A Tree-Shaped Tableau for Checking the Satisfiability of Signal Temporal Logic with Bounded Temporal Operators

2025-07-29

AI Summary

This work addresses the satisfiability verification problem for Signal Temporal Logic (STL) specifications with bounded temporal operators under discrete-time semantics. We propose a tree-shaped one-pass tableau method that exploits semantic redundancy inherent in large time intervals of STL formulas to accelerate reasoning. The method supports consistency checking, requirement signal generation, and equivalence/implication verification between STL formulas, and is extendable to Metric Temporal Logic (MTL). By leveraging discrete-time STL semantics, our approach employs a dedicated one-pass inference mechanism—bypassing expensive SMT solving or first-order logic encodings. Experimental evaluation on multiple benchmarks demonstrates that our method significantly outperforms state-of-the-art SMT- and first-order-logic-based approaches, particularly under stringent timing constraints, achieving superior scalability and efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Signal Temporal Logic (STL) is a widely recognized formal specification language to express rigorous temporal requirements on mixed analog signals produced by cyber-physical systems (CPS). A relevant problem in CPS design is how to efficiently and automatically check whether a set of STL requirements is logically consistent. This problem reduces to solving the STL satisfiability problem, which is decidable when we assume that our system operates in discrete time steps dictated by an embedded system's clock. This paper introduces a novel tree-shaped, one-pass tableau method for satisfiability checking of discrete-time STL with bounded temporal operators. Originally designed to prove the consistency of a given set of STL requirements, this method has a wide range of applications beyond consistency checking. These include synthesizing example signals that satisfy the given requirements, as well as verifying or refuting the equivalence and implications of STL formulas. Our tableau exploits redundancy arising from large time intervals in STL formulas to speed up satisfiability checking, and can also be employed to check Mission-Time Linear Temporal Logic (MLTL) satisfiability. We compare our tableau with Satisfiability Modulo Theories (SMT) and First-Order Logic encodings from the literature on a benchmark suite, partly collected from the literature, and partly provided by an industrial partner. Our experiments show that, in many cases, our tableau outperforms state-of-the-art encodings.

Problem

• Efficiently check STL satisfiability for CPS
• Develop tree-shaped tableau for bounded temporal operators
• Improve performance over SMT and FOL methods

Innovation

• Tree-shaped tableau for STL satisfiability
• Exploits redundancy in large time intervals
• Outperforms SMT and First-Order Logic