First-Order LTLf Synthesis with Lookback (Extended Version)

2025-08-25

AI Summary

Existing LTLfMT-based reactive synthesis approaches cannot handle cross-temporal variable comparisons (“backtracking”), severely limiting their expressiveness and practical applicability. This paper introduces the first reactive synthesis framework supporting full LTLfMT with backtracking. We propose a strategy induction method based on first-order logic constraint modeling and symbolic search, yielding a sound and complete synthesis algorithm for bounded strategy lengths. Our work identifies several new decidable subclasses of LTLfMT synthesis and unifies and generalizes prior decidability results. Experimental evaluation demonstrates that the framework significantly enhances specification modeling capability and synthesis feasibility in canonical application domains—including AI planning and business process management—while preserving formal correctness guarantees.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Reactive synthesis addresses the problem of generating a controller for a temporal specification in an adversarial environment; it was typically studied for LTL. Driven by applications ranging from AI to business process management, LTL modulo first order-theories over finite traces (LTLfMT) has recently gained traction, where propositional variables in properties are replaced by first-order constraints. Though reactive synthesis for LTLf with some first-order features has been addressed, existing work in this direction strongly restricts or excludes the possibility to compare variables across instants, a limitation that severely restricts expressiveness and applicability. In this work we present a reactive synthesis procedure for LTLfMT, where properties support "lookback" to model cross-instant comparison of variables. Our procedure works for full LTLfMT with lookback, subsuming the fragments of LTLfMT for which realizability was studied earlier. However, the setting with cross-instant comparison is inherently highly complex, as realizability is undecidable even over decidable background theories. Hence termination of our approach is in general not guaranteed. Nevertheless, we prove its soundness, and show that it is complete if a bound on the strategy length exists. Finally, we show that our approach constitutes a decision procedure for several relevant fragments of LTLfMT, at once re-proving known decidability results and identifying new decidable classes.

Problem

• Reactive synthesis for LTLfMT with cross-instant variable comparison
• Overcoming limitations in expressiveness from restricted lookback capabilities
• Addressing undecidability challenges in realizability over background theories

Innovation

• LTLfMT synthesis with cross-instant variable comparison
• Sound procedure for full LTLfMT with lookback support
• Decision procedure for several decidable LTLfMT fragments