Infinite Trace Objectives with Finite Trace Techniq...

Infinite Trace Objectives with Finite Trace Techniq... | AI Research

Key Takeaways

This paper presents a method to translate Linear Temporal Logic (LTL) formulas into LTLf+, a logic that extends finite-trace reasoning to infinite traces.
Recent work has introduced LTLf+, which lifts the finite-trace logic LTLf to infinite traces.
LTLf+ has the same expressive power as LTL, yet it retains most of the crucial advantages of its base logic LTLf.
Most reasoning in LTLf+ rests on finite automata on finite words, for which we have not only a canonical minimal representation but also an efficient determinization procedure.
In this work we present the first translation from LTL to LTLf+.

Paper AbstractExpand

Linear Temporal Logic (LTL) is one of the most widely adopted languages for specifying temporal extended objectives in AI, with applications ranging from reactive synthesis to stochastic planning in Markov decision processes and reinforcement learning. Traditionally, solving any of these problems requires translating the LTL specification to a nondeterministic automata on infinite words and then determinizing it, a step that is notoriously difficult in theory and in practice. Recent work has introduced LTLf+, which lifts the finite-trace logic LTLf to infinite traces. LTLf+ has the same expressive power as LTL, yet it retains most of the crucial advantages of its base logic LTLf. Most reasoning in LTLf+ rests on finite automata on finite words, for which we have not only a canonical minimal representation but also an efficient determinization procedure. In this work we present the first translation from LTL to LTLf+. We first normalize an LTL formula into the syntactic reactivity fragment of the Manna-Pnueli hierarchy, to create the general fragment-based shape of LTLf+. We then present linear translations for each individual component of that fragment. As a consequence of this translation, the expanding body of techniques developed for LTLf+ now becomes available to many AI problems currently formulated in LTL. We further show that this comes at no asymptotic cost, as the pipeline from LTL to automaton via LTLf+ remains doubly exponential.

This paper presents a method to translate Linear Temporal Logic (LTL) formulas into LTLf+, a logic that extends finite-trace reasoning to infinite traces. By performing this translation, researchers can apply efficient, automaton-based techniques—such as deterministic finite automata (DFA) minimization—to complex AI problems like reactive synthesis and stochastic planning that were previously restricted by the computational difficulty of traditional LTL methods.

Translating LTL to LTLf+

The authors provide the first constructive translation from LTL to LTLf+. The process begins by normalizing an LTL formula into the syntactic reactivity fragment of the Manna-Pnueli hierarchy. Once in this form, the researchers apply linear translations to each individual component. This approach allows LTL formulas to be expressed as Boolean combinations of LTLf+ leaf formulas, which correspond to the four basic classes of the Manna-Pnueli hierarchy: safety, guarantee, recurrence, and persistence.

Leveraging Finite Trace Techniques

LTLf+ is designed to bridge the gap between finite-trace logic (LTLf) and infinite-trace requirements. Because LTLf+ reasoning relies on finite automata on finite words, it benefits from existing tools that are highly efficient. Specifically, DFAs can be minimized to improve performance, a task that is notoriously difficult when working with the nondeterministic automata on infinite words typically required for standard LTL. By using LTLf+ as an intermediate language, the authors enable these DFA-based optimizations to be applied to the full expressive power of LTL.

Computational Cost

The authors demonstrate that this translation does not increase the asymptotic complexity of the reasoning process. The pipeline from an LTL formula to an automaton remains doubly exponential, whether solved through the traditional LTL route or via the new LTLf+ translation. This ensures that the benefits of using LTLf+—such as improved scalability and access to DFA-based algorithms—are gained without incurring additional theoretical costs.

Why This Matters

Strategic reasoning tasks, such as reactive synthesis and planning for temporally extended goals, have long been limited by the complexity of determinizing nondeterministic Büchi automata. By establishing LTLf+ as a practical intermediate language, this work allows the growing ecosystem of techniques developed for LTLf to be applied to broader AI specifications. This provides a path to improve the performance and scalability of LTL-based synthesis methods, bringing them closer to the efficiency currently seen in model checking.