Propagators Skill

"The Art of the Propagator" — Radul & Sussman, 2009

Core Concept

Propagators are autonomous machines that:

1. Watch cells for new information
2. Compute derived values
3. Add information to other cells
4. Repeat until fixpoint

  ┌──────┐         ┌──────┐
  │cell A│────────▶│cell B│
  └──────┘  prop   └──────┘
      │                │
      │    ┌──────┐    │
      └───▶│cell C│◀───┘
           └──────┘

No control flow. Information flows until nothing new can be derived.

Why It's Strange

1. Bidirectional — constraints work both ways
2. Monotonic — cells only gain information, never lose it
3. Mergeable — conflicting info produces refined info (or contradiction)
4. Concurrent — all propagators run "simultaneously"

Cell Lattice

Cells hold values from a join-semilattice:

        ⊤ (contradiction)
       /|\
      / | \
     /  |  \
   3.14  e  √2
     \  |  /
      \ | /
       \|/
        ⊥ (nothing)

• ⊥ = "I know nothing"
• Value = "I know this specific thing"
• ⊤ = "Contradiction! Conflicting claims"

Basic Operations

;; Create cells
(define-cell a)
(define-cell b)
(define-cell c)

;; Add propagator: c = a + b
(p:+ a b c)

;; Set values (can be in any order!)
(add-content a 3)
(add-content b 4)

;; c automatically becomes 7
(content c)  ; → 7

;; BIDIRECTIONAL: set c, derive a!
(add-content c 10)
(add-content b 4)
(content a)  ; → 6 (inferred!)

Partial Information

;; Intervals
(define-cell x)
(add-content x (make-interval 0 10))   ; x ∈ [0, 10]
(add-content x (make-interval 5 15))   ; x ∈ [5, 10] (intersection!)

;; Symbolic
(add-content x 'positive)
(add-content x 7)  ; Consistent: 7 is positive

;; Contradiction
(add-content x 'negative)  ; → ⊤ (7 is not negative!)

Implementation

Minimal Propagator in Python

class Cell:
    def __init__(self):
        self.content = Nothing()
        self.neighbors = []  # Propagators to notify
    
    def add_content(self, value):
        merged = merge(self.content, value)
        if merged != self.content:
            self.content = merged
            self.alert_propagators()
    
    def alert_propagators(self):
        for prop in self.neighbors:
            schedule(prop)

class Propagator:
    def __init__(self, inputs, output, func):
        self.inputs = inputs
        self.output = output
        self.func = func
        for cell in inputs:
            cell.neighbors.append(self)
    
    def run(self):
        values = [c.content for c in self.inputs]
        if all(v.is_known() for v in values):
            result = self.func(*[v.value for v in values])
            self.output.add_content(result)

# Adder propagator (a + b = c, bidirectional)
def make_adder(a, b, c):
    Propagator([a, b], c, lambda x, y: x + y)
    Propagator([a, c], b, lambda x, z: z - x)
    Propagator([b, c], a, lambda y, z: z - y)

Scoped Propagators (Gay.jl)

# From your codebase: scoped_propagators.jl
abstract type ScopedPropagator end

struct ConeUp <: ScopedPropagator      # ↑ Bottom-up (colimit)
    cells::Vector{Cell}
end

struct DescentDown <: ScopedPropagator  # ↓ Top-down (limit)
    cells::Vector{Cell}
end

struct AdhesionHoriz <: ScopedPropagator  # ↔ Beck-Chevalley
    left::Vector{Cell}
    right::Vector{Cell}
end

GF(3) Integration

# Triadic propagator network
struct TriadicCell
    trit::Int  # -1, 0, +1
    value::Any
    neighbors::Vector{Propagator}
end

# Conservation: sum of connected cells = 0 (mod 3)
function verify_gf3(cells::Vector{TriadicCell})
    sum(c.trit for c in cells) % 3 == 0
end

Dependency-Directed Backtracking

When contradiction (⊤) is reached:

(define-cell x)
(define-cell y)

;; Track provenance
(add-content x (supported 5 '(assumption-1)))
(add-content y (supported 7 '(assumption-2)))

;; Contradiction!
(add-content x (supported 10 '(assumption-3)))

;; System identifies: assumption-1 OR assumption-3 must go
;; Backtrack to consistent state

Applications

Domain	Use Case
CAD	Constraint-based modeling
Physics	Unit conversion, equations
Type inference	Bidirectional typing
Planning	Constraint satisfaction
Pricing	Epistemic arbitrage

Relationship to Other Models

Model	Propagators
Dataflow	Similar but propagators are bidirectional
Constraint Logic	Propagators = constraint propagation
Reactive	Similar but propagators reach fixpoint
SAT/SMT	Unit propagation is a propagator

Literature

1. Radul & Sussman (2009) - "The Art of the Propagator"
2. Steele (1980) - "The Definition and Implementation of Constraint Languages"
3. Apt (1999) - "The Essence of Constraint Propagation"

Related Skills

• epistemic-arbitrage - Uses scoped propagators
• constraint-logic - Logical foundation
• dataflow - One-way version
• interaction-nets - Another "no control" model

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Neighbor Awareness (Co-Occurrence Patterns)

Basin Affinity

From interaction_entropy.duckdb skill co-occurrence analysis:

skill: propagators
basin: NEUTRAL
avg_basin_energy: 1.0
interleave_role: generator (+1)

Co-Occurring Skills (Constraint Partners)

Skills frequently invoked together in propagator networks:

Skill	Role	Trit	Affinity Pattern
gay-mcp	Generator	+1	Color cells by value
duckdb-temporal-versioning	Generator	+1	Store cell states
datalog-fixpoint	Coordinator	0	Fixpoint iteration
specter-acset	Coordinator	0	Navigate cell networks
unworld	Coordinator	0	Seed-derived constraints
sheaf-cohomology	Validator	-1	Verify cell consistency
three-match	Validator	-1	GF(3) conservation

GF(3) Triad Partners

Natural skill groupings that satisfy GF(3) conservation (sum = 0):

propagators (+1) ⊗ datalog-fixpoint (0) ⊗ sheaf-cohomology (-1) = 0 ✓
propagators (+1) ⊗ specter-acset (0) ⊗ three-match (-1) = 0 ✓
propagators (+1) ⊗ unworld (0) ⊗ moebius-inversion (-1) = 0 ✓
propagators (+1) ⊗ acsets (0) ⊗ temporal-coalgebra (-1) = 0 ✓

Basin Transition Flows

Energy flow patterns in constraint propagation:

NEUTRAL → NEUTRAL:  LATERAL ↔  energy_delta =  0.000  (propagating)
NEUTRAL → PLUS:     RISE ↑     energy_delta = +0.382  (information gain)
NEUTRAL → MINUS:    DESCENT ↓  energy_delta = -0.382  (contradiction)

Interleave Topology

Position in the constraint satisfaction pipeline:

Level 1: ⊕ generator  (propagators)                 NEUTRAL basin  [PROPAGATE]
Level 2: ○ coordinator (datalog-fixpoint)           NEUTRAL basin  [FIXPOINT]
Level 3: ○ coordinator (specter-acset)              NEUTRAL basin  [NAVIGATE]
Level 4: ⊖ validator   (sheaf-cohomology)           NEUTRAL basin  [VERIFY]

Upstream Skills (Constraint Producers)

Skills that produce constraints for propagation:

Skill	Constraint Type	Propagation Pattern
acsets	ACSet schema	Cell per part, morphism propagators
datalog-fixpoint	Derived relations	Rule → propagator
gay-mcp	Color constraints	Trit conservation
unworld	Seed-derived	Chain constraints

Downstream Skills (Fixpoint Consumers)

Skills that consume propagator fixpoints:

Skill	Usage Pattern	Output
sheaf-cohomology	Verify consistency	H¹ = 0 check
three-match	Verify GF(3)	Conservation proof
specter-acset	Navigate result	Selected values
duckdb-temporal-versioning	Store fixpoint	Persistent state

Skill Invocation Chains

Common multi-skill sequences observed:

;; Constraint satisfaction pipeline
(-> (acsets :define-schema)
    (propagators :build-network)
    (datalog-fixpoint :run-to-fixpoint)
    (sheaf-cohomology :verify-consistency))

;; Bidirectional type inference
(-> (propagators :type-cells)
    (specter-acset :navigate-types)
    (three-match :verify-gf3))

;; Epistemic arbitrage
(-> (propagators :scoped-network)
    (gay-mcp :color-by-confidence)
    (duckdb-temporal-versioning :store-arbitrage))

;; Triadic cell network
(-> (gay-mcp :tripartite-seeds)
    (propagators :triadic-cells)
    (three-match :verify-balance))

MCP Tool Coordination

When invoked via MCP, coordinates with:

mcp_neighbors:
  - tool: acset_colim
    relation: "cell structure from ACSets"
    direction: upstream
  - tool: datalog_query
    relation: "rule-based propagators"
    direction: upstream
  - tool: sheaf_verify
    relation: "verify cell consistency"
    direction: downstream
  - tool: gay_mcp
    relation: "color cells by value"
    direction: downstream
  - tool: duckdb_query
    relation: "store fixpoint states"
    direction: downstream

Propagator Network Example

# Full pipeline with neighbor coordination
def propagate_with_neighbors(constraints, initial_values):
    # Build propagator network
    cells = {}
    propagators = []

    for var in constraints.variables:
        cells[var] = Cell()

    for constraint in constraints.all:
        prop = build_propagator(constraint, cells)
        propagators.append(prop)

    # Set initial values (from upstream)
    for var, value in initial_values.items():
        cells[var].add_content(value)

    # Run to fixpoint (like datalog-fixpoint)
    while schedule.has_work():
        prop = schedule.pop()
        prop.run()

    # Verify via sheaf-cohomology (downstream)
    for cell_name, cell in cells.items():
        if cell.content == CONTRADICTION:
            # Dependency-directed backtracking
            deps = cell.get_dependencies()
            sheaf_cohomology.report_obstruction(cell_name, deps)

    # Color cells via gay-mcp (downstream)
    for i, (name, cell) in enumerate(cells.items()):
        cell.color = gay_mcp.color_at(seed, i)

    # Store via duckdb (downstream)
    duckdb_insert(db, "propagator_fixpoints", (
        num_cells=len(cells),
        num_propagators=len(propagators),
        reached_fixpoint=True,
        timestamp=now()
    ))

    return cells

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Skill Name: propagators Type: Constraint Propagation Generator Trit: +1 (PLUS - Generator) GF(3): Forms valid triads with coordinators (0) and validators (-1) Applications: Bidirectional constraints, type inference, epistemic arbitrage, CAD modeling