Backward through Time, Algebraically
Introduces an algebra-generic evaluation engine for Linear Temporal Logic (LTL) that supports differentiation, enabling soft-valued training signals from temporal logic formulas Proposes an executable specification of semantic algebras for LTL evaluation, allowing multiple differentiable semantics to coexist without upfront commitment to a single algebra Demonstrates that each differentiable LTL algebra represents a distinct tradeoff in how it approximates boolean satisfaction in both forward an
Analysis
TL;DR
- Introduces an algebra-generic evaluation engine for Linear Temporal Logic (LTL) that supports differentiation, enabling soft-valued training signals from temporal logic formulas
- Proposes an executable specification of semantic algebras for LTL evaluation, allowing multiple differentiable semantics to coexist without upfront commitment to a single algebra
- Demonstrates that each differentiable LTL algebra represents a distinct tradeoff in how it approximates boolean satisfaction in both forward and backward passes
- Releases the implementation as the
telosPyTorch library, providing audited forward and backward behavior for various algebras - Addresses the gap between discrete temporal logic and continuous neural/adaptive systems by making goal formula satisfaction a differentiable training signal
Why It Matters
This work bridges formal verification and differentiable programming, enabling practitioners to use temporal logic constraints as trainable objectives in neural policies, adaptive controllers, and sequence models. For AI researchers working on neuro-symbolic integration or differentiable logic, it provides a principled, extensible foundation rather than ad-hoc shallow embeddings. The algebra-generic design means the community can develop and compare semantic variants within a unified framework.
Technical Details
- Algebra-generic LTL evaluation engine: Instead of hardcoding a single semantics, the paper defines an abstract interface for LTL algebras and builds an evaluation engine that operates over any conforming algebra, specified via an executable specification
- Differentiable semantics for temporal operators: Each algebra defines how temporal operators (next, until, etc.) are approximated with continuous values and their gradients, with the key insight that "each algebra is a choice of which direction to disappoint, and how"
- PyTorch implementation (telos library): All algebras are implemented and audited for both forward evaluation and backward gradient behavior, with the full codebase released as a reusable library
- Cross-domain applicability: The framework targets neural policies, adaptive controllers, and sequence models where boolean satisfaction is insufficient and gradient-based optimization is required
Industry Insight
- Neuro-symbolic AI systems will benefit from principled differentiable logic layers; practitioners should evaluate which LTL algebra best matches their gradient quality requirements before adopting
- The algebra-generic approach suggests a broader pattern: formal systems can be made differentiable through parameterized semantic choices rather than monolithic re-embeddings, a strategy applicable to other logics beyond LTL
- As temporal constraints become more common in safety-critical ML systems (robotics, autonomous control), standardized differentiable logic libraries like
teloswill reduce fragmentation and accelerate safe deployment
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