Signal Temporal Logic

Falsification searches for counterexamples to formal specifications in cyber-physical systems (CPS). With specifications written in Signal Temporal Logic (STL), falsification can be formulated as a robustness optimization problem, traditionally tackled with black-box search algorithms. In parallel, large language models (LLMs) have recently emerged as surprisingly effective optimizers when coupled with iterative prompting. In this work, we connect these ideas and introduce LLM-Falsifier, an LLM-based approach that falsifies specifications by minimizing the STL robustness degree. Beyond generic prompt-based optimization, our key idea is to expose the LLM to semantic information that is natural for language models but absent from standard numerical optimizers, including natural-language input and output names, output trajectories, and critical-time witnesses for the minimum robustness value. These additions enable smarter and more sample-efficient robustness search. On the ARCH-COMP falsification benchmarks, LLM-Falsifier is shown to outperform existing falsification tools based on a range of optimization paradigms, from surrogate-based and Bayesian optimization to search-based testing, on 14 of 21 specifications when measured by the average number of simulations required to find a counterexample.

Ensuring the satisfaction of Signal Temporal Logic (STL) specifications under uncertainty is challenging, as reachability-based monitoring provides guarantees but does not indicate how to restore satisfaction when it becomes indeterminate. A key difficulty is identifying which uncertain components actually affect global satisfaction, especially for nested formulas. This paper introduces a logical dependency tracking framework that propagates uncertainty through the STL structure and captures the causal contribution of reachable sets to satisfaction. By associating markers to uncertain predicates and propagating them via three-valued semantics, we extract in milliseconds a compact Disjunctive Normal Form (DNF) of sufficient constraints, avoiding combinatorial enumeration. As an application, we formulate control correction as a minimum-effort optimization problem. Using zonotopic reachability, the derived constraints are enforced via linear programming, yielding corrections that guarantee STL satisfaction under bounded uncertainty and provide certified probabilistic bounds in the stochastic case. We demonstrate the approach on a nonlinear system with nested STL specifications, showing that dependency tracking enables efficient and formally guaranteed correction. The tracking implementation is available at https://github.com/Antoine-Bst/STL-Three-Valued-Clause-Filtering/.

Automated mining of formal specifications is vital for verifying real-time systems. However, existing passive learning approaches remain restricted to deterministic specifications or limited fragments of Timed Regular Expressions (TRE). To our knowledge, this paper presents the first framework to tackle \emph{precise} passive learning for an expressive timed logic, \emph{Metric Interval Temporal Logic} (MITL) without relying on predefined templates or restricted logic fragments. Our approach formally reduces the timed learning problem into a scalable untimed one. By identifying quantitative timing differences between positive and negative traces, we synthesise precise timed constraints and inject them as new Boolean atomic propositions. This embeds timing into the alphabet, delegating the complex formula evaluation to highly optimised, off-the-shelf untimed LTL tools. Crucially, our framework is complete, guaranteeing a separating specification can always be found. We evaluate our implementation across several benchmarks, demonstrating the effectiveness of our approach.

Translating natural language instructions into machine-interpretable formal specifications enables robots and autonomous systems to plan, reason, and formally verify their behavior. However, existing translation models typically generate a specification for every input, even when the result is unreliable or fails to capture the user's intent, creating risks in safety-critical applications. Inspired by selective conformal prediction, we propose a selective translation framework that not only generates formal specifications but also determines when they can be trusted. Reliability is scored by two complementary black-box signals, the fidelity of the specification back-translated into natural language and the dispersion of repeated translations under exact semantic equivalence, which fail on different errors and jointly separate incorrect translations more sharply than either alone. Conformal risk control calibrates this score into a decision that accepts a specification or abstains, with a distribution-free bound on the rate at which incorrect specifications are accepted for execution, and a conformal anomaly detector on instruction embeddings screens out-of-distribution inputs before any translation is attempted. The proposed framework is general across formal specification languages, with experiments on Signal Temporal Logic (STL), Linear Temporal Logic (LTL), and geometric Spatio-Temporal Logic (SpaTiaL) demonstrating improved translation reliability, robustness under the evaluated cross-tier shifts, and effective uncertainty-aware abstention. This work establishes a foundation for trustworthy natural language interfaces by enabling AI systems to recognize when generated specifications may not be reliable.

Temporal logic (TL) provides a compositional language for the formulation of long horizon robotic tasks, but existing TL-conditioned trajectory generators can sidestep perception-to-symbol binding by encoding exact object geometry in the task graph. We introduce \emph{Vision-TL-Action}, which generates action trajectories from multi-view images, a coordinate-free TL syntax graph, and the robot initial state. TL-node tokens and spatial visual tokens are fused through bidirectional cross-attention, and the resulting representation conditions a flow-matching trajectory generator. Visual tokens are augmented only with normalized image-plane locations and camera-view identifiers, while a training-only predicate-to-region objective encourages grounding to referenced objects. Consistent with prior work in this domain, we evaluate the model using Success@$K$, the fraction of tasks for which at least one of K sampled trajectories satisfies the TL specification. On Panda task, our model achieves 67.45% Success@1024, compared with 59.11% for the oracle-state baseline. On AntMaze task, it achieves 96.35% Success@256, comparable to the oracle result of 96.88%. Resolution and intervention studies show that spatial detail depends on semantic grounding and predicate identity affects both attention and performance. These results demonstrate a direct mapping from visual observations and structured TL goals to action trajectories without requiring object geometry at inference. Code is available at https://github.com/AricLau07/vision-tl-action.

We present a unified trajectory planning and control framework for the satisfaction of Signal Temporal Logic (STL) specifications defined over convex predicates. At the planning layer, STL tasks are encoded as time-varying convex sets in configuration space, specifically designed so that forward invariance of the system with respect to these sets implies satisfaction of the specification with a prescribed robustness margin. This representation is then lifted to the joint time--configuration space and combined with the Graphs of Convex Sets (GCS) framework, yielding a shortest-path formulation of the planning problem over convex spatio-temporal sets. Trajectories are parameterized by B-splines, which enable continuous-time enforcement of STL satisfaction, collision avoidance, and smoothness constraints. At the control layer, the same time-varying sets used for planning are exploited to design a feedback controller that tracks the planned trajectory while prioritizing satisfaction of the STL specification during execution in the presence of tracking errors and model mismatch. We validate the proposed approach in simulation and in real-world experiments on space robotic platforms.

Behavior Trees (BTs) are widely adopted for complex task execution in robotics, providing modular, reactive control but lacking formal guarantees. However, existing correct-by-construction synthesis from Linear Temporal Logic (LTL) cannot express quantitative timing constraints. This letter synthesizes correct-by-construction BTs from Signal Temporal Logic (STL) specifications. The workspace is modeled as a timed transition system and abstracted into a zone graph, and an augmented state space tracking both logical progress and timing constraints is introduced. A hierarchical fixed-point algorithm computes winning sets for an STL fragment encompassing safety, reachability, response, recurrence, and persistence, yielding BT subtrees with a runtime constraint function. Correctness guarantees are proven and complexity bounds are derived. Simulations demonstrate specification satisfaction with strictly positive robustness, and a physical quadrotor experiment with six STL specifications validates practical deployability.

Vision-language-action (VLA) models have shown impressive generalization, but often lack interpretability and can struggle to follow precise natural language instructions that encode spatial, temporal, and logical requirements. We propose a hierarchical framework that uses Signal Temporal Logic (STL) as a shared representation connecting high-level language understanding with low-level robot execution. A high-level policy leverages a VLM to decompose language instructions into high-level subtasks, generate STL specifications for each subtask, and choose a low-level policy for executing each subtask. The STL specifications translate language-derived intent into precise constraints, and the low-level policy selection determines whether those constraints are enforced directly through STL-guided model-predictive control or monitored during execution of a learned policy for perceptually complex, or contact-rich behaviors. By integrating STL into plan validation, low-level policy, subtask monitoring, and replanning, our framework enables language-derived plans to be checked, optimized, and revised at runtime using a common formal structure. We evaluate the approach on a real-world tabletop domain, demonstrating how formal specifications can improve the precision, reliability, and interpretability of language-conditioned robot planning.

Signal Temporal Logic (STL) is increasingly used to describe interpretable objectives and constraints for optimal control and learning methods, especially when no target time series data is available. In this work, we propose to synthesize parameters for nonlinear systems that robustly satisfy continuous-time STL specifications for uncertain initial conditions. To this end, we use gradient-based optimization along with set-based reachability verification to efficiently learn in high-dimensional parameter spaces while providing provable satisfaction guarantees for the optimized parameters. We demonstrate the effectiveness and scalability of our method on three systems with up to 18 parameter dimensions.

Reinforcement learning (RL) for quadruped locomotion commonly depends on fixed, hand-crafted, and Markovian reward functions that limit both interpretability of learned policies and lack explicit control over gait behaviors. We introduce a framework where distinct gaits are specified using parameterized constraints expressed in Signal Temporal Logic (STL). These include safety bounds, gait synchronization constraints, command tracking, and actuation bounds. From these specifications, we develop a reward shaping mechanism that provides learning agents a dense, continuous reward landscape that encodes desired behavior. We define parametric STL templates for three speed regimes (walking-trot, trot, bound), calibrate their parameters from reference rollouts, and compute rewards from using smooth approximations of STL robustness over the rollouts. The generated rewards can be used to provide shaped gradients compatible with Proximal Policy Optimization (PPO). We instantiate the approach on Google's Barkour quadruped robot in MuJoCo XLA (MJX). We use parallelization within the simulator to improve training speeds and use domain randomization to robustify learned policies. We show that compared to a baseline of hand-crafted rewards, the STL-shaped rewards yield tighter velocity tracking and more stable training. Videos can be found on our project website: https://stl-locomotion.github.io/.

Formal specification is a powerful tool to guide the learning process and provides significant advantages over reward shaping: (1) mathematical rigor; (2) expressiveness to specify objectives and constraints, and (3) the ability to define tactics to achieve objectives. However, these benefits remain largely unexplored in the context of Multi-Agent Reinforcement Learning (MARL). This paper introduces HyPOLE, a novel framework for MARL under partial observability, where learning is guided by the expressive power of the so-called hyperproperties and, in particular, the temporal logic HyperLTL. We integrate Centralized Training for Decentralized Execution (CTDE) techniques with HyPOLE to synthesize decentralized policies, and our evaluation on SMAC, MessySMAC, and WildFire benchmark demonstrates clear advantages over baselines.

Spatio-Temporal Logic with Graph Operators (STL-GO) extends Signal Temporal Logic (STL) to multi-agent systems via graph operators that count neighboring agents satisfying a property, together with multi-agent quantifiers. While Boolean semantics for STL-GO are well-defined, quantitative semantics have not yet been developed and existing quantitative semantics for spatio-temporal logics such as STREL cannot capture the counting constraints in STL-GO's graph operators. We develop quantitative semantics for STL-GO as a layered algebraic construction that separates temporal aggregation from graph-operator aggregation (governed by an abstract accumulator with a monotone fold and readout). We prove that soundness and completeness reduce to monotonicity conditions on these components. We implement the framework and evaluate it on two multi-agent environments: a 2D bounded region with stochastic Dubins-car dynamics and a 3D Earth-satellite system, under four semantic instantiations (Boolean, min-max, signed-deficit, and a hybrid), demonstrating the tradeoffs between accumulator choices and reporting scalability in the number of agents and time horizon.

Diffusion policies enable multimodal robot behavior but offer limited ability to choose among behavior modes at inference time, even though such control is desirable in human-robot settings. Prior solutions to this lack of control have utilized Signal Temporal Logic (STL) to express human intentions and provide corresponding guidance for diffusion policy inference. However, these approaches can only guide diffusion policies that jointly generate future actions and states, increasing both complexity and runtime. We propose a novel guidance method for action-only diffusion policies that uses a separate learned world model to enable differentiable evaluation of STL robustness, with its gradient then injected into the diffusion process. This steers behavior toward constraint satisfaction without retraining, improving constraint adherence while preserving task performance. On the Can Transport task from Robomimic, our method maintains 100% task success while reducing constraint violations from over 80% for baseline methods to 4%. We also discuss extensions toward improved robustness and more complex constraints.

As robots move from controlled environments into real-world settings, it becomes increasingly crucial to ensure that they perform as expected. A key step toward that goal is a rigorous specification of the desired robot behavior, capturing intricate temporal, spatial, and logical requirements. Complementing this, plan and control synthesis methods are needed to fulfill these specifications with provable guarantees. This manuscript presents temporal logics - particularly linear and signal temporal logic - as expressive specification languages for robot behavior over time. We then discuss principles of formal synthesis, from discrete graph- and game-based approaches to sampling-based motion planning, trajectory optimization, and control-certificate-based synthesis. Finally, we outline challenges in deploying formal synthesis in real-world robotics, emphasizing the interplay between modeling fidelity, computational tractability, and the types of rigorous guarantees that can be achieved.

Autonomous vehicles must plan trajectories that satisfy a multitude of requirements on safety, passenger comfort, and compliance with traffic rules. However, in safety-critical scenarios, it is not always possible to satisfy all requirements simultaneously, necessitating their prioritization based on importance. At the same time, in these safety-critical scenarios, the uncertainty in trajectory predictions of the surrounding traffic, such as other vehicles and pedestrians, should be explicitly accounted for. In this work, we propose an uncertainty-aware trajectory planning framework that incorporates a predefined lexicographic ordering over Signal Temporal Logic (STL) specifications that stays valid under uncertainty. We implement this formulation with Model Predictive Path Integral (MPPI) control and we demonstrate the effectiveness of our method on simulation scenarios, showing that our framework efficiently handles conflicting objectives under realistic multi-modal uncertainty.

Autonomous robots operating in uncertain environments must satisfy complex temporal and safety specifications despite stochastic dynamics and sensing noise. While Signal Temporal Logic (STL) offers robustness measures for gradient-based optimization, existing extensions either lack differentiability or ignore belief-space uncertainty. We introduce pdSTL (probabilistic differentiable Signal Temporal Logic), a framework that unifies probabilistic semantics with differentiable robustness over belief trajectories. pdSTL employs interval-valued probabilistic semantics to compute conservative satisfaction bounds, propagated compositionally through the STL syntax tree. We formulate the temporal robustness evaluation as a recurrent, LSTM-style unfolding of STL operators, enabling linear-time, differentiable monitoring suitable for end-to-end trajectory optimization. We validate pdSTL on simulated obstacle avoidance, lane-change maneuvers, and real-world Crazyflie quadcopter flight experiments under aerodynamic disturbances. Results demonstrate that pdSTL achieves efficient optimization with formal probabilistic guarantees, significantly outperforming deterministic differentiable STL in maintaining safety margins under real-world uncertainty.

We present HierSVA, an integrated suite that combines a pipeline, dataset, and benchmark for LLM-driven hierarchical hardware formal verification. HierSVA-SP pairs an RTL preprocessing toolchain with an LLM-in-the-loop formal verification flow to produce reference SystemVerilog Assertions (SVA) on hierarchical RTL. Applying it to BaseJump STL yields HierSVA-DS, a dataset of 342 modules, with hierarchy metadata and depths 0--9, accompanied by a deep subset of 28 module-bug pairs with natural-language specifications and bug variants. HierSVA-B decomposes assertion quality into six metric axes: syntax correctness, assertion proof success rate, vacuity, specification faithfulness, mutation coverage, and formal core coverage. Applying HierSVA-B to twelve recent LLMs reveals three findings. First, the module-level compile rate is 67.1%; among generated assertions in evaluable runs, 82.1% prove non-vacuously, but the corresponding assertion sets detect only 70.2% of eligible injected faults and cover 36.2% of the formal core. Second, on 211 evaluable model--module entries in the deep subset, assertion sets flag buggy RTL with 0.87 recall, but 40% of predicted-buggy outcomes are false positives on correct RTL, limiting precision to 0.60. Third, agentic mode improves S1-style provability and strength metrics, but gains plateau and oscillate. Codes and artifacts are available at \href{https://github.com/HierSVAAnon/HierSVACodeAndArtifacts}{https://github.com/HierSVAAnon/HierSVACodeAndArtifacts}. Dataset is available at \href{https://huggingface.co/datasets/AnonymousHierSVA/HierSVA}{https://huggingface.co/datasets/AnonymousHierSVA/HierSVA}.

The development of temporal extensions of Answer Set Programming (ASP) has led to the emergence of non-monotonic linear-time (TEL), dynamic (DEL), and metric (MEL) temporal equilibrium logics. However, the inherent rigidity of highly optimized ASP systems often hinders the rapid exploration and implementation of alternative logical designs. In this work, we propose a flexible meta-programming framework that operationalizes the semantics of varied temporal logics through a unified, declarative framework. Our approach extends standard ASP meta-programming by augmenting clingo's theory grammar with formal type specifications and nesting capabilities. To ensure semantic correctness, we introduce a transformation pipeline that protects nested modalities from stable-model-based simplifications during grounding. We demonstrate the extensibility of our framework by implementing meta-encodings for TEL, MEL, and DEL. We provide a comprehensive account of TEL and highlight the key features for managing the interval constraints of MEL and the Fischer-Ladner closure in DEL. Finally, we introduce the metasp system, a versatile tool that encapsulates this workflow.

Signal Temporal Logic (STL), has recently seen extensive development, owing to its rich expressivenes for autonomous planning and control. Nevertheless, existing verification and control synthesis methods are limited with respect to the complexity and degree of nesting of the formulae. In this work, we propose a novel approach to STL based on an operator acting on reachability value functions. This constitutes a new theoretical framework for handling complex multi-nested formulae while at the same time providing tools for on-line control synthesis. In contrast to focusing on the design of STL-based reachability (or control barrier) functions, we develop operator-based nesting rules directly. Our method's expressiveness is demonstrated both theoretically, where necessary and sufficient conditions for STL formula satisfaction are extracted, as well as in simulations with complex fragments.

This paper investigates continuous-time motion planning under Signal Temporal Logic (STL) specifications. The goal is to generate smooth robot trajectories that satisfy high-level logical and timing requirements while respecting low-level motion constraints. To this end, we propose an efficient framework that combines timed-automata reasoning with graphs of convex sets (GCS). An STL specification is first represented by a timed automaton, which is then coupled with a convex decomposition of the configuration space to form a joint transition system encoding both task progress and region occupancy. Based on this joint transition system, the STL motion-planning problem is reformulated as a shortest-path problem over a GCS, whose solution induces a smooth Bézier-spline trajectory satisfying the STL specification, smoothness requirements, and velocity bounds. We establish the soundness of the proposed formulation and analyze its computational complexity, showing that, once the timed automaton and convex decomposition are fixed, the convex relaxation scales polynomially with the configuration-space dimension and the Bézier degree. We further develop a compact timed-automaton construction for an expressive STL fragment using dedicated templates and Boolean composition. Numerical experiments on low-dimensional benchmarks, a $3$-D quadrotor, a $30$-DoF humanoid, and a hardware experiment on a UR-3 robot arm demonstrate that the proposed method efficiently solves complex STL motion-planning problems and produces smooth executable trajectories.

Runtime monitoring of autonomous systems traditionally relies on mapping continuous sensor observations to discrete logical propositions defined over low-dimensional state variables. This abstraction breaks down in perception-driven settings, where such mappings require additional learned modules that are often computationally expensive, brittle, and semantically misaligned. In this work, we propose Embedding Temporal Logic (ETL), a temporal logic that performs monitoring directly in learned embedding spaces. ETL defines predicates through distances between observed embeddings and target embeddings derived from reference observations. This formulation allows specifications to capture high-level perceptual concepts, such as similarity to visual goals or avoidance of semantic regions, that are difficult or impossible to express using traditional predicates. By composing these predicates with temporal operators, ETL naturally expresses temporally extended and sequential perceptual behaviors. We introduce ETL monitors for evaluating specifications over bounded embedding traces, along with a conformal calibration procedure that provides reliable and safety-oriented predicate evaluation. We evaluate our approach across multiple manipulation environments to show that ETL achieves strong empirical agreement with ground-truth semantics, including accurate monitoring of temporally composed behaviors.

Robot swarms promise scalable assistance in complex and hazardous environments. Task planning lies at the core of human-swarm collaboration, translating the operator's intent into coordinated swarm actions and helping determine when validation or intervention is required during execution. In long-horizon missions under dynamic scenarios, however, reliable task planning becomes difficult to maintain: emerging events and changing conditions demand continual adaptation, and sustained operator oversight imposes substantial cognitive burden. Existing LLM-based planning tools can support plan generation, yet they remain susceptible to invalid task orderings and infeasible robot actions, resulting in frequent manual adjustment. Here we introduce a neuro-symbolic framework for long-horizon human-swarm collaboration that tightly melds verifiable task planning with context-grounded LLM reasoning. We formalize mission goals and operational rules as temporal logic formulas and admissible task orderings as task automata. Conditioned on these formal constraints and live perceptual context, LLMs generate executable subtask sequences that satisfy mission rules and remain grounded in the current scene. An uncertainty-aware scheduler then assigns subtasks across the heterogeneous swarm to maximize parallelisms while remaining resilient to disruptions. An event-triggered interaction protocol further limits operator involvement to sparse, high-level confirmation and guidance. Deployment on a heterogeneous robotic fleet yields similar results while remaining robust to hardware-specific actuation and communication uncertainties. Together, these results support a formal and scalable paradigm for reliable and low-overhead human-swarm collaboration in dynamic environments

Signal Temporal Logic (STL) is an expressive formal language for specifying spatio-temporal requirements over real-valued, real-time signals. It has been widely used for the verification and synthesis of autonomous systems and cyber-physical systems. In practice, however, users often express their requirements in natural language rather than in structured STL formulas, making natural-language-to-STL translation a critical yet challenging task. Manual specification requires temporal-logic expertise and cannot scale, while prompting commercial LLM APIs incurs substantial token costs and may expose sensitive system requirements to third-party services, raising privacy concerns for industrial deployment. To address these challenges, we present \textsc{ReasonSTL}, a tool-augmented framework that adapts local open-source language models for natural-language-to-STL generation. \textsc{ReasonSTL} decomposes the translation process into explicit reasoning, deterministic tool calls, and structured formula construction. We further introduce process-rewarded training to supervise both tool-use trajectories and final formulas, together with \textsc{STL-Bench}, a bilingual, computation-aware benchmark grounded in real-world signals. Experiments show that a 4B model trained with \textsc{ReasonSTL} achieves state-of-the-art performance in both automatic metrics and human evaluations, demonstrating that \textsc{ReasonSTL} provides a transparent, low-cost, and privacy-preserving alternative for formal specification drafting.

We propose an architecture for integrating high-level, human-provided safety rules and operator-aligned semantic preferences into autonomous robot navigation in unstructured outdoor environments. In our approach, natural-language rules are translated into Signal Temporal Logic (STL) specifications that guide planning and navigation during runtime. Persistent, environment-centric rules and terrain preferences are grounded into a 2D cost map, while temporally dynamic requirements are expressed as STL specifications to be monitored during runtime. We hypothesize the use of Vision-Language Models (VLMs) for zero-shot scene understanding, enabling mapping between human instructions, semantic features, and environmental constraints. Within this framework, we construct an illustrative navigation model that is designed to satisfy a set of STL-encoded specifications and soft operator preferences through formal satisfaction metrics embedded into environmental properties and runtime monitoring.

In this work, a novel method for planar task and motion planning based on hybrid modeling is proposed. By virtue of a discrete variable which models local constraint satisfaction and enables local feasibility analysis, the proposed control architecture unifies planning with control design. Concurrently, control barrier functions are designed on a transformed disk version of the original nonconvex and geometrically complex robotic workspace, thus amending the issue of deadlocks. Simulations of the proposed method indicate effective handling of multiple overlapping spatio-temporal tasks even in the face of input saturation.

Several applications demand the timely detection of critical situations, such as threats to safety and transparency, over high-velocity streams of symbolic events. This demand has motivated the development of (i) event specification languages, which define composite events via temporal patterns over simpler events, and (ii) stream reasoning frameworks, evaluating patterns expressed in these languages. However, event specification languages are typically studied in isolation, complicating their comparison in terms of expressivity and obscuring the scope of their associated stream reasoners. To mitigate this issue, we map practical fragments of prominent event specification languages into Temporal Datalog->-, a temporal Datalog with stratified negation and no future dependencies. To support efficient stream reasoning over Temporal Datalog->-, we propose Streaming Trigger Graphs, an extension of a state-of-the-art technique for Datalog materialisation. Our approach yields a uniform composite event recognition mechanism that has the potential to generalise across a wide range of practical event specification languages.

We study feedback motion planning for continuous-time stochastic nonlinear systems under signal temporal logic (STL) specifications. We propose a framework that synthesizes control policies for chance-constrained STL trajectory optimization problems, with the goal of ensuring that the closed-loop stochastic system satisfies a given STL formula with high probability (e.g., 99.99%). Our approach is based on a predicate erosion strategy that transforms the intractable stochastic problem into a deterministic STL trajectory optimization problem with tightened STL formula constraints. The amount of erosion is determined by a probabilistic reachable tube (PRT) that bounds the deviation between the stochastic trajectory and an associated nominal trajectory. To compute such bounds, we leverage contraction theory and feedback design, and develop several tracking controllers. This yields a complete feedback motion planning pipeline which can be implemented by numerical optimizations. We demonstrate the efficacy and versatility of the proposed framework through simulations on several robotic systems and through experiments on a real-world quadrupedal robot, and show that it is less conservative and achieves higher specification satisfaction probability than representative baselines.

Signal Temporal Logic (STL) offers verifiable task specifications and is crucial for safety-critical control. Yet STL planning remains challenging: exact optimization-based methods are often too slow, and learning-based methods struggle to generalize across varying environments. We propose a zero-shot STL planning solver for variable-map environments that generates feasible trajectories without retraining. By integrating a map-conditioned Transformer architecture with a lightweight heuristic, our approach effectively handles complex disjunctive (OR) subformulas. Furthermore, we leverage Transitive Reinforcement Learning (TRL) to ensure consistent temporal grounding and logical coherence across decomposed sub-tasks. Experiments on dynamic semantic maps with diverse obstacle layouts demonstrate consistent gains, highlighting the framework's superior zero-shot generalization to changing environments and broad STL coverage.

While Bellman equations for basic reach, avoid, and reach-avoid problems are well studied, the relationship between value optimality and policy optimality becomes subtle in the undiscounted infinite-horizon setting, particularly for more complicated tasks. Greedily maximizing the Q-function can produce policies that indefinitely defer task completion for reach-avoid problems, or equivalently, Until specifications, even when the value function is optimal. Building upon recent results decomposing the value function for temporal logic (TL) into a graph of constituent value functions, we construct non-Markovian policies based on state history that avoid this pathology and prove their optimality with respect to the quantitative robustness score for nested Until, Globally, and Globally-Until specifications. We further show how the Q function can serve as a safety filter for complex TL specifications, extending prior results beyond simple avoid or reach-avoid tasks.

We introduce a new family of temporal logics intended for specifications in motion planning (MP). It builds upon the signal temporal logic (STL), which is a linear-time logic over real-valued signals that possess quantitative semantics and thus became popular in the areas of cyber-physical systems, robotics, and specifically robot MP. However, in contrast to STL, the proposed logic works with paths as first-class citizens, separating the concerns of geometry and of logic. This in turn leads to simpler and more understandable formulae, and a more refined notion of satisfaction being able to reflect also preferences over behaviours. Technically, the logic is built on fuzzy, time-varying signal constraints. As a consequence of this expressivity, it is (i) more usable for human-given specifications in MP and (ii) more amenable to learning specifications from demonstrations than other logics. The former is important for the traditional style of verification in robot MP; the latter is becoming recognized as crucial for mining data-given tasks and controller synthesis in human-aware MP. We expose the advantages of our proposed logic on examples and show the versatility and flexibility of the framework on a number of scenarios. Finally, we give a learning algorithm with a prototype implementation and discuss the possibilities of model checking and monitoring.

Motion planning for autonomous vehicles often requires satisfying multiple conditionally conflicting specifications. In situations where not all specifications can be met simultaneously, minimum-violation motion planning maintains system operation by minimizing violations of specifications in accordance with their priorities. Signal temporal logic (STL) provides a formal language for rigorously defining these specifications and enables the quantitative evaluation of their violations. However, a total ordering of specifications yields a lexicographic optimization problem, which is typically computationally expensive to solve using standard methods. We address this problem by transforming the multi-objective lexicographic optimization problem into a single-objective scalar optimization problem using non-uniform quantization and bit-shifting. Specifically, we extend a deterministic model predictive path integral (MPPI) solver to efficiently solve optimization problems without quadratic input cost. Additionally, a novel predicate-robustness measure that combines spatial and temporal violations is introduced. Our results show that the proposed method offers an interpretable and scalable solution for lexicographic STL minimum-violation motion planning within a single-objective solver framework.

Signal Temporal Logic (STL) is a powerful language for specifying temporally structured robotic tasks. Planning executable trajectories under STL constraints remains difficult when system dynamics and environment structure are not analytically available. Existing methods typically either assume explicit models or learn task-specific behaviors, limiting zero-shot generalization to unseen STL tasks. In this work, we study offline STL planning under unknown dynamics using only task-agnostic trajectory data. Our central design philosophy is to separate logical reasoning from trajectory realization. We instantiate this idea in DAG-STL, a hierarchical framework that converts long-horizon STL planning into three stages. It first decomposes an STL formula into reachability and invariance progress conditions linked by shared timing constraints. It then allocates timed waypoints using learned reachability-time estimates. Finally, it synthesizes trajectories between these waypoints with a diffusion-based generator. This decomposition--allocation--generation pipeline reduces global planning to shorter, better-supported subproblems. To bridge the gap between planning-level correctness and execution-level feasibility, we further introduce a rollout-free dynamic consistency metric, an anytime refinement search procedure for improving multiple allocation hypotheses under finite budgets, and a hierarchical online replanning mechanism for execution-time recovery. Experiments in Maze2D, OGBench AntMaze, and the Cube domain show that DAG-STL substantially outperforms direct robustness-guided diffusion on complex long-horizon STL tasks and generalizes across navigation and manipulation settings. In a custom environment with an optimization-based reference, DAG-STL recovers most model-solvable tasks while retaining a clear computational advantage over direct optimization based on the explicit system model.