The Geometry of Time: Horizon-Independent Feasibility and Repair for STL

2026-09-19

AI Summary

This study addresses the challenges of physical infeasibility and combinatorial explosion due to discretization in control synthesis for Signal Temporal Logic (STL). To overcome these issues, this work proposes a geometric decision procedure that translates temporal constraints into continuous-space backward reachable sets. By leveraging the analytical inverse of Bhat-Bernstein integrals alongside Farkas duality certificates, the method isolates conflicting constraints and performs automatic repairs. Its core contribution lies in achieving the first horizon-independent feasibility determination, thereby circumventing exponential complexity growth. Experimental evaluations on six-dimensional unmanned aerial vehicle models demonstrate sub-millisecond execution efficiency. The proposed approach significantly outperforms mainstream optimization-based encodings while exhibiting strong robustness against deeply nested logical specifications.

Abstract

Signal Temporal Logic control synthesis frequently encounters physical infeasibility due to actuator limits or flawed task deadlines. Standard optimization methods model time by discretizing the horizon, which leads to exponential computational growth and prevents the extraction of continuous temporal adjustments. This paper presents a geometric decision procedure that evaluates physical feasibility completely independently of the temporal horizon length. The method operates by transforming explicit temporal logic constraints into continuous spatial backward reachable sets evaluated at time zero. It analytically inverts the Bhat-Bernstein settling-time integral to map temporal windows into continuous spatial boundaries, reducing the feasibility check to a local matrix and vector inclusion evaluation. When a specification is infeasible, the procedure extracts a Farkas dual certificate to isolate conflicting constraints and identifies the maximum geometric spatial gap. It then analytically inverts the system's dynamic expansion to map this largest geometric gap into an exact, closed-form temporal delay, precisely fixing the boundary deficit to restore physical realizability. We formally prove the strict soundness, mathematically bounded completeness, and horizon-independent scalability of this procedure. Experimental evaluations on six-dimensional drone kinematics demonstrate sub-millisecond execution times, massive speedups over state-of-the-art optimization encodings, and computational immunity to deeply nested logical formulas.

Problem

• Signal Temporal Logic
• Control Synthesis
• Physical Feasibility
• Temporal Discretization
• Specification Repair

Innovation

• Signal Temporal Logic
• Geometric Decision Procedure
• Backward Reachable Sets
• Farkas Dual Certificate
• Feasibility Repair