Finite State Machine Controller

• Finite State Machine (FSM) Controllers are a computational model defined by a finite set of states, inputs, and transitions that execute precise control logic.
• They are widely applied in robotics, embedded systems, and swarm technologies to implement reliable, deterministic behaviors with formal verification.
• FSM controller design employs manual, AI-assisted, and security-guided methodologies to enhance code synthesis, rapid prototyping, and system safety.

A finite state machine (FSM) controller is a fundamental computational model for representing and implementing control logic in both hardware and software systems. FSM controllers are defined by a finite set of discrete states, a collection of input symbols or events, a transition function that determines the next state based on the current state and input, and often an output function that prescribes system actions or responses. In robotics, embedded systems, digital hardware, and other automation domains, FSM controllers offer a rigorous framework for encoding behavioral logic, ensuring analyzable, reliable, and deterministic operations.

1. Formal Mathematical Definitions

FSM controllers are formalized in multiple variants to suit different control requirements:

• The basic (synchronous) FSM can be defined as the tuple

$M = (S,\, I,\, O,\, \delta,\, \gamma,\, s_0)$

where - $S = \{s_0,\, \dotsc,\, s_n\}$ is the set of states, - $I$ is the set of input symbols, - $O$ is the set of output symbols, - $\delta: S \times I \rightarrow S$ is the transition function, - $\gamma$ is the output function ($\gamma: S \times I \rightarrow O$ for Mealy machines, $\gamma: S \rightarrow O$ for Moore machines), - $s_0 \in S$ is the initial state (Hu et al., 18 Aug 2025, Gan et al., 7 Dec 2024).

• Hierarchical FSMs (HFSMs) extend this model by structuring multiple interconnected Mealy machines into a tree, allowing local state refinement, with transitions and outputs propagating and abstracting over the hierarchy. The HFSM control algorithm achieves linear-time preprocessing and near-logarithmic query for optimal planning between states (Stefansson et al., 2023).
• Extended FSMs (EFSMs) augment classical FSMs with program variables, guards, and actions, as in formal SDL modeling. EFSM transitions depend on variable valuations and enable supervisory control synthesis by mapping guards to conflicting event pairs (Wang et al., 2016).

2. FSM Controller Design Methodologies

FSM controllers are synthesized and implemented using several methodologies:

• Manual FSM Design and Modification: Conventional FSM controllers are often hand-designed, requiring developers to enumerate all states, transitions, and output logic. This approach, while precise, is time-consuming and error-prone for complex systems.
• Language-Driven and AI-Assisted FSM Generation: Recent work demonstrates the utility of LLMs to automate FSM code modifications through natural language instructions, as in ChatFSM (Gan et al., 7 Dec 2024). LLM pipelines extract FSM definitions, summarize required changes, and synthesize updated code structures, yielding high correctness and an order-of-magnitude speed-up over manual editing.
• Security-Guided Generation: SecFSM introduces a security-oriented knowledge graph (FSKG) to guide LLMs in synthesizing secure FSMs in Verilog. Vulnerabilities such as DeadState (unreachable states) and overflow are detected by graph-driven pre-analysis, prompting assertion and guard insertion during code generation. This approach significantly raises secure code pass rates in SoC designs (Hu et al., 18 Aug 2025).
• Automatic Code Generation from Formal Models: RoboChart establishes a direct, semantics-preserving pipeline from formal state-machine specifications (enriched with timers, concurrency, and modular interfaces) to simulation-ready controller code, eliminating manual translation errors and supporting formal verification (Li et al., 2017).

3. Applications in Robotics, Hardware, and Swarm Systems

FSM controllers underpin diverse application domains:

• Microrobotic Swarms: Group-based control schemes for large-scale swarms (e.g., MicroStressBots) utilize physical on-board FSMs for distributed collision avoidance. Local accessibility and global controllability are achieved under group-based control signals, with a two-phase FSM switching between random-walk exploration and pseudo-inverse goal-seeking. Probabilistic completeness of this FSM is mathematically proven for collision-free configuration spaces (Li et al., 2022).
• Hierarchical Planning in Large-Scale Systems: HFSM controllers efficiently solve optimal state-to-state planning by exploiting system hierarchy, yielding orders-of-magnitude speed-up over flat Dijkstra or contraction hierarchy methods and enabling rapid reconfiguration of control plans upon model changes (Stefansson et al., 2023).
• FPGA and ASIC Hardware Control: FSMs provide deterministic, high-speed logic for digital circuits, e.g., auto-billing vending machine controllers implemented on FPGAs. Mealy-type models are mapped directly to hardware using VHDL or Verilog, with design flexibility and timing accuracy surpassing traditional CMOS or single-electron logic (Monga et al., 2012, Hu et al., 18 Aug 2025).
• Robotic Manipulation and Locomotion: FSM controllers drive bipedal and quadrupedal robots by encoding behavioral states (e.g. walking, grasping, walling). Contact-aware FSMs can be modulated by learned policies to achieve robust, adaptive behaviors under perturbation (Liu et al., 2021, Singh et al., 2022, Kannapiran et al., 26 Oct 2025). Serialization of FSMs into token streams enables transformer-based imitation learning conditioned on spatial and language observations for long-horizon manipulation with near-perfect success rates (Mu et al., 7 Mar 2025).
• Autonomous Vehicle Safety: Evolving FSMs (e-FSMs) are deployed as online risk recognition and decision-revision modules. By clustering observation space into dynamically generated states and learning stochastic transition matrices, e-FSMs can override unsafe actions proposed by underlying controllers (e.g., IDM or deep RL agents), reducing car-following failure rates and mitigating latent collision risks (Han et al., 2019, Han et al., 2020).

4. Collision Avoidance and Probabilistic Guarantees

• The group-based collision-avoidance FSM for microrobot swarms alternates between random walk (exploration) and pseudo-inverse control phase (directed movement toward the goal). Mathematical lemmas ensure that, given adjacent visibility sets covering the collision-free configuration space, the random walk phase has a uniformly bounded probability of progressing toward intermediate states. Repeated trials guarantee probabilistic completeness: with probability 1, the controller will eventually reach the desired intermediate state, and then the pseudo-inverse phase can drive the system to the goal without further collisions (Li et al., 2022).

| Phase | Description | |------------------|------------------------------------------------------------------------| | RANDOM_WALK | Group-based random exploration; toggles to pseudo-inverse upon collision| | PSEUDO_INVERSE | Directed control toward intermediate or goal; toggles back if collision |

The absence of dead-ends in configuration space is assumed; progress is quantified by lower-bound probabilities per transition step.

5. Verification, Security, and Formal Guarantees

• FSM controllers, especially those implemented for safety-critical or security-sensitive hardware, are formalized to enable both functional and security analyses. RoboChart allows CSP-based refinement and temporal logic analysis, with ongoing work on code-generator soundness and probabilistic extensions (Li et al., 2017).
• Security vulnerabilities (e.g., DeadState, overflow) are addressed by graph-guided analysis and assertion insertion during code synthesis, as in SecFSM. DeepSeek-R1 evaluations show substantial advances in secure FSM code generation, with pass rates increasing from 40% to 84% on stringent test cases (Hu et al., 18 Aug 2025).

6. Limitations, Extensions, and Open Directions

• FSM controller efficacy is bounded by state-space coverage and modeling fidelity. Cascade errors can arise if coverage is non-exhaustive, emphasizing the value of serialized FSMs and systematic demonstration generation for manipulation tasks (Mu et al., 7 Mar 2025).
• Most frameworks currently address deterministic transitions and discrete states; ongoing work extends FSMs with probabilistic, stochastic, and hierarchical constructs (e.g., e-FSMs and HFSMs).
• Integration with reinforcement learning, formal verification tools, and natural language engineering continues to broaden FSM controller capabilities, enabling scalable swarm control, dynamic action revision, and agile hardware code generation (Han et al., 2020, Gan et al., 7 Dec 2024, Kannapiran et al., 26 Oct 2025).

7. Representative FSM Controller Architectures

Application Domain	FSM Variant	Key Features	Reference
Microrobot Swarms	On-board FSM	Group-based random walk, two-phase	(Li et al., 2022)
SoC Security	LLM + KG-guided	Knowledge-graph-driven Verilog	(Hu et al., 18 Aug 2025)
Hierarchical Planning	HFSM	Fast optimal planning, reconfigurable	(Stefansson et al., 2023)
Vending Machines	Mealy FSM	Auto-billing, hardware synthesis	(Monga et al., 2012)
Quad/Biped Locomotion	Contact FSM	Policy modulation, robust transitions	(Liu et al., 2021, Singh et al., 2022)
RL-based AV Control	e-FSM	Online clustering, risk recognition	(Han et al., 2019, Han et al., 2020)
Decentralized Swarms	CRW FSM, DQN-FSM	Emergent walling, adaptive logic	(Kannapiran et al., 26 Oct 2025)

Controllers exhibit specialized extension in each domain, with FSM variants tuned to application-specific requirements and research objectives.

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Finite state machine controllers remain a central tool for orchestrating complex, reliable, and analyzable control logic in robotics, hardware, and automated systems. Continued research advances their scalability, safety, verifiability, and ease of synthesis, with security, probabilistic, and machine learning augmentation shaping the frontier of FSM controller design.