Automata-Guided Planning

• Automata-guided planning is a methodology that uses automata abstractions to represent system dynamics, task specifications, and constraints for synthesizing optimal plans.
• It leverages product constructions and shortest path algorithms to combine multiple automata models, reducing state-space complexity via abstraction and heuristic search.
• Widely applicable in robotics, autonomous driving, multi-agent systems, and partially observable domains, its modular design supports scalability and formal correctness.

Automata-guided planning is a methodology for synthesizing optimal or correct-by-construction plans by representing planning problems—such as dynamics, specifications, preferences, uncertainty, and temporal or epistemic constraints—through automata-theoretic abstractions. The central principle is the compilation or reinterpretation of planning-relevant entities (transition systems, specifications, agent capabilities, user relaxations, etc.) as appropriate classes of automata (e.g., deterministic/nondeterministic/weighted automata, transducers, timed automata, multiplicity automata), then solving the planning task as an automata-theoretic search or optimization problem, typically via shortest-path algorithms, product construction, or symbolic reasoning.

1. Automata-Theoretic Representations in Planning

Automata-guided planning leverages various automata representations matched to the structural features of the planning domain:

• System Dynamics and Motion Models: Discrete transition systems, weighted automata, and motion primitive automata encode state transitions and cost structures for mobile agents or robots (Kamale et al., 2021, Pedrosa et al., 25 Jan 2024).
• Task Specifications: Temporal logic specifications (LTL, sc-LTL, MITL, SITL, STL) are compiled into deterministic finite automata (DFA), timed automata, or timed signal transducers that precisely accept sequences (or signals) satisfying the specification (Kamale et al., 2021, Lindemann et al., 2019, Kamale et al., 20 Nov 2025).
• Preferences and Relaxations: User-defined relaxation preferences are encoded as weighted finite-state edit systems or edit automata, permitting permissible substitutions, deletions, or relaxations with associated penalties (Kamale et al., 2021, Kamale et al., 20 Nov 2025).
• Multi-Agent Systems: Agent capabilities, constraints, and failure modes are modeled as ε-augmented nondeterministic automata (ε₀-NFA), with system-level composition via modular automata operations (Tziola et al., 2022).
• Uncertainty and Partial Observability: Predictive state representations of POMDPs are encoded as multiplicity automata, enabling planning even in hidden-state domains (Even-Dar et al., 2012).
• Symbolic and Quantitative Semantics: Symbolic automata with transitions guarded by continuous predicates, and weighted automata over semirings, allow integration of continuous planning and differentiable objectives (Balakrishnan et al., 15 Oct 2024).

A defining feature is the use of product constructions or synchronized parallel composition, fusing the automata models for the agent dynamics, specification, and task constraints into a single global automaton.

2. Core Algorithms and Solution Strategies

Automata-guided planning relies on classical and modern algorithms grounded in automata theory:

• Product Automaton Construction: The synthesis of a global automaton typically forms the synchronous product of (i) the agent transition system, (ii) the specification automaton, and (iii) any auxiliary automata representing edits, preferences or supervisory constraints (Kamale et al., 2021, Kamale et al., 20 Nov 2025).
• Weighted Shortest Path Search: Planning is framed as finding a minimal-cost or admissible path (sequence of transitions) from an initial state to an accepting state in the global automaton, leveraging Dijkstra's algorithm, A*, or parametric variants for Pareto frontier recovery (Kamale et al., 2021, Pedrosa et al., 25 Jan 2024, Kamale et al., 20 Nov 2025).
• Heuristic Search Over Large Spaces: Automata product constructions may yield large state spaces. Recent frameworks leverage admissible or weighted heuristic functions precomputed via backward BFS or similar abstractions, achieving substantial reductions in memory and computation with bounded suboptimality (Kamale et al., 20 Nov 2025).
• Multiobjective and Relaxed Planning: Bi- or multi-objective optimization is handled by formulating cost vectors (e.g., travel cost and relaxation penalty) and either solving weighted-sum variants or directly enumerating Pareto-optimal runs (Kamale et al., 2021, Pedrosa et al., 25 Jan 2024).
• Symbolic or Gradient-based Optimization: Quantitative or robustness-driven objectives are encoded as automaton matrix operators over continuous state spaces, enabling planning via automatic differentiation and gradient-based optimization rather than explicit enumeration (Balakrishnan et al., 15 Oct 2024).
• Model Checking and Goal Reasoning: Model checking engines, especially in the context of high-level task networks (GTN), invoke LTL/CTL-based product automata to synthesize plans as witness traces and support dynamic decision-making under new goals or contingencies (Bride et al., 2019).

3. Applications and Exemplary Frameworks

Automata-guided planning is implemented and evaluated across diverse domains:

Domain	Automata Paradigm	Key Features
Temporal-logic robot motion planning	DFA, WFSE, product automaton	Relaxed specifications, user preferences, shortest/best path synthesis
Autonomous driving, trajectory planning	Motion primitive automata, universal automaton	Multiobjective Pareto frontiers, receding-horizon adaptive planning
Multi-agent systems	ε₀-NFA modular composition	Agent capabilities, constraints, on-the-fly failure/integration
Partially observable planning	Multiplicity automata, PSR	Planning complexity exponential in PSR rank, not state space size
Spatio-temporal logic planning	Timed Signal Transducer, system abstraction	Avoids product-space blowup, synthesizes logic-based controllers
UI and software agent guidance	Symbolic UTG (UI Transition Graph) automaton	Symbolic pathfinding composes with LLMs for high-level instruction generation
Epistemic/knowledge-based planning	DFA, synchronous transducers	Decidability and synthesis for epistemic goals with complexity guarantees

Representative frameworks and case studies include the weighted three-way product method for relaxed temporal logic tasks (Kamale et al., 2021), optimization-based motion primitive automata for real-time autonomous driving (Pedrosa et al., 25 Jan 2024), and the Agent+P system for LLM-guided UI automation leveraging symbolic planning on the UI Transition Graph (Ma et al., 7 Oct 2025).

4. Scalability, Complexity, and Optimization Considerations

A central consideration in automata-guided planning is state-space and computational complexity:

• Product Space Size: Explicit construction of global product automata yields a state space scaling as the product of component sizes (e.g., $|X|\cdot|Z_E|\cdot|S_A|$ in the three-way product), but careful design allows reachability-driven sparse materialization (Kamale et al., 2021).
• Heuristic Optimization: Use of admissible heuristics (e.g., minimal steps to goal in automaton space, weighted A*), precomputed over automaton graphs, substantially reduces expansion rates while preserving near-optimality bounds (Kamale et al., 20 Nov 2025).
• Abstraction and Pruning: In continuous-time and spatio-temporal settings, product blowup is mitigated by symbolic abstraction and pruning, embedding system-consistency checks within modified specification automata (as in the combined TST construction for spatial predicates) (Lindemann et al., 2019).
• Multi-Agent Modularization: Modular automata-algebraic operations (union, subtraction, parallel concatenation) allow incremental composition of complex agent, constraint, and environmental models (Tziola et al., 2022).
• PSR/Mixed Observability Complexity: In partially observable domains, planning can be executed in time exponential only in the predictive-state (PSR) rank of the system, not the size of the hidden-state space, yielding quasi-polynomial algorithms for structured instances (Even-Dar et al., 2012).
• Empirical Results: Execution times and memory scale linearly with underlying model size for realistic instances, with dramatic efficiency benefits from heuristic strategies (e.g., order-of-magnitude reductions in nodes examined and search time for large urban graphs or multi-agent factories) (Kamale et al., 2021, Kamale et al., 20 Nov 2025, Tziola et al., 2022).

5. Advanced Features: Relaxation, Preferences, Quantitative and Knowledge Constraints

Modern automata-guided approaches extend classical models via:

• Specification Relaxation: Weighted finite-state edit systems (WFSE) and edit automata encode permissible relaxations (substitutions, deletions, insertions), with penalties for deviation from the nominal specification, enabling graceful degradation under infeasibility or user preferences (Kamale et al., 2021, Kamale et al., 20 Nov 2025).
• Multiobjective Planning and Runtime Modulation: Universal automata for motion primitives can be dynamically reduced or reconfigured to prioritize different cost metrics (e.g., safety vs. time vs. energy) at runtime, enabling adaptation to evolving priorities without full recomputation (Pedrosa et al., 25 Jan 2024).
• Gradient-based Planning with Symbolic Automata: Matrix semiring representations of symbolic automata, combined with differentiable predicate maps and dynamics, enable direct, scalable, and differentiable planning under complex regular-language objectives, supporting both open-loop and receding-horizon frameworks (Balakrishnan et al., 15 Oct 2024).
• Formal Guarantees and Model Checking: Model checking techniques over product automata with LTL or CTL specifications ensure correctness, optimality, and counterexample-driven refinement within integrated planning and goal-reasoning frameworks (Bride et al., 2019).
• Epistemic and Protocol Synthesis: Regular automata representations of histories in dynamic epistemic logic enable decidability and synthesis results for epistemic planning and protocol synthesis, with explicit complexity characterizations relative to knowledge-nesting depth (Aucher et al., 2014).
• Dynamic, On-the-Fly Planning: Modular compositionality in agent/task automata, as well as reconfigurable and incrementally constructed automata, supports responsive and context-native integration of new preferences, failure modes, or environmental events (Tziola et al., 2022, Kamale et al., 2021).

6. Limitations, Future Directions, and Theoretical Foundations

While automata-guided planning confers significant expressiveness, transparency, and control-theoretic rigor, several limitations and extension avenues remain:

• State Explosion: While mitigated by abstraction, pruning, and implicit search, certain products (especially in high-dimensional continuous or large multi-agent domains) may yet suffer from exponential blowup.
• Global vs. Local Optima: Gradient-based planning with symbolic automata may converge only to local optima, lacking completeness guarantees (Balakrishnan et al., 15 Oct 2024).
• Requirement of (Piecewise-)Smoothness: Differentiable semantics in symbolic or STL-based automata assume smooth predicates and system dynamics.
• Expressive Boundaries: Infinite-horizon or ω-regular properties, as well as stochastic and risk-aware planning, are not yet fully subsumed by mainstream automata-guided methods, though integration with Markov models and fixed-point unrolling is possible (Kamale et al., 2021, Balakrishnan et al., 15 Oct 2024).
• Computational Complexity: In knowledge-based planning, the power of automata techniques is tempered by multi-exponential complexity in the epistemic-nesting depth (Aucher et al., 2014). For POMDPs, efficiency hinges on low PSR-rank structure (Even-Dar et al., 2012).
• Integration with Learning: Recent proposals envision embedding automaton-derived reward/penalty signals as differentiable learning objectives within deep RL or cross-entropy frameworks (Balakrishnan et al., 15 Oct 2024).

Altogether, automata-guided planning constitutes a general, modular, and unifying paradigm for correct-by-construction planning across a spectrum of domains, offering provable guarantees, transparent relaxation, and algorithmic scalability via systematic exploitation of automata-theoretic structure.