# Protocol Inheritance: Communication Type System

**Authors:** Research Stack Team  
**Date:** April 2026  
**Domain:** TTM Layer M (Lean Semantics - Protocol)  
**OTOM Version:** 2.2

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## Abstract

Protocol Inheritance provides a type-safe communication framework for OTOM agents. It formalizes message passing, protocol composition, and inheritance hierarchies with verification guarantees.

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## 1. Protocol Structure

### 1.1 Protocol Type

```
Protocol P = {
  messages : Set Message,
  states : Set State,
  transitions : State × Message → State,
  invariants : Set (State → Bool)
}
```

### 1.2 Message Format

$$m = (\text{header}, \text{payload}, \text{signature})$$

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## 2. Inheritance Hierarchy

### 2.1 Subtyping

$$P_{\text{child}} <: P_{\text{parent}} \iff \forall m \in P_{\text{parent}}.\text{messages}, m \in P_{\text{child}}.\text{messages}$$

### 2.2 Liskov Substitution

$$P_{\text{child}} <: P_{\text{parent}} \implies \forall s.\text{verify}_{P_{\text{parent}}}(s) \implies \text{verify}_{P_{\text{child}}}(s)$$

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## 3. Protocol Composition

### 3.1 Sequential Composition

$$P_1 ; P_2 = \{(s_1, m, s_2) \mid (s_1, m, s) \in P_1 \land (s, m, s_2) \in P_2\}$$

### 3.2 Parallel Composition

$$P_1 \parallel P_2 = \{(s_1 \times s_2, m_1 \oplus m_2, s_1' \times s_2')\}$$

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## 4. Verification

### 4.1 Protocol Compliance

$$\text{compliant}(a, P) = \forall m \in a.\text{messages}, m \in P.\text{messages}$$

### 4.2 State Consistency

$$\text{consistent}(s, P) = \forall i \in P.\text{invariants}, i(s)$$

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## 5. Bridge Protocol

For domain transfer:

$$P_{\text{bridge}}(D_1, D_2) = \{(s_1, m, s_2) \mid s_1 \in D_1 \land s_2 \in D_2\}$$

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## 6. Implementation

**Lean 4 Modules:**
- `Protocol.lean` — Core protocol system
- `OmniNetwork.lean` — Network layer
- `Pbacs.lean` — Protocol-based actor coordination

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## 7. Theorems

### 7.1 Type Safety

$$\text{wellTyped}(m, P) \land \text{compliant}(a, P) \implies \text{safe}(a, m)$$

### 7.2 Composition Safety

$$\text{safe}(P_1) \land \text{safe}(P_2) \implies \text{safe}(P_1 ; P_2)$$

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## 8. References

- Honda, K., et al. (2008). Multiparty asynchronous session types.
- Research Stack, OTOM Ontology v2.2.