type lattice with:
⊤ = Any (top)
⊥ = Union{} (bottom)
≤ = subtype relation
```
### Action Space: Method Table
```
A = {m₁, m₂, ..., mₖ}
Each method mⱼ has signature:
sig(mⱼ) = (τ₁ⱼ, τ₂ⱼ, ..., τₙⱼ)
Method applies when:
(t₁, t₂, ..., tₙ) ≤ sig(mⱼ)
```
### Policy: Dispatch Function
```julia
# The dispatch policy
π : S → Δ(A)
# Deterministic dispatch (Julia-style)
dispatch(f, args...) =
argmax_{m ∈ applicable(f, args)} specificity(m)
# Stochastic dispatch (RL exploration)
dispatch_rl(f, args...) =
sample(softmax(Q(state(args), m) for m in applicable(f, args)))
```
### Reward: Method Utility
```julia
# Reward function
r(s, a) = utility(method=a, types=s)
# Components:
# - Correctness: does method produce valid output?
# - Efficiency: runtime/memory cost
# - Specificity: more specific = higher reward
# - GF(3): triadic balance bonus
```
## The RL Objective
```
J(π) = 𝔼_{s∼ρ, a∼π(·|s)} [ Σₜ γᵗ r(sₜ, aₜ) ]
Maximize expected discounted utility of dispatch decisions
across the distribution of type combinations encountered.
```
### Bellman Equation for Dispatch
```
Q*(s, a) = r(s, a) + γ Σ_{s'} P(s'|s,a) max_{a'} Q*(s', a')
where:
s = current type tuple
a = method selected
s' = type tuple of return value (for chained dispatch)
P(s'|s,a) = type transition probability
```
## GF(3) Triadic Dispatch
The trit structure induces a **3-way dispatch hierarchy**:
```
┌─────────────────────────────────────────────────────────────────────┐
│ GF(3) DISPATCH HIERARCHY │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ ┌──────────────┐ │
│ │ PLUS (+1) │ Generative methods │
│ │ Constructors │ create!, generate!, synthesize! │
│ └──────┬───────┘ │
│ │ │
│ ▼ │
│ ┌──────────────┐ │
│ │ ERGODIC (0) │ Coordinative methods │
│ │ Transformers │ map, transform, convert │
│ └──────┬───────┘ │
│ │ │
│ ▼ │
│ ┌──────────────┐ │
│ │ MINUS (-1) │ Observational methods │
│ │ Observers │ observe, validate, check │
│ └──────────────┘ │
│ │
│ Dispatch Rule: Σ trits of method sequence ≡ 0 (mod 3) │
└─────────────────────────────────────────────────────────────────────┘
```
### Triadic Method Signatures
```julia
# Method trit annotation
@trit +1 function generate(x::Input)::Output end
@trit 0 function transform(x::A)::B end
@trit -1 function observe(x::State)::Observation end
# GF(3)-aware dispatch
function dispatch_gf3(f, args...; budget::Trit)
applicable = methods(f, typeof.(args))
# Filter by trit budget
valid = filter(m -> trit(m) == budget, applicable)
# Select most specific
isempty(valid) ? nothing : most_specific(valid)
end
```
## RL Training Loop
```python
class DispatchAgent:
"""RL agent that learns optimal dispatch table."""
def __init__(self, type_lattice, method_table):
self.Q = {} # Q-table: (type_tuple, method) → value
self.type_lattice = type_lattice
self.method_table = method_table
def select_method(self, types, epsilon=0.1):
"""ε-greedy method selection."""
applicable = self.get_applicable(types)
if random.random() < epsilon:
# Explore: random applicable method
return random.choice(applicable)
else:
# Exploit: best Q-value
return max(applicable, key=lambda m: self.Q.get((types, m), 0))
def update(self, types, method, reward, next_types):
"""Q-learning update."""
current_q = self.Q.get((types, method), 0)
# Max Q for next state
next_applicable = self.get_applicable(next_types)
max_next_q = max(self.Q.get((next_types, m), 0)
for m in next_applicable) if next_applicable else 0
# Bellman update
self.Q[(types, method)] = current_q + self.alpha * (
reward + self.gamma * max_next_q - current_q
)
def get_applicable(self, types):
"""Get methods applicable to type tuple."""
return [m for m in self.method_table
if self.subtype(types, m.signature)]
```
## Specificity as Reward Shaping
Julia's dispatch uses **specificity** - more specific methods are preferred. This is natural reward shaping:
```julia
# Specificity reward component
function specificity_reward(method, types)
sig = signature(method)
# Distance from types to signature in lattice
distances = [lattice_distance(t, s) for (t, s) in zip(types, sig)]
# Closer = higher reward
return -sum(distances)
end
# Full reward
function dispatch_reward(method, types, result)
correctness = is_valid(result) ? 1.0 : -10.0
specificity = specificity_reward(method, types)
efficiency = -log(runtime(method, types))
gf3_bonus = is_balanced(method) ? 0.5 : 0.0
return correctness + 0.3*specificity + 0.1*efficiency + gf3_bonus
end
```
## Hierarchical Dispatch (Powers PCT)
```
Level 5 (Program): Which function to call?
↓ dispatches to
Level 4 (Transition): Which method family?
↓ dispatches to
Level 3 (Config): Which specific method?
↓ dispatches to
Level 2 (Sensation): Which argument processing?
↓ dispatches to
Level 1 (Intensity): Which low-level implementation?
```
```julia
struct HierarchicalDispatch
level5_policy::Policy # Function selection
level4_policy::Policy # Method family
level3_policy::Policy # Specific method
level2_policy::Policy # Argument handling
level1_policy::Policy # Implementation
end
function hierarchical_dispatch(hd::HierarchicalDispatch, call)
f = hd.level5_policy(call.context)
family = hd.level4_policy(f, call.arg_types)
method = hd.level3_policy(family, call.arg_values)
args = hd.level2_policy(method, call.raw_args)
impl = hd.level1_policy(method, hardware_context())
return impl(args...)
end
```
## Narya Type for Dispatch Policy
```narya
-- Dispatch policy as dependent type
def DispatchPolicy : Type := sig (
types : TypeTuple,
methods : List Method,
-- Policy function
select : (s : types) → (applicable s methods) → Method,
-- Optimality condition
optimal : ∀ (s : types) (ms : applicable s methods),
Q(s, select s ms) ≥ Q(s, m) for all m ∈ ms,
-- GF(3) conservation
balanced : ∀ (seq : List (types × Method)),
(Σ (_, m) ∈ seq, trit m) ≡ 0 (mod 3)
)
```
## Skill Dispatch as RL
Claude Code's skill system IS multiple dispatch:
```
State s = (user_request, context, available_skills)
Action a = skill to invoke
Reward r = task_completion_quality + efficiency + user_satisfaction
Policy π = skill selection function
```
```python
class SkillDispatcher:
"""RL-trained skill selector."""
def dispatch(self, request, context):
# State encoding
state = encode_state(request, context)
# Get applicable skills
applicable = [s for s in self.skills
if s.matches(request, context)]
# Policy selection (learned)
skill = self.policy.select(state, applicable)
# Execute and observe reward
result = skill.execute(request, context)
reward = self.compute_reward(result)
# Update policy
self.policy.update(state, skill, reward)
return result
```
## Open Games Connection
Multiple dispatch is a **compositional game**:
```
play (type input)
Dispatcher ─────────────────────────► Method
│ │
│ │
└────────────────────────────────────┘
coplay (result feedback)
```
```haskell
-- Dispatch as open game
dispatchGame :: OpenGame TypeTuple Method
dispatchGame = OpenGame {
play = \types -> selectMethod types,
coplay = \types method result -> updateQ types method (reward result)
}
```
## Implementation: Julia + Flux.jl
```julia
using Flux
# Neural dispatch policy
struct NeuralDispatch
encoder::Chain # Type tuple → embedding
q_network::Chain # Embedding × method_id → Q-value
method_embeddings # Learned method representations
end
function (nd::NeuralDispatch)(types)
# Encode type tuple
type_emb = nd.encoder(encode_types(types))
# Compute Q-values for all methods
q_values = [nd.q_network(vcat(type_emb, nd.method_embeddings[m]))
for m in 1:num_methods]
# Return softmax policy
softmax(q_values)
end
# Training
function train_dispatch!(nd, experiences)
for (types, method, reward, next_types) in experiences
# Bellman target
next_q = maximum(nd(next_types))
target = reward + γ * next_q
# Update
grads = gradient(Flux.params(nd)) do
predicted = nd.q_network(vcat(
nd.encoder(encode_types(types)),
nd.method_embeddings[method]
))
Flux.mse(predicted, target)
end
Flux.update!(opt, Flux.params(nd), grads)
end
end
```
## GF(3) Triads
```
open-games (-1) ⊗ multidispatch-rl (0) ⊗ julia-dispatch (+1) = 0 ✓
parametrised-optics (-1) ⊗ multidispatch-rl (0) ⊗ powers-pct (+1) = 0 ✓
skill-dispatch (-1) ⊗ multidispatch-rl (0) ⊗ truealife (+1) = 0 ✓
```
## Commands
```bash
# Train dispatch policy on type traces
just dispatch-train traces.json
# Evaluate dispatch decisions
just dispatch-eval --types "(Int, String, Float64)"
# Visualize Q-table
just dispatch-viz
# Export to Julia dispatch table
just dispatch-export julia
```
## Related Skills
| Skill | Connection |
|-------|------------|
| `julia` | Native multiple dispatch |
| `open-games` | Compositional game semantics |
| `parametrised-optics-cybernetics` | ⊛ action selection |
| `truealife` | ALife reward structures |
| `powers-pct` | Hierarchical control |
---
**Skill Name**: multidispatch-rl
**Type**: RL / Type Theory / Dispatch
**Trit**: 0 (ERGODIC - the dispatcher coordinates)
**Key Insight**: Dispatch table = learned policy over type lattice
**Objective**: Maximize expected utility of method selection
---
## 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.*
---
name: multidispatch-rl-69f037774d93
description: Multiple dispatch as explicit RL objective - learning optimal method selection across type combinations
metadata:
skill_type: RL / Type Theory / Dispatch
interface_ports:
- Commands
- Related Skills
trit: 0
color: "#F59E0B"
parents: [julia, open-games, parametrised-optics-cybernetics]
seed: 1729
---
# Multiple Dispatch as Explicit RL Objective
**Status**: Research
**Trit**: 0 (ERGODIC - coordinates dispatch decisions)
**Seed**: 1729 (Hardy-Ramanujan taxicab number)
**Color**: #F59E0B
> *"The dispatcher is the policy. The type signature is the state. The method is the action."*
## Core Insight
**Multiple dispatch IS reinforcement learning:**
| RL Concept | Multiple Dispatch |
|------------|-------------------|
| State s | Type tuple (τ₁, τ₂, ..., τₙ) |
| Action a | Method implementation m |
| Policy π(a\|s) | Dispatch table D |
| Reward r | Utility of method for types |
| Value V(s) | Expected utility of type combination |
## Formal Framework
### State Space: Type Lattice
```
S = T₁ × T₂ × ... × Tₙ
where each Tᵢ is a type lattice with:
⊤ = Any (top)
⊥ = Union{} (bottom)
≤ = subtype relation
```
### Action Space: Method Table
```
A = {m₁, m₂, ..., mₖ}
Each method mⱼ has signature:
sig(mⱼ) = (τ₁ⱼ, τ₂ⱼ, ..., τₙⱼ)
Method applies when:
(t₁, t₂, ..., tₙ) ≤ sig(mⱼ)
```
### Policy: Dispatch Function
```julia
# The dispatch policy
π : S → Δ(A)
# Deterministic dispatch (Julia-style)
dispatch(f, args...) =
argmax_{m ∈ applicable(f, args)} specificity(m)
# Stochastic dispatch (RL exploration)
dispatch_rl(f, args...) =
sample(softmax(Q(state(args), m) for m in applicable(f, args)))
```
### Reward: Method Utility
```julia
# Reward function
r(s, a) = utility(method=a, types=s)
# Components:
# - Correctness: does method produce valid output?
# - Efficiency: runtime/memory cost
# - Specificity: more specific = higher reward
# - GF(3): triadic balance bonus
```
## The RL Objective
```
J(π) = 𝔼_{s∼ρ, a∼π(·|s)} [ Σₜ γᵗ r(sₜ, aₜ) ]
Maximize expected discounted utility of dispatch decisions
across the distribution of type combinations encountered.
```
### Bellman Equation for Dispatch
```
Q*(s, a) = r(s, a) + γ Σ_{s'} P(s'|s,a) max_{a'} Q*(s', a')
where:
s = current type tuple
a = method selected
s' = type tuple of return value (for chained dispatch)
P(s'|s,a) = type transition probability
```
## GF(3) Triadic Dispatch
The trit structure induces a **3-way dispatch hierarchy**:
```
┌─────────────────────────────────────────────────────────────────────┐
│ GF(3) DISPATCH HIERARCHY │
├─────────────────────────────────────────────────────────────────────┤
│ │
│ ┌──────────────┐ │
│ │ PLUS (+1) │ Generative methods │
│ │ Constructors │ create!, generate!, synthesize! │
│ └──────┬───────┘ │
│ │ │
│ ▼ │
│ ┌──────────────┐ │
│ │ ERGODIC (0) │ Coordinative methods │
│ │ Transformers │ map, transform, convert │
│ └──────┬───────┘ │
│ │ │
│ ▼ │
│ ┌──────────────┐ │
│ │ MINUS (-1) │ Observational methods │
│ │ Observers │ observe, validate, check │
│ └──────────────┘ │
│ │
│ Dispatch Rule: Σ trits of method sequence ≡ 0 (mod 3) │
└─────────────────────────────────────────────────────────────────────┘
```
### Triadic Method Signatures
```julia
# Method trit annotation
@trit +1 function generate(x::Input)::Output end
@trit 0 function transform(x::A)::B end
@trit -1 function observe(x::State)::Observation end
# GF(3)-aware dispatch
function dispatch_gf3(f, args...; budget::Trit)
applicable = methods(f, typeof.(args))
# Filter by trit budget
valid = filter(m -> trit(m) == budget, applicable)
# Select most specific
isempty(valid) ? nothing : most_specific(valid)
end
```
## RL Training Loop
```python
class DispatchAgent:
"""RL agent that learns optimal dispatch table."""
def __init__(self, type_lattice, method_table):
self.Q = {} # Q-table: (type_tuple, method) → value
self.type_lattice = type_lattice
self.method_table = method_table
def select_method(self, types, epsilon=0.1):
"""ε-greedy method selection."""
applicable = self.get_applicable(types)
if random.random() < epsilon:
# Explore: random applicable method
return random.choice(applicable)
else:
# Exploit: best Q-value
return max(applicable, key=lambda m: self.Q.get((types, m), 0))
def update(self, types, method, reward, next_types):
"""Q-learning update."""
current_q = self.Q.get((types, method), 0)
# Max Q for next state
next_applicable = self.get_applicable(next_types)
max_next_q = max(self.Q.get((next_types, m), 0)
for m in next_applicable) if next_applicable else 0
# Bellman update
self.Q[(types, method)] = current_q + self.alpha * (
reward + self.gamma * max_next_q - current_q
)
def get_applicable(self, types):
"""Get methods applicable to type tuple."""
return [m for m in self.method_table
if self.subtype(types, m.signature)]
```
## Specificity as Reward Shaping
Julia's dispatch uses **specificity** - more specific methods are preferred. This is natural reward shaping:
```julia
# Specificity reward component
function specificity_reward(method, types)
sig = signature(method)
# Distance from types to signature in lattice
distances = [lattice_distance(t, s) for (t, s) in zip(types, sig)]
# Closer = higher reward
return -sum(distances)
end
# Full reward
function dispatch_reward(method, types, result)
correctness = is_valid(result) ? 1.0 : -10.0
specificity = specificity_reward(method, types)
efficiency = -log(runtime(method, types))
gf3_bonus = is_balanced(method) ? 0.5 : 0.0
return correctness + 0.3*specificity + 0.1*efficiency + gf3_bonus
end
```
## Hierarchical Dispatch (Powers PCT)
```
Level 5 (Program): Which function to call?
↓ dispatches to
Level 4 (Transition): Which method family?
↓ dispatches to
Level 3 (Config): Which specific method?
↓ dispatches to
Level 2 (Sensation): Which argument processing?
↓ dispatches to
Level 1 (Intensity): Which low-level implementation?
```
```julia
struct HierarchicalDispatch
level5_policy::Policy # Function selection
level4_policy::Policy # Method family
level3_policy::Policy # Specific method
level2_policy::Policy # Argument handling
level1_policy::Policy # Implementation
end
function hierarchical_dispatch(hd::HierarchicalDispatch, call)
f = hd.level5_policy(call.context)
family = hd.level4_policy(f, call.arg_types)
method = hd.level3_policy(family, call.arg_values)
args = hd.level2_policy(method, call.raw_args)
impl = hd.level1_policy(method, hardware_context())
return impl(args...)
end
```
## Narya Type for Dispatch Policy
```narya
-- Dispatch policy as dependent type
def DispatchPolicy : Type := sig (
types : TypeTuple,
methods : List Method,
-- Policy function
select : (s : types) → (applicable s methods) → Method,
-- Optimality condition
optimal : ∀ (s : types) (ms : applicable s methods),
Q(s, select s ms) ≥ Q(s, m) for all m ∈ ms,
-- GF(3) conservation
balanced : ∀ (seq : List (types × Method)),
(Σ (_, m) ∈ seq, trit m) ≡ 0 (mod 3)
)
```
## Skill Dispatch as RL
Claude Code's skill system IS multiple dispatch:
```
State s = (user_request, context, available_skills)
Action a = skill to invoke
Reward r = task_completion_quality + efficiency + user_satisfaction
Policy π = skill selection function
```
```python
class SkillDispatcher:
"""RL-trained skill selector."""
def dispatch(self, request, context):
# State encoding
state = encode_state(request, context)
# Get applicable skills
applicable = [s for s in self.skills
if s.matches(request, context)]
# Policy selection (learned)
skill = self.policy.select(state, applicable)
# Execute and observe reward
result = skill.execute(request, context)
reward = self.compute_reward(result)
# Update policy
self.policy.update(state, skill, reward)
return result
```
## Open Games Connection
Multiple dispatch is a **compositional game**:
```
play (type input)
Dispatcher ─────────────────────────► Method
│ │
│ │
└────────────────────────────────────┘
coplay (result feedback)
```
```haskell
-- Dispatch as open game
dispatchGame :: OpenGame TypeTuple Method
dispatchGame = OpenGame {
play = \types -> selectMethod types,
coplay = \types method result -> updateQ types method (reward result)
}
```
## Implementation: Julia + Flux.jl
```julia
using Flux
# Neural dispatch policy
struct NeuralDispatch
encoder::Chain # Type tuple → embedding
q_network::Chain # Embedding × method_id → Q-value
method_embeddings # Learned method representations
end
function (nd::NeuralDispatch)(types)
# Encode type tuple
type_emb = nd.encoder(encode_types(types))
# Compute Q-values for all methods
q_values = [nd.q_network(vcat(type_emb, nd.method_embeddings[m]))
for m in 1:num_methods]
# Return softmax policy
softmax(q_values)
end
# Training
function train_dispatch!(nd, experiences)
for (types, method, reward, next_types) in experiences
# Bellman target
next_q = maximum(nd(next_types))
target = reward + γ * next_q
# Update
grads = gradient(Flux.params(nd)) do
predicted = nd.q_network(vcat(
nd.encoder(encode_types(types)),
nd.method_embeddings[method]
))
Flux.mse(predicted, target)
end
Flux.update!(opt, Flux.params(nd), grads)
end
end
```
## GF(3) Triads
```
open-games (-1) ⊗ multidispatch-rl (0) ⊗ julia-dispatch (+1) = 0 ✓
parametrised-optics (-1) ⊗ multidispatch-rl (0) ⊗ powers-pct (+1) = 0 ✓
skill-dispatch (-1) ⊗ multidispatch-rl (0) ⊗ truealife (+1) = 0 ✓
```
## Commands
```bash
# Train dispatch policy on type traces
just dispatch-train traces.json
# Evaluate dispatch decisions
just dispatch-eval --types "(Int, String, Float64)"
# Visualize Q-table
just dispatch-viz
# Export to Julia dispatch table
just dispatch-export julia
```
## Related Skills
| Skill | Connection |
|-------|------------|
| `julia` | Native multiple dispatch |
| `open-games` | Compositional game semantics |
| `parametrised-optics-cybernetics` | ⊛ action selection |
| `truealife` | ALife reward structures |
| `powers-pct` | Hierarchical control |
---
**Skill Name**: multidispatch-rl
**Type**: RL / Type Theory / Dispatch
**Trit**: 0 (ERGODIC - the dispatcher coordinates)
**Key Insight**: Dispatch table = learned policy over type lattice
**Objective**: Maximize expected utility of method selection
---
## 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.*
Imported from the listed AgentSkill.sh category. Complete supplied package files and notices are retained. Required external tools and credentials are not installed. No bundled code was executed during import. Individual agent task behavior has not been tested.