Introduction to Finite State Machine Theory

The Theory of Computation, also referred to as Automata Theory, is a cross-disciplinary field of study encompassing both mathematical and computational theory. This theory concerns itself with the foundational principles of computational logic and their application to simple machines. A Finite State Machine, commonly known as an Automaton, is the fundamental unit of Automata Theory. Through the use of Automata, researchers are able to gain greater insight into how computers conduct mathematical computations and address problems.

The study of the dynamic behaviour of discrete systems is a driving force in the progression of the theory of computing. The term “automaton,” which is closely linked to “automation,” is the source of the concept of “automata.” The theory of computing presents the basis for fundamental ideas such as symbols, strings, and alphabets. To gain further insight into finite state machines, please read on.

An explanation of the concept of a finite state machine.

In system design, a finite state machine can be a helpful conceptual tool for processing a series of inputs, each of which will change the state of the system. Once all of the relevant data has been taken into consideration, it is important to pay attention to the output of the system’s ultimate conclusion. Additionally, it is essential to verify that the provided sequence of inputs was valid.

An FSA, FSM, or state machine is a model of computing in mathematics. It’s a theoretical machine with a limited number of possible states.

Depending on the input stimuli, a Finite State Machine (FSM) may transition between its defined states. Transitioning is the process by which one state is transformed into another. The defining characteristics of an FSM include its states, inputs, and the initial state that causes each transition.

There are two different types of finite state machines: deterministic finite state machines (DFSMs) and non-deterministic finite state machines (NFSMs). The process of creating a DFSM is essentially the same as constructing an NFSM. Both types of finite state machines consist of a set of states, a set of inputs, and a set of transitions that describe how the machine transitions between its states. Additionally, both types of machines can be used to model a wide variety of problems and processes.

In the modern world, many everyday technologies operate in a manner similar to state machines, executing pre-defined actions in response to a series of events. This type of technology is responsible for the automated operations of countless devices, from mundane household appliances to sophisticated medical equipment.

Finite state machines are a type of computing system that can be seen in everyday life. Examples include vending machines that provide items depending on the combination of coins that are inserted, traffic lights that allow drivers to determine which vehicles wait, move, and stop in a specific order, and elevators which halt at predetermined levels based on the needs of the passengers. All of these examples demonstrate the ability of finite state machines to take input and produce an output based on the data received.

When compared to other types of computing, such as the Works machine, the finite state machine’s processing capabilities are significantly lower. This is due to the fact that the finite state machine’s memory is limited by the number of states it possesses, and thus there are certain computations that the Works machine can handle and the finite state machine cannot.

The Finite State Machine and Its Parts

The finite state machine is comprised of the following parts:

The Beginning

The beginning point of our system is the initial state. A direction arrow is often included in the illustration.


The collection of all the correct information we provide is called the alphabet, or the language.

Comprised of all the recognised states

When we have a catalogue of states, we can assume that the number of available configurations of our systems is restricted. At any given moment, only a single state will be in effect. The circles in the diagram serve to illustrate this idea.

Contextualised collection of agreeable conditions

Accepting states form a subset of the set of all possible states and are used to determine the validity of the data that has been analysed. A double circle is commonly used to represent an accepting state and signifies its acceptance.


The movement of the machine from one state to another is regulated by transitions, which are essentially laws. These transitions are most often represented as a line connecting two distinct locations.

The Binary Parser in Action

We can construct a finite state machine that is capable of reading and interpreting a binary sequence in which a 0 follows each 1. For instance, the input sequence 1010 would be considered valid while inputs 1101010 and 1011001 would not. The resulting finite state machine would be structured as follows:

State 1, State 2, and Error make up the three states of our finite state machine. You may find yourself in one of the following situations:

  • If you’re in state 1 and you obtain a 1, you’ll go to state 2.
  • In the event that the outcome of state 2 is 0, transitioning back to state 1 is the next logical step.
  • If we’re in state 1 and the outcome is 0, we’ll transition to the error state.
  • If we’re in state 2 and we receive a 1, you’ll enter the error state as well.

Because there is no way to leave the error state, any input in the error state immediately terminates the operation.

A Python implementation of a finite state machine’s creation steps

Many software programs make use of Finite State Machines (FSM) to manage and control their internal states. It is possible to use code written in Python to validate user input against a predefined set of transitions and states, enabling the development of a finite state machine.

In order to achieve this, the Transition class will include a match() method. This method can be utilised to determine if the current state and input of the Finite State Machine (FSM) allow for a transition to the next state that is specified. After the FSM class is initialised, the add transitions() function must be called.

This technique guarantees that only valid states are implemented in the transition regulations. To confirm that your machine will be accepted, utilise the accepts() function with the input parameters provided.

Illustrations of Finite State Machines

Finite State Machines (FSMs) are highly versatile and have a wide range of applications beyond text parsing and software systems. To gain a better understanding of the practical applications of FSMs, it is worth exploring various real-world examples.

  1. There are traffic signals.

    The most basic use of FSM is a traffic light system. Let’s take a look at each of the main parts and figure out what they mean:

    States: The three possible states of a traffic light are green, yellow, and red.
    In the beginning: Green.
    In States Where Acceptance Is permitted: Due of the traffic light’s infinite duration, this model has no valid transitions.
    Alphabet: Seconds are represented by the positive integers.
    • The rule of thumb is to wait six minutes after the light turns green before proceeding to the next state.
    • When the yellow light is on, stay still for 10 seconds, and then switch to red.
    • When the light becomes red, we have to stop for 120 seconds before moving on to the green stage.
  2. Robotic Intelligence that Wants to Kill You

    It is also worth noting that Enemy Artificial Intelligence (AI) is another example of where finite state machines (FSM) can be utilised. As an example, let us consider the mechanics of a parol from an action game. To build a complete FSM, the following components are essential:

    States: They may either Patrol, Reload, Attack, Die, or Take Cover from enemy fire.
    Situation in the outset: The guard role necessitates that its default state be Patrol.
    Countries that accept: Since a deceased enemy bot cannot take orders, we will use the Deceased state as our accepting state.
    Alphabet: It is possible to represent different states of the world in a video game, such as “full ammo,” “low ammo,” “full health,” “player approaches,” “player runs,” and “no health,” through the use of string constants.
    Transitions: Unlike a simple traffic light, this model has many states, therefore we’ll be moving between them one at a time.
    • Patrol
      • Change to Attach when a player is nearby.
      • If our vitality runs gone, we’ll have to enter the condition of Deceased.
    • Reload
      • When your ammunition is all used up, go to the Attack mode.
      • If your health is at danger, you should visit your state’s Take Cover website.
      • If your health bar is empty, you must enter the condition of Deceased.
    • Attack
      • If your ammunition is running short, you should visit the state Reload.
      • Visit the state’s Take Cover website if your health status is dangerously low.
      • Visit the State Patrol after the player manages to evade capture.
      • If your health bar is empty, you must enter the condition of Deceased.
    • Hunker Down
      • The Attack state is entered when all health bars are full.
      • If your ammunition is running short, you should visit the state Reload.
      • When your health reaches zero, you must enter the condition of Deceased.

Here’s how it looks in the FSM notation:


The implementation of Finite State Machines (FSMs) has been praised for its simplicity, yet it is also this very quality that presents a limitation: FSMs are incapable of representing systems with a variable number of states. Previously, FSMs that always returned a set value of 10 were deemed satisfactory, however, this is no longer the case.

It may not be possible to create a finite state machine (FSM) if the given input is a string of any number of 1s followed by the same number of 0s. This is because a finite state machine is only capable of recognising the current state it is in. Therefore, it would not be able to determine the number of 1s and 0s in the string without being provided additional information.

Prior to the introduction of the Pumping Lemma, it was impossible to gain a comprehensive understanding of the fundamental components of any programming language. It became impossible to create a finite-state machine, and there was no way to categorise the language linguistically.

In order to model and understand complex systems, the finite state machine is an effective theoretical framework. This framework requires knowing the initial state, the accepting state, and the rules for transitioning between the different stages. By understanding these elements, it is possible to determine if a given set of inputs will be accepted.

The Pumping Lemma has demonstrated that, due to the finite number of states, finite state machines cannot be used to represent all possible systems. Despite this limitation, finite state machines are becoming increasingly popular for a variety of practical applications.


  1. In the field of computational theory, what is the definition of a finite state machine?

    A Finite-State Automaton (FSA), also known as a Finite-State Machine (FSM), is a mathematical representation of a computational process. It is a hypothetical model that can exist in any of a large number of discrete states at any given instant. In the fields of automata theory and computational theory, finite-state machines are an important area of study and research.
  2. Can you explain how TOC is used?

    Here’s an example of how TOC is used:
    • The Biological Complexities of Natural Selection
    • Use of Deterministic Finite Automata in Compiler Construction
    • Computer-limitation models, such as the Halting issue
    • Existence Is a Game
    • Universe-wide modelling
    • Computing hardware and difficulties in the real world
    • Algorithms with a Super-Recursive Structure
    • Utilises algorithmic frameworks developed by machines
    • Situations under which
      • The Term “Video Games”
      • There are traffic signals.
      • Concession stands
      • Recognising spoken language
      • Digital Signal Processor Controllers
      • Analysis of natural language
  3. Why do we refer to computational devices with a limited number of states as “finite machines

    The concept of the finite state machine was developed as a computational model to describe a system’s current state while it awaits a transition to take place. A transition is a series of operations that bring about a specific result. It is called finite because the machine is only capable of being in one state at a time, although it may shift between states to execute various tasks.

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