Introduction to Finite State Machine Theory

Automata Theory, widely known as the Theory of Computation, is an interdisciplinary domain of study involving both mathematical and computational theory. This theory is focused on the fundamental principles of computational logic and how they relate to simple machines. The Finite State Machine, also called an Automaton, serves as the cornerstone of Automata Theory. By using Automata, scholars can attain a deeper understanding of how digital devices perform mathematical operations and solve problems.

The investigation of the dynamic properties of discrete systems is a key factor in the advancement of computing theory. The term “automaton” is closely linked to “automation” and serves as the origin of the concept of “automata.” The fundamental concepts of computing theory such as symbols, strings, and alphabets are established based on this theory. If you wish to delve deeper into finite state machines, please continue reading.

An elucidation of the notion of a finite state machine.

Finite state machine is a valuable conceptual tool for handling a series of inputs, each of which modifies the system’s state during system design. After taking into account all relevant information, it is crucial to monitor the system’s final output. Moreover, it is imperative to confirm that the given input sequence was valid. For further information about finite state machines, visit

An FSA, FSM, or state machine is a mathematical computing model. It is a theoretical machine with a finite number of possible states.

A Finite State Machine (FSM) can switch between its defined states based on input stimuli. This transition is the process of changing one state to another. The essential features of an FSM include its states, inputs, and the initial state that triggers each transition.

Deterministic Finite State Machines (DFSMs) and Non-deterministic Finite State Machines (NFSMs) are the two main types of finite state machines. Constructing a DFSM is basically the same as creating an NFSM. Both types of machines contain a set of states, a set of inputs, and a set of transitions that determine how the machine moves between its states. Furthermore, both machines are suitable for modeling various issues and procedures.

In today’s world, numerous routine technologies function similarly to state machines, carrying out predetermined actions in response to a sequence of events. This type of technology automates the operations of a multitude of devices, from common household appliances to advanced medical equipment.

Everyday life features finite state machines in various computing systems, such as vending machines that dispense products based on the combination of inserted coins, traffic lights that enable drivers to determine which vehicles wait, move, or halt in a specific order, and elevators that stop on predetermined floors based on the passengers’ requirements. These instances provide evidence of the ability of finite state machines to process input and produce output based on the received data.

The processing capacity of finite state machines is notably lower than other types of computing, such as the Works machine. This disparity results from the fact that the finite state machine’s memory is constrained by the number of states it has, therefore, there are specific computations that the Works machine can perform but the finite state machine cannot.

The Components of the Finite State Machine

The finite state machine consists of the subsequent components:

The Start

The initial state is where our system commences. The illustration frequently features a directional arrow.


The set of accurate information we supply is denoted as the alphabet or the language.

Including all the Recognised States

Having a list of states allows us to limit the number of potential arrangements of our systems. At any given point, only one state operates. The circles depicted in the diagram emphasize this notion.

Contextual Collection of Acceptable Conditions

Accepting states are a subset of the complete set of possible states and are employed to assess the legitimacy of the scrutinized data. An accepting state is usually depicted with a double circle, which indicates its acceptance.


Transitions, which are essentially rules, govern the progression of the machine from one state to another. The majority of the time, these transitions are depicted as a line linking two separate locations.

The Functioning of Binary Parser

We can build a finite state machine that can read and interpret a binary sequence in which each 1 is followed by a 0. In other words, the input sequence 1010 would be valid, while the inputs 1101010 and 1011001 would not. The resulting finite state machine would be organized in the following manner:

Our finite state machine is comprised of State 1, State 2, and Error. You may encounter one of the following scenarios:

  • When in State 1, receiving a 1 will take you to State 2.
  • If State 2 outputs 0, returning to State 1 is the subsequent action.
  • When in State 1, receiving 0 as output will lead to a transition to the Error state.
  • If you’re in State 2 and you receive a 1, you’ll also transition to the Error state.

As there is no means of exiting the Error state, any input received in this state instantly halts the operation.

Creation Steps for a Finite State Machine in Python

Several software applications utilize Finite State Machines (FSM) to manage and regulate their internal states. With the use of Python, it is feasible to verify user input against a predetermined set of transitions and states, allowing for the construction of a finite state machine.

To accomplish this task, the Transition class should have a match() function. The purpose of this function is to determine whether the present state and input of the Finite State Machine (FSM) permit a transition to the stated next state. Once the FSM class is instantiated, the add_transitions() function must be invoked.

This procedure certifies that only permitted states are incorporated into the transition rules. To verify that your machine is valid, employ the provided input parameters with the accepts() function.

Examples of Finite State Machines

Finite State Machines (FSMs) are incredibly versatile and applicable beyond text parsing and software systems. It is worthwhile to investigate multiple real-world instances to better comprehend the practical uses of FSMs.

  1. Traffic signals utilize FSM.

    The fundamental use of FSM is a traffic light system. We can understand the significance of each primary component by examining it:

    States: The three potential states of a traffic light are green, yellow, and red.
    Starting point: Green.
    Permissible Transition States: As the traffic light continually displays, this model does not permit any valid transitions.
    Alphabet: The positive integers represent seconds.
    • The general guideline is to wait for six minutes after the light switches to green before moving on to the next state.
    • During the yellow light phase, stay put for 10 seconds before switching to red.
    • Upon the light turning red, we must pause for 120 seconds before commencing the green stage.
  2. Hazardous Robotic Intelligence

    Finite state machines (FSM) can also be used in Enemy Artificial Intelligence (AI), and as an example, let us consider the mechanics of a patrol from an action game. To create a comprehensive FSM, the following elements are crucial:

    States: The states can be Patrol, Reload, Attack, Die, or Take Cover to protect against enemy fire.
    Initial Situation: A guard’s default state must be Patrol.
    Permissible States: The Dead state serves as the allowable state since a dead enemy bot is incapable of following orders.
    Alphabet: In a video game, string constants can be used to represent various world states, such as “full ammo,” “low ammo,” “full health,” “player approaches,” “player runs,” and “no health.”
    Transitions: As opposed to a simple traffic light, this model has numerous stages and, as a result, must be moved through one stage at a time.
    • Patrol

      • Switch to Attack mode upon sensing the presence of a player nearby.
      • If our health is depleted, we must transition to the Dead state.
    • Reload

      • Upon running out of ammunition, switch to Attack mode.
      • If your health is in peril, seek refuge in the Take Cover state.
      • If your health gauge is depleted, you must transition to the Dead state.
    • Attack

      • If your ammunition is running low, seek the Reload state.
      • If your health is critically low, seek shelter in the Take Cover state.
      • Upon the player successfully evading capture, return to the Patrol state.
      • If your health bar becomes empty, you must transition to the Dead state.
    • Take Cover

      • Switch to the Attack state when all health bars are restored to full capacity.
      • When low on ammunition, go to the Reload state for replenishment.
      • When your health level hits zero, you must transition to the Dead state.

Here’s an example in FSM notation:


While Finite State Machines (FSMs) are applauded for their easy implementation, their simplicity can also pose a drawback: FSMs cannot depict systems with varying numbers of states. In the past, FSMs that consistently yielded a fixed value of 10 were deemed satisfactory, but this approach is no longer adequate.

Constructing a finite state machine (FSM) might be unfeasible if the input provided is a sequence of any count of 1s followed by an identical number of 0s. The reason is that an FSM can only identify the present state it is in. Consequently, without extra information, it cannot determine the quantity of 1s and 0s in the string.

Before the advent of the Pumping Lemma, it was unfeasible to acquire a thorough comprehension of the basic constituents of any programming language. Constructing a finite-state machine was out of the question, and there was no way to classify the language linguistically.

To model and grasp intricate systems, the finite state machine constitutes a potent theoretical framework. This framework necessitates familiarity with the starting state, the accepting state, and the guidelines for shifting between the disparate phases. By grasping these factors, it is possible to ascertain whether a specified set of inputs will be approved or not.

The Pumping Lemma has shown that, owing to their finite number of states, finite state machines cannot depict every conceivable system. Nevertheless, these machines are gaining traction for numerous pragmatic applications.


  1. In computational theory, what does a finite state machine refer to?

    A Finite-State Automaton (FSA), or a Finite-State Machine (FSM) as it is also called, represents a computational mechanism mathematically. It is a theoretical model that can exist in any of numerous distinct states at any point in time. In the realms of automata theory and computational theory, finite-state machines constitute a crucial arena of investigation and exploration.
  2. What is the application of TOC?

    Take a look at this sample usage of TOC:
    • The Intricacies of Natural Selection in Biology
    • The Role of Deterministic Finite Automata in Building Compilers
    • Models of Computer Constraints, Such as the Halting Problem
    • Life as a Game of Existence
    • Modelling Across the Universe
    • Computing Hardware and Real-World Challenges
    • Super-Recursive Algorithms
    • Uses Algorithmic Frameworks Created by Machines
    • Scenarios Where
      • The Phrase “Video Games”
      • Traffic Signals Exist.
      • Food and Beverage Stands
      • Identification of Spoken Language
      • Controllers for Digital Signal Processors
      • Natural Language Analysis
  3. What is the Reason for Referring to Computational Devices with Limited States as “Finite Machines?”

    The idea of the finite state machine was created as a computational framework for describing the current state of a system while it is waiting for a transition to happen. A transition is a sequence of actions that produces a certain outcome. It is labelled “finite” because the machine can only be in one state at a time, even though it can move between states to perform different functions.

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