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103 lines
2.2 KiB
Markdown
103 lines
2.2 KiB
Markdown
# Protocol Inheritance: Communication Type System
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**Authors:** Research Stack Team
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**Date:** April 2026
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**Domain:** TTM Layer M (Lean Semantics - Protocol)
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**OTOM Version:** 2.2
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---
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## Abstract
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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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---
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## 1. Protocol Structure
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### 1.1 Protocol Type
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```
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Protocol P = {
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messages : Set Message,
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states : Set State,
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transitions : State × Message → State,
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invariants : Set (State → Bool)
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}
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```
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### 1.2 Message Format
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$$m = (\text{header}, \text{payload}, \text{signature})$$
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---
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## 2. Inheritance Hierarchy
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### 2.1 Subtyping
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$$P_{\text{child}} <: P_{\text{parent}} \iff \forall m \in P_{\text{parent}}.\text{messages}, m \in P_{\text{child}}.\text{messages}$$
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### 2.2 Liskov Substitution
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$$P_{\text{child}} <: P_{\text{parent}} \implies \forall s.\text{verify}_{P_{\text{parent}}}(s) \implies \text{verify}_{P_{\text{child}}}(s)$$
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---
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## 3. Protocol Composition
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### 3.1 Sequential Composition
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$$P_1 ; P_2 = \{(s_1, m, s_2) \mid (s_1, m, s) \in P_1 \land (s, m, s_2) \in P_2\}$$
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### 3.2 Parallel Composition
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$$P_1 \parallel P_2 = \{(s_1 \times s_2, m_1 \oplus m_2, s_1' \times s_2')\}$$
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---
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## 4. Verification
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### 4.1 Protocol Compliance
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$$\text{compliant}(a, P) = \forall m \in a.\text{messages}, m \in P.\text{messages}$$
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### 4.2 State Consistency
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$$\text{consistent}(s, P) = \forall i \in P.\text{invariants}, i(s)$$
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---
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## 5. Bridge Protocol
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For domain transfer:
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$$P_{\text{bridge}}(D_1, D_2) = \{(s_1, m, s_2) \mid s_1 \in D_1 \land s_2 \in D_2\}$$
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---
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## 6. Implementation
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**Lean 4 Modules:**
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- `Protocol.lean` — Core protocol system
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- `OmniNetwork.lean` — Network layer
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- `Pbacs.lean` — Protocol-based actor coordination
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---
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## 7. Theorems
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### 7.1 Type Safety
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$$\text{wellTyped}(m, P) \land \text{compliant}(a, P) \implies \text{safe}(a, m)$$
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### 7.2 Composition Safety
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$$\text{safe}(P_1) \land \text{safe}(P_2) \implies \text{safe}(P_1 ; P_2)$$
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---
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## 8. References
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- Honda, K., et al. (2008). Multiparty asynchronous session types.
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- Research Stack, OTOM Ontology v2.2.
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