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Differentiable SpaTiaL: Symbolic Learning and Reasoning with Geometric Temporal Logic for Manipulation Tasks

Licheng Luo, Kaier Liang, Cristian-Ioan Vasile, Mingyu Cai

Abstract

Executing complex manipulation in cluttered environments requires satisfying coupled geometric and temporal constraints. Although Spatio-Temporal Logic (SpaTiaL) offers a principled specification framework, its use in gradient-based optimization is limited by non-differentiable geometric operations. Existing differentiable temporal logics focus on the robot's internal state and neglect interactive object-environment relations, while spatial logic approaches that capture such interactions rely on discrete geometry engines that break the computational graph and preclude exact gradient propagation. To overcome this limitation, we propose Differentiable SpaTiaL, a fully tensorized toolbox that constructs smooth, autograd-compatible geometric primitives directly over polygonal sets. To the best of our knowledge, this is the first end-to-end differentiable symbolic spatio-temporal logic toolbox. By analytically deriving differentiable relaxations of key spatial predicates--including signed distance, intersection, containment, and directional relations--we enable an end-to-end differentiable mapping from high-level semantic specifications to low-level geometric configurations, without invoking external discrete solvers. This fully differentiable formulation unlocks two core capabilities: (i) massively parallel trajectory optimization under rigorous spatio-temporal constraints, and (ii) direct learning of spatial logic parameters from demonstrations via backpropagation. Experimental results validate the effectiveness and scalability of the proposed framework.Code Available: https://github.com/plen1lune/DiffSpaTiaL

Differentiable SpaTiaL: Symbolic Learning and Reasoning with Geometric Temporal Logic for Manipulation Tasks

Abstract

Executing complex manipulation in cluttered environments requires satisfying coupled geometric and temporal constraints. Although Spatio-Temporal Logic (SpaTiaL) offers a principled specification framework, its use in gradient-based optimization is limited by non-differentiable geometric operations. Existing differentiable temporal logics focus on the robot's internal state and neglect interactive object-environment relations, while spatial logic approaches that capture such interactions rely on discrete geometry engines that break the computational graph and preclude exact gradient propagation. To overcome this limitation, we propose Differentiable SpaTiaL, a fully tensorized toolbox that constructs smooth, autograd-compatible geometric primitives directly over polygonal sets. To the best of our knowledge, this is the first end-to-end differentiable symbolic spatio-temporal logic toolbox. By analytically deriving differentiable relaxations of key spatial predicates--including signed distance, intersection, containment, and directional relations--we enable an end-to-end differentiable mapping from high-level semantic specifications to low-level geometric configurations, without invoking external discrete solvers. This fully differentiable formulation unlocks two core capabilities: (i) massively parallel trajectory optimization under rigorous spatio-temporal constraints, and (ii) direct learning of spatial logic parameters from demonstrations via backpropagation. Experimental results validate the effectiveness and scalability of the proposed framework.Code Available: https://github.com/plen1lune/DiffSpaTiaL

Paper Structure

This paper contains 29 sections, 2 theorems, 15 equations, 8 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. Differentiable Temporal Logic and Robustness
  4. Differentiable Collision Detection and Geometry
  5. Learning and Inference of Logical Specifications
  6. Problem Formulation
  7. Signal Temporal Logic
  8. SpaTiaL: Geometry-based Spatial Predicates
  9. Differentiable Problem Formulation
  10. Methodology
  11. Overview of Tensorized Spatial Semantics
  12. Differentiable Penetration via Smooth SAT
  13. Smooth Approximations of Geometric Distances
  14. Compositional Spatial Predicates
  15. Enclosure and Containment
  16. ...and 14 more sections

Key Result

Lemma 1

For any vector $\mathbf{x} \in \mathbb{R}^N$ and temperature $\tau > 0$: $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure A1: Overview of Differentiable SpaTiaL. Our framework replaces discrete geometry engines with a fully tensorized architecture, enabling end-to-end trajectory optimization under formal specifications.
  • Figure D1: Smooth spatial and temporal operators form a differentiable robustness computation graph.
  • Figure D2: Differentiable penetration depth via Smooth SAT. Exact discrete overlap is relaxed into a smooth, differentiable repulsion field.
  • Figure D3: Differentiable Signed Distance Field (SDF) via boundary sampling. Soft aggregation ensures informative gradients in both colliding and non-colliding states.
  • Figure D4: Geometric interpretation of compositional predicates (left: EnclIn, right: leftOf). Spatial relations are resolved as differentiable algebraic operations over vertex tensors.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Definition 1: SpaTiaL Atomic Predicates pek2023spatial
  • Definition 2: Quantitative Semantics
  • Lemma 1
  • Theorem 1
  • proof