Document
Composition Model
**Verifiable Semantic Execution Layer (VSEL)**
Verifiable Semantic Execution Layer (VSEL)
1. Purpose
This document defines how independently correct VSEL systems can be composed without violating semantic correctness.
It specifies:
- composition semantics
- cross-system state interactions
- invariant preservation across boundaries
- proof and verification composition
- failure modes of multi-system interaction
The core requirement is:
Composition must preserve semantic correctness, not just local correctness.
2. Composition Definition
Let:
[ \mathcal{M}_A = (S_A, I_A, T_A, O_A) ]
[ \mathcal{M}_B = (S_B, I_B, T_B, O_B) ]
The composed system is:
[ \mathcal{M}{A \circ B} = (S{AB}, I_{AB}, T_{AB}, O_{AB}) ]
Where:
- ( S_{AB} \subseteq S_A \times S_B )
- ( T_{AB} \subseteq T_A \cup T_B \cup T_{cross} )
Composition introduces cross-transitions:
[ T_{cross} \subseteq S_A \times S_B \times \Sigma_{cross} \times S_A \times S_B ]
3. Composition Objectives
A composed system is correct if:
[
ValidTrace_{AB}(\tau) \Rightarrow
\begin{cases}
ValidTrace_A(\tau_A)
ValidTrace_B(\tau_B)
\end{cases}
]
and:
[ Preserve(G_A \cup G_B \cup G_{cross}) ]
Where:
- ( G_{cross} ) are cross-system invariants
4. Composition Types
4.1 Parallel Composition
Systems evolve independently:
[ (s_A, s_B) \rightarrow (s'_A, s'_B) ]
Constraint:
- no shared state
- no cross-dependencies
4.2 Sequential Composition
Output of one system feeds another:
[ s_A \xrightarrow{\sigma_A} s'_A \Rightarrow \sigma_B = f(s'_A) ]
4.3 Shared-State Composition
Systems operate over shared state:
[ S_{shared} \subseteq S_A \cap S_B ]
Requires strict synchronization.
4.4 Cross-Triggered Composition
Transitions in one system trigger transitions in another:
[ (s_A, s_B) \xrightarrow{\sigma_A} (s'_A, s'_B) ]
5. State Composition
The composed state is:
[ s_{AB} = (s_A, s_B, S_{shared}) ]
Constraints:
- shared state must be consistent
- no conflicting updates
5.1 State Consistency Condition
[ C_{shared}^A = C_{shared}^B ]
If violated:
- composition is invalid
5.2 Derived State Alignment
Derived values must match across systems:
[ D_A(shared) = D_B(shared) ]
6. Cross-System Transitions
A cross-transition:
[ (s_A, s_B, \sigma_{cross}) \rightarrow (s'_A, s'_B) ]
Must satisfy:
[ Apply_A(s_A, \sigma_A) = s'_A ]
[ Apply_B(s_B, \sigma_B) = s'_B ]
Where:
[ \sigma_{cross} = (\sigma_A, \sigma_B) ]
7. Cross-System Invariants
Composition introduces new invariants.
7.1 Resource Conservation Across Systems
[ Total_A + Total_B = constant ]
Prevents:
- double-spend across boundaries
7.2 State Synchronization
[ State_A(shared) = State_B(shared) ]
7.3 Causal Consistency
[ Order_A \Rightarrow Order_B ]
Prevents:
- inconsistent ordering
7.4 Authorization Consistency
Permissions must be consistent across systems.
8. Trace Composition
Let:
[ \tau_A, \tau_B ]
Combined trace:
[ \tau_{AB} = Merge(\tau_A, \tau_B) ]
Requirements:
- ordering preserved
- cross-dependencies respected
8.1 Trace Validity
[ ValidTrace_{AB}(\tau_{AB}) ]
Must imply:
- individual validity
- cross-invariant preservation
9. Proof Composition
Proofs may be combined:
[ \pi_{AB} = Combine(\pi_A, \pi_B, \pi_{cross}) ]
Where:
- ( \pi_A ): proof of system A
- ( \pi_B ): proof of system B
- ( \pi_{cross} ): proof of cross-invariants
9.1 Cross-Proof Requirements
Must enforce:
[ Consistency(s_A, s_B) ]
9.2 Recursive Composition
Proofs may embed verification of other proofs.
Ensures:
- scalability
- compositional correctness
10. Verification of Composition
Verifier must check:
- individual proofs
- cross-invariants
- state consistency
[ Verify(\pi_{AB}) \Rightarrow ValidTrace_{AB}(\tau_{AB}) ]
11. Failure Modes
11.1 Invariant Violation Across Systems
Each system is locally valid, globally invalid.
11.2 State Divergence
Shared state differs between systems.
11.3 Ordering Mismatch
Different systems observe different orders.
11.4 Double-Spend Across Domains
Resources duplicated via composition.
11.5 Proof Inconsistency
Proofs individually valid but mutually incompatible.
11.6 Partial Composition Validation
Only one system validated.
12. Composition Safety Conditions
Composition is safe if:
[ \forall \tau_{AB}: ValidTrace_{AB}(\tau_{AB}) ]
and:
[ Preserve(G_A \cup G_B \cup G_{cross}) ]
13. Isolation Guarantees
Systems may enforce isolation:
- no shared state
- no cross-effects
Isolation reduces complexity but limits composability.
14. Compositional Soundness
[ Valid(\mathcal{M}_A) \land Valid(\mathcal{M}B) \land Compatible \Rightarrow Valid(\mathcal{M}{A \circ B}) ]
Compatibility must be explicitly defined.
15. Compositional Completeness
All intended interactions must be representable in:
[ T_{cross} ]
Otherwise:
- system behavior is incomplete
16. Boundary Definition
Each system must define:
- input/output boundaries
- shared state boundaries
- trust boundaries
17. Upgrade and Evolution
Composition must handle:
- version mismatches
- protocol upgrades
- backward compatibility
Explicit versioning required.
18. Closing Statement
Composition is where correctness goes to die quietly.
VSEL enforces that:
correctness must survive interaction, not just isolation.