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Shield Synthesis for LTL Modulo Theories
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Extends shield synthesis from propositional LTL to LTL modulo theories by combining Boolean abstraction with reactive LTL_T synthesis, giving controller-based and winning-region-based shields with optimization via MaxSMT.
Problem
Shielding for safe DNN/DRL controllers is limited to Boolean-atomic-proposition LTL specifications, precluding rich data domains such as integers and reals.
Approach
Lift shielding to LTL modulo theories by applying Boolean abstraction to obtain an equi-realizable LTL spec, then either using a Boolean controller plus a theory-level provider (controller-based shield) or a winning region with a find component (WR-based shield); optimize overrides via MaxSMT to choose the closest safe output.
Key results
- Boolean step averages under 0.21 microseconds versus over 0.47 microseconds for Wu et al. (roughly twice as fast).
- Final output step averages about 0.157 ms versus 0.557 ms for Wu et al. on the 13 specifications.
- The approach also handles T_Z (linear integer arithmetic) and optimized variants (smallest/greatest and closest-to-candidate outputs), which Wu et al.'s monolithic method cannot provide for T_Z.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed Boolean step instead of Wu et al.'s Boolean step | LTL modulo theories shield synthesis · 13 specifications | Boolean-step runtime: less than 0.21μs vs over 0.47μs | Helped |
| Use the proposed rich-data output step instead of Wu et al.'s output step | LTL modulo theories shield synthesis · 13 specifications | rich-data output runtime: about 0.157ms vs 0.557ms | Helped |
| Use the full valid-reaction abstraction rather than a feasible subset that omits non-minimal reactions | Winning-region shield · LTL modulo theories safety specifications | permissivity | Helped |
| Omit the e0+ reaction from the abstraction (ablation) | Winning-region shield · example with input x:0 and candidate output y:2 | candidate output preservation | Hurt |
| Use a Boolean controller rather than the winning region | LTL modulo theories shield · LTL modulo theories safety specifications | permissivity | Hurt |
| Minimize the distance between the safe correction and the unsafe candidate using soft constraints | LTL modulo theories shield · integer arithmetic example, unsafe y:4 | correction distance | Helped |
Surprising findings
- Using a winning region is less intrusive than using a Boolean controller, but winning regions are harder to interpret and slower to compute.
- Ignoring valid reactions that are subsumed by others (minimal valid reactions) yields smaller formulas but a less permissive shield.
Limitations
- Method requires the theory to be decidable in the exists-forall fragment for Boolean abstraction to apply.
- Winning region computation can be expensive and may not terminate, especially for liveness specifications.
- Evaluation is limited to a comparison on 13 specifications with timing only; no DRL system experiments in the main text.
Experimental setup
Benchmarks: 13 specifications from Wu et al. (phi_0 to phi_12)
Metrics: runtime in microseconds for Boolean step and theory step
Baselines: Wu et al. shielding for rich-data specifications
Builds on: Boolean abstraction for LTL modulo theories reactive LTL_T synthesis shielding (Alshiekh et al.) bounded synthesis
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