ffc labeling and semantics

Labeling & Semantic Energy Accounting in Focus‑Flow Computation

**purpose ** Provide a concise, self‑contained reference on (1) the minimal node/edge labeling scheme (atom / pair / function + 5 rewrite ops + 3 edge types) and (2) how those edge labels connect to the four free‑energy terms (spring, diffusion, context, entropy).

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1 Node kinds (universality primitives)

kind	payload	why it’s enough
atom	integer / byte‑string	base data, constants
pair	two edge slots left,right	builds lists, trees, maps
function	pointer to body sub‑graph + argument port	encodes λ‑terms / SK combinators

1.1 Graph rewrite ops (5)

1. construct (atom/pair/function) 2. destruct (left/right access) 3. apply (connect function→arg) 4. rewrite (beta‑substitute, returns new node) 5. delete (remove unreachable)

These five ops give full Turing power while remaining local & deterministic.

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2 Edge types – the semantic layer

Only three labels are needed to map structural links onto energy terms:

label	semantic intuition	goes into energy
h‑edge	“is‑a / part‑of” taxonomic constraint	spring (E_{spring}) (hierarchy smoothing)
d‑edge	reference / citation / transport path	diffusion (E_{diff}) (mass transport cost)
c‑edge	transient vote / query / context injection	context term (C_i)

(All other relations — causal, vote, ref, meta, etc.— are aliases that map onto one of these three for energy accounting. You can refine later by splitting labels.)

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3 Semantic energy accounting

Goal. Attach a physical‑style “cost” to each edge so probabilistic focus flow has meaningful gradients.

3.1 Spring energy (hierarchy)

For node i:

$$E_{spring,i}=\sum _{j\in N_h(i)}{k}_h{w}_{ij}(p_i-p_j{)}^2$$

N_h(i) = neighbours via h‑edges; k_h = stiffness constant.

3.2 Diffusion energy (transport)

$$E_{diff,i}=\sum _{j\in N_d(i)}{k}_d{w}_{ij}|p_i-p_j|$$

N_d(i) over d‑edges; weight inverse to edge distance.

3.3 Context potential

$$C_i=-c_i{p}_i,c_i=\sum _{(i,\cdot )\in E_c}{w}_{ic}$$

E_c = c‑edges connected to i. Higher votes/queries push probability up.

3.4 Entropy term

Standard Gibbs entropy over node probabilities keeps exploration.

Semantic energy accounting = mapping edge labels → coefficients (k_h,k_d,c_i) so each local update only needs neighbour information.**

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4 Putting it together – local update

$$\Delta p_i=\eta (\underset{spring}{\underbrace{\sum _{j\in N_h}{k}_h{w}_{ij}(p_j-p_i)}}+\underset{diffusion}{\underbrace{\sum _{j\in N_d}{k}_d{w}_{ij}\text{sgn}(p_j-p_i)}}+\gamma c_i-T(1+\log p_i))$$

Conservation enforced every k ticks ⇒ (∑ p_i = 1).

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5 Why only three edge types?

Separating concerns:

• Structure/execution handled by node kinds + 5 ops.
• Semantics/energy handled by edge label → coefficient.
  This keeps the core rewrite engine minimal while still letting us expand semantics later: split h‑edge into isa vs part or d‑edge into ref vs trans by assigning different constants without touching algorithms.

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6 Example

1. Add node “Cat” atom.
2. Add h‑edge Cat→Animal (weight 1).
3. Add d‑edge Cat→Wikipedia‑Cat (weight 0.5).
4. User asks “What is a Cat?” → adds a c‑edge (query) into Cat.
   Focus flow pushes mass to Cat and its neighbourhood; spring keeps Cat near Animal; diffusion pulls in pages; entropy prevents collapse.

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7 Take‑aways

• Minimal labeling = comprehension + scalability.
• Semantic energy accounting links edge labels to physical‑style costs.
• Together with node primitives, the system is both Turing‑complete and meaning‑aware without bloating the protocol.