Differentiable Ternary Temporal Logic Semantics with Polynomial Surrogate Networks

2026-10-03

AI Summary

This study addresses the limitations of Boolean temporal logic in quantifying uncertainty and the lack of differentiability in three-valued logic for gradient-based robot control. To overcome these challenges, this work proposes a probabilistically relaxed three-valued temporal logic framework. By employing polynomial surrogate networks to compute both semantic values and gradients, it constructs the first differentiable three-valued temporal logic, bridging discrete logical reasoning with continuous gradient optimization while preserving exact decision-making capabilities. Experimental evaluations demonstrate that the proposed framework effectively facilitates intermediate verification and sequencing of subtasks in robotic manipulator scenarios, ensuring both correctness and timeliness across offline planning and online control settings.

Abstract

Temporal logic provides designers a formal reasoning tool for specifying complex spatio-temporal behaviors and tasks for autonomous systems. Many techniques exist for control synthesis according to temporal logic specifications, spanning a large variety of systems, including robotics. However, the vast majority of implementations are limited to Boolean temporal logics. Boolean temporal logic does not have an innate mechanism for quantifying abstention, rather verdicts are either definitively \textit{True} or \textit{False}. Recent work demonstrates that ternary logic is a capable formalism for reasoning about robotic behaviors, with the natural ability to classify uncertainty with the \textit{Unknown} literal. Ternary temporal logic is still in its infancy, with work limited to synthesis for linear systems and monitoring. This work proposes a probabilistic relaxation of ternary temporal logic that allows for value and gradient computation through the same network while retaining exact verdicts on actual data. The relaxed network can be used as an objective in gradient-based control synthesis for both offline and online robotic control applications. We demonstrate the utility of our framework with two robotic manipulator tasks, one offline and one online, involving intermediate checking and sequencing of subtasks to demonstrate the benefit of our approach in terms of correct and timely execution.

Problem

• Temporal Logic
• Ternary Logic
• Control Synthesis
• Differentiable Semantics
• Robotic Control

Innovation

• Ternary Temporal Logic
• Differentiable Semantics
• Polynomial Surrogate Networks
• Gradient-based Control Synthesis
• Probabilistic Relaxation