Ternary Logic Encodings of Temporal Behavior Trees with Application to Control Synthesis

2026-04-13

AI Summary

This work addresses the lack of a computable formal model in existing behavior trees that can support online control synthesis while guaranteeing satisfaction of temporal safety specifications at runtime. To bridge this gap, the paper introduces three-valued signal temporal logic—augmented with an Unknown truth value—into the behavior tree framework, thereby enabling formal evaluation of specifications over partially observable trajectories. The authors develop a formal control synthesis framework that encodes linear dynamical systems via mixed-integer linear programming, simultaneously ensuring correctness with respect to temporal logic specifications and optimizing performance. Experimental results demonstrate that the proposed approach effectively generates optimal control strategies satisfying complex temporal requirements, confirming its practicality and efficacy in formal control synthesis.

Technology Category

• Intelligent Robots: Behavior Learning & Control
• Humans and AI: Human-Aware Planning and Behavior Prediction
• Planning, Routing, and Scheduling: Temporal Planning

Application Category

• 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: LLM based quality controls for crowd work
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

Behavior Trees (BTs) provide designers an intuitive graphical interface to construct long-horizon plans for autonomous systems. To ensure their correctness and safety, rigorous formal models and verification techniques are essential. Temporal BTs (TBTs) offer a promising approach by leveraging existing temporal logic formalisms to specify and verify the executions of BTs. However, this analysis is currently limited to offline post hoc analysis and trace repair. In this paper, we reformulate TBTs using a ternary-valued Signal Temporal Logic (STL) amenable for control synthesis. Ternary logic introduces a third truth value \textit{Unknown}, formally capturing cases where a trajectory has neither fully satisfied or dissatisfied a specification. We propose mixed-integer linear encodings for partial trajectory STL and TBTs over ternary logic allowing for correct-by-construction control strategies for linear dynamical systems via mixed-integer optimization. We demonstrate the utility of our framework by solving optimal control problems.

Problem

• Temporal Behavior Trees
• Signal Temporal Logic
• Control Synthesis
• Ternary Logic
• Formal Verification

Innovation

• Ternary Logic
• Signal Temporal Logic
• Behavior Trees
• Control Synthesis
• Mixed-Integer Linear Programming