Research-Stack/6-Documentation/papers/OTOM/11_Protocol_Inheritance.md

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.