Document
Constraint Derivation
**Verifiable Semantic Execution Layer (VSEL)**
Verifiable Semantic Execution Layer (VSEL)
1. Purpose
This document defines how formal semantics are transformed into constraint systems suitable for proof generation.
It establishes a deterministic, verifiable transformation:
[ \mathcal{D}: SIR \rightarrow C ]
Where:
- ( SIR ) is the semantic intermediate representation
- ( C ) is the constraint system
The core requirement is:
Constraints must encode semantic validity, not just computational execution.
2. Constraint Objective
Given a trace ( \tau ), constraints must enforce:
[ SatisfiesConstraints(\tau) \Leftrightarrow ValidTrace(\tau) ]
This implies two properties:
2.1 Soundness
[ SatisfiesConstraints(\tau) \Rightarrow ValidTrace(\tau) ]
No invalid execution may satisfy constraints.
2.2 Completeness
[ ValidTrace(\tau) \Rightarrow SatisfiesConstraints(\tau) ]
All valid executions must be representable.
3. Constraint Domains
Constraints are derived over:
- state variables
- inputs
- transition relations
- invariants
- trace structure
Each domain must be explicitly mapped.
4. State Constraints
Let ( s = (C, D, E, \tau) )
Constraints enforce:
4.1 Canonical State Validity
[ C \in ValidC ]
Includes:
- structural checks
- domain constraints
- encoding correctness
4.2 Derived State Consistency
[ D = Derive(C) ]
Derived values must be recomputed, not trusted.
4.3 Environment Validity
[ E \in ValidE ]
All environmental assumptions must be constrained.
4.4 Metadata Consistency
[ \tau_{i+1} = Update(\tau_i) ]
Ensures trace continuity.
5. Input Constraints
For each input ( \sigma ):
5.1 Canonical Form
[ \sigma = Canonical(\sigma) ]
Reject ambiguous encodings.
5.2 Authorization
[ Auth(\sigma) = true ]
Signature or permission checks must be fully constrained.
5.3 Domain Validity
[ \sigma \in \Sigma ]
Input must belong to valid input space.
6. Transition Constraints
For each transition:
[ (s, \sigma) \rightarrow s' ]
Constraints must enforce:
[ Apply(s, \sigma) = s' ]
6.1 Functional Correctness
Each field in ( s' ) must be derived from:
- ( s )
- ( \sigma )
No unconstrained fields allowed.
6.2 Mutation Scope
Only allowed fields may change:
[ Diff(s, s') \subseteq AllowedFields ]
6.3 No Implicit State
All state changes must be explicit in constraints.
If a value changes and is not constrained, that is a vulnerability.
7. Invariant Constraints
All invariants must be encoded.
7.1 Local Invariants
[ L(s, \sigma, s') ]
Example:
- resource conservation
- bounded changes
7.2 Global Invariants
[ G(s) ]
Example:
- valid structure
- commitment consistency
7.3 Temporal Invariants
[ TInv(\tau) ]
Example:
- monotonic counters
- no rollback
8. Trace Constraints
Constraints must enforce:
8.1 Transition Linking
[ s_{i+1} = Apply(s_i, \sigma_i) ]
8.2 Ordering
[ i < j \Rightarrow Order(i, j) ]
No reordering unless explicitly modeled.
8.3 Completeness
All transitions must be present.
No skipped states.
9. Observable Constraints
For each transition:
[ Obs(s, \sigma, s') = o ]
Constraints must enforce:
[ o = ComputeObservable(s, \sigma, s') ]
Ensures external outputs are correct.
10. Constraint Generation Rules
The derivation process must be:
- deterministic
- formally specified
- auditable
10.1 Rule-Based Mapping
Each SIR construct maps to constraint templates.
Example:
- assignment → equality constraint
- arithmetic → arithmetic circuit
- condition → boolean constraint
10.2 No Manual Constraint Injection
Constraints must not be handwritten outside the derivation system.
Manual constraints introduce semantic drift.
11. Underconstraint Prevention
The system must prevent:
11.1 Unused Variables
Every variable must be constrained.
11.2 Free Witness Values
Witness assignments must be uniquely determined.
11.3 Partial Constraints
All branches must be constrained, not only the executed one.
12. Constraint Completeness Verification
The system must validate:
[ ValidTrace(\tau) \equiv SatisfiesConstraints(\tau) ]
Using:
- differential testing
- symbolic execution
- adversarial witness generation
13. Cross-Layer Consistency
13.1 Semantic Mapping Alignment
[ \mu_{Tr}(\tau_c) = \tau_f ]
Constraints must apply to ( \tau_f ).
13.2 Execution Alignment
[ Apply_{impl} = Apply_{constraint} ]
13.3 Proof Alignment
[ Verify(\pi) \Rightarrow SatisfiesConstraints(\tau) ]
14. Failure Modes
14.1 Underconstraint
Invalid execution satisfies constraints.
14.2 Overconstraint
Valid execution cannot be proven.
14.3 Constraint Drift
Constraints diverge from formal semantics.
14.4 Witness Ambiguity
Multiple witnesses represent same constrained output.
15. Minimal Validation Conditions
A valid constraint system must satisfy:
- soundness
- completeness
- determinism
- semantic alignment
16. Closing Statement
Constraints are not the system.
They are a projection of the system.
If that projection is incomplete, you are proving a shadow and calling it truth.