Automaton-Based Plan Supervision

• Automaton-based plan supervision is a formal framework that uses symbolic automata to specify, enforce, and verify task execution with formal temporal and safety guarantees.
• It integrates quantitative objectives through matrix encodings and gradient-based optimization, supporting reinforcement learning and scalable control.
• Applications span hybrid motion planning, LLM-based agent planning, and discrete event systems, ensuring both practical execution and formal correctness.

Automaton-based plan supervision formalizes the specification, enforcement, and verification of task execution or agent behavior using automata-theoretic representations. This approach unifies high-level logical objectives and temporal sequencing with the algorithmic and runtime properties of automata, supporting both symbolic synthesis and efficient, scalable supervision mechanisms. Automaton-based supervisors are prevalent in contemporary research on motion planning, discrete event systems, temporal logic control, reinforcement learning, and AI-driven planning. The following sections elucidate the core formal models, quantitative embedding, algorithmic techniques, and operational guarantees that underpin automaton-based plan supervision.

1. Symbolic Automata as Plan Supervisors

Automaton-based supervision encodes desired plan properties and task structures as symbolic automata. A generic symbolic automaton used for plan supervision over continuous or hybrid state spaces is defined as

$\mathcal{A} = (\Sigma, Q, Q_0, Q_F, \Delta)$

where

• $\Sigma$ is typically the system or robot state space $S^n$,
• $Q$ is the finite set of automaton locations,
• $Q_0 \subseteq Q$ are initial states, $Q_F \subseteq Q$ are accepting (goal) states,
• $\Delta : Q \times Q \to \Phi$ assigns each transition a predicate $\varphi \in \Phi$ over $\Sigma$.

The predicates are built from atomic smooth conditions (e.g., $\mu(x) \geq 0$ for region or safety properties) closed under conjunction and disjunction. A finite input trace $\xi = (x_0, ..., x_L) \in \Sigma^*$ induces an automaton run if and only if each $x_t$ upholds the transition predicate, with acceptance if $q_L \in Q_F$ (Balakrishnan et al., 2024).

This abstract structure generalizes the supervisory automata used in diverse frameworks: Büchi automata for reactive manipulation under temporal logic (Vasilopoulos et al., 2020), pushdown automata for LLM-driven plan validation (Li et al., 2024), DFAs for reward or preference structure in RL (Alinejad et al., 17 Oct 2025), and Mealy transducers or discrete-event automata for plant-level supervisory control (Felli et al., 2016, Mohamadkhani et al., 2023, Partovi et al., 2018).

2. Quantitative Objectives and Matrix Encodings

Recent advancements address the challenge of integrating symbolic automata with optimization and learning. The matrix-operator architecture in (Balakrishnan et al., 2024) expresses every automaton as a sequence of weighted transition matrices

$A(x)_{i,j} = \lambda(x, \Delta(q_i, q_j)) \in K$

where $\lambda$ evaluates the logical predicate at $x$ in a semiring $K$. The cost or "generalized robustness" of a trajectory is computed as

$$w_{\mathcal{A}}(\xi) = \alpha^\top A(x_0) \cdots A(x_L) \beta$$

with $\alpha, \beta$ encoding initial and accepting locations. In max-plus or min-max semirings, this construction generalizes signed distances, STL robustness, and similar quantitative temporal objectives.

For reinforcement learning, the automaton state (typically product with MDP state) supports the construction of non-Markovian reward functions: a DFA processes the trace, and trajectory-level preferences or subgoal completions are converted to dense or trajectory-wise reward signals (Alinejad et al., 17 Oct 2025).

3. Algorithmic Approaches and Differentiable Planning

Automaton-based plan supervision enables integration with modern optimization and control toolchains. The matrix-product encoding lends itself to automatic differentiation for gradient-based planning:

• Compute the cost function $w_{\mathcal{A}}(\xi)$ as a compute graph.
• Use chain-rule expansion to differentiate with respect to control sequences $u_t$, utilizing adjoint matrix products and derivative propagation.
• This enables trajectory optimization via standard first-order optimizers (PyTorch, JAX).

The two principal planning approaches are:

• Open-loop optimization: solve for the entire plan by gradient descent over a fixed horizon.
• Model Predictive Control (receding horizon): at each cycle, forward-propagate automaton state summaries, solve for the next segment, and replan after each step. The automaton state reduces trajectory history storage to $O(|Q|)$ per step (Balakrishnan et al., 2024).

In the context of reinforcement learning, automaton supervision supports both static and dynamic variants:

• Static: fit a reward function (via margin ranking loss) to automaton-induced preferences and optimize policies using classical RL.
• Dynamic: iteratively refine both policy and reward function using automaton feedback and trajectory comparisons until convergence, yielding near-optimal policies for temporally extended tasks (Alinejad et al., 17 Oct 2025).

4. Applications: Hybrid Control, RL, and LLM Plan Validation

Automaton-based plan supervision is widely deployed:

Hybrid planning and manipulation

Symbolic automata derived from LTL or task-tuned logics orchestrate the composition of symbolic subtasks (e.g., MOVE, GRASP, RELEASE) and coordinate their continuous realization in environments with unknown obstacles or object configurations. Reactive supervisors fuse discrete automaton progress graphs with geometric controllers, achieving completeness and reactivity in unstructured domains (Vasilopoulos et al., 2020).

LLM-based agent planning

In Formal-LLM (Li et al., 2024), complex planning constraints (tool-call typing, sequencing, admissible structures) are compiled as pushdown automata. During LLM-based plan generation, the automaton supervises each token or plan step, restricting generations to valid prefixes, filtering out invalid actions, and backtracking from dead-ends. This achieves 100% valid plan generation and significant task metric improvement compared to unconstrained LLM planners.

Epistemic protocol synthesis

Automata-theoretic techniques in dynamic epistemic logic use finite automata to represent all valid event-model plans, supporting both offline synthesis and runtime plan monitoring (Aucher et al., 2014).

Supervisor synthesis in stochastic and partially observable domains

History-dependent automaton supervisors (za-DFA) synthesized via L*-style learning wrap POMDPs, ensuring satisfaction of PCTL-type formal requirements with minimal controllers. This leverages membership queries and three counterexample oracles to automatically learn supervisors that guarantee bounded-horizon temporal logic satisfaction (Zhang et al., 2017).

Flexible manufacturing and discrete event systems

Hierarchical or modular supervisory automata coordinate sequences of manufacturing ‘activities’ while enforcing both logical sequencing and tight timing (e.g., via max-plus linear timing analysis), providing time- and behavior-preserving execution architectures (Mohamadkhani et al., 2023). Classic DES supervisory control synthesizes memoryless or Mealy-machine supervisors that enforce a specification as the controlled language, with extensions for preferences and weighted objectives (Felli et al., 2016).

Reactive open systems

Reactive supervisors for open input-output automata are synthesized as positional, game-theoretic strategies, ensuring the system achieves the specification for all possible environment behaviors, with necessary and sufficient conditions characterized by automata-theoretic controllability, closure, and realization (Partovi et al., 2018).

A summary table of key automaton types and their application contexts:

Automaton type	Application domain	Supervisory role
Symbolic automaton	Hybrid motion planning	Quantitative path scoring/optimization
Büchi/NBA	Task & manipulation planning	LTL task decomposition
DFA / product automata	RL, POMDP, temporal logic control	History encoding, trajectory ranking
Pushdown automaton	LLM-based planning agents	Constraint validation, plan decoding
Finite transducer	Discrete event/FMS supervision	Event-driven sequencing

5. Scalability, Complexity, and Practical Implementation

Automaton-based supervision typically incurs polynomial or exponential complexity with respect to the size of the automaton, temporal nesting, or horizon:

• Matrix-based gradient planning: per-step $O(|Q|^2)$, overall $O(K H |Q|^2)$ for $K$ iterations, horizon $H$ (Balakrishnan et al., 2024).
• Large automata with deeply-nested goals or extensive state spaces can expose exponential blowup, yet reduction via Q-vector summarization, progress graphs, or product state coalescence mitigates this.
• Learning-based supervision in POMDPs and RL adds dimensions due to trajectory enumeration, preference comparison, and reward fitting; sample efficiency hinges on reward expressivity, preference alignment, and representation dimension (Alinejad et al., 17 Oct 2025, Zhang et al., 2017).

Limitations include nonconvexity (local minima for gradient search), reliance on differentiable dynamics (for gradient methods), controller growth for deeply temporally-structured tasks, and no guarantee of global optimality unless task structure is simple or the automaton small.

Implementation leverages GPU-accelerated matrix and autodiff frameworks (e.g. PyTorch, JAX), scalable game solvers for synthesis, and standard model checking for verification.

6. Correctness, Formal Guarantees, and Runtime Monitoring

Supervisory automata provide strong formal guarantees:

• The trajectory or sequence produced by a supervised controller is accepted by the automaton only if the underlying plan satisifies the specification (temporal logic, safety/liveness, protocol, or grammar) (Vasilopoulos et al., 2020, Mohamadkhani et al., 2023, Aucher et al., 2014, Partovi et al., 2018).
• For continuous domains under smooth controllers, hybrid proofs ensure both completeness of plan execution and collision/safety guarantees under plausible assumptions (e.g., obstacle separation) (Vasilopoulos et al., 2020).
• In RL, under standard preference-consistency, representational, and exploration assumptions, learned policies are $\varepsilon$-optimal with respect to non-Markovian objectives tracked by the supervisor (Alinejad et al., 17 Oct 2025).
• Automated plan validation using model-checking ensures that counterexamples are caught and plans refined until all behaviors entailed by the supervisor automaton satisfy the user-specified LTL properties (Yang et al., 2022).
• Real-time monitoring is achieved by tracking the evolution of the plant or agent alongside the automaton; any deviation from the automaton’s accepted language is instantly detected as a non-conforming prefix or error (Aucher et al., 2014, Mohamadkhani et al., 2023).

7. Automated Plan Extraction and Natural-Language Integration

Recent advances enable the extraction and synthesis of plan supervisors directly from informal task specifications, particularly in natural-language domains. The GLM2FSA procedure (Yang et al., 2022) demonstrates that high-level task knowledge can be elicited from large generative LLMs, parsed, and formalized as FSAs that are then subject to automated model-checking and refinement. The resulting automata bridge the gap between natural-language user requirements and fully-specified, formally verifiable plan supervisors, generalizing across commonsense and expert tasks.

In LLM-integrated systems, developer requirements specified as context-free grammars are compiled to a pushdown automaton whose transitions are used at each planning step to prune the LLM’s allowed choices, guaranteeing that every generated plan is both admissible and executable (Li et al., 2024).

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Automaton-based plan supervision amalgamates symbolic logic, quantitative optimization, formal verification, and runtime execution. It supports a unified language for expressing complex temporal and structural constraints, guarantees correctness of agent behaviors and task realization, and offers scalable computational procedures for planning, learning, and monitoring. This paradigm is foundational in contemporary research spanning hybrid systems, discrete event control, reinforcement learning, AI planning, and autonomous robotics (Balakrishnan et al., 2024, Vasilopoulos et al., 2020, Alinejad et al., 17 Oct 2025, Mohamadkhani et al., 2023, Aucher et al., 2014, Felli et al., 2016, Yang et al., 2022).