Finite Automaton Simulator
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Finite Automaton Simulator

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Easily create and simulate state machines using this JS library. Import into your own server side or browser based JS application.

FSA Example

Visualization of an FSA


Add the latest version of fas-js to your package.json:

npm install fas-js --save-dev

Import the ES6 module:

var fas_js = require('fas-js');

Import into HTML file

<script src=""></script>


A finite automaton is a formally defined state machine, a concept that can then be expanded on to build more complex and powerful computational machines. I highly recommend the book "Introduction to the Theory of Computation" by Michael Sipser if you want to learn more about state machines and related concepts.

A Finite State Automaton (FSA) is defined as a 5-tuple (Q, Σ, δ, q0, F) where:

  1. Q is a finite set called states
  2. Σ is a finite set called alphabet
  3. δ: Q x Σ → Q is the transition function
  4. q0 ∈ Q is the start state
  5. F ⊆ Q is the set of accept states

An FSA (a type of state machine) recognizes exactly one language. More formally, if A is the set of all strings that a given machine M accepts, then we can say that A is the language of machine M, or L(M) = A. When we say that M recognizes A, it means that the machine M accepts all possible strings within A. This library gives you tools to simulate a given input string w to determine if w ∈ A.

An FSA can be conceptualized as a Directed Graph, and more speifically, an Oriented Graph. It's often visualized in this way for teaching and demonstration (see below) purposes. It is also important to understand Sets and their logical operators when working with FSAs.


First, define your FSA according to the definition above.

* Q cannot be empty, and each state name must be unique
* Type: string[]
const states = ["q1", "q2"];

* Σ cannot be empty, and can be passed in as one string (each character will be interpreted as a separate symbol) or a string array
* Cannot contain duplicate symbols
* Type: string | string[]
const alphabet = "01";

* δ is an array of Objects that define the transitions for each alphabet symbol, for each state
* size of Q x size of Σ = size of transition set
* Each transition must contain: an origin state, a destination state, and an input symbol
* Type: Array<Object>
const tfunc = [ // from=origin_state, to=dest_state, input=symbol
    { from: "q1", to: "q2", input: "1" },
    { from: "q2", to: "q1", input: "0" },
    { from: "q2", to: "q2", input: "1" },
    { from: "q1", to: "q1", input: "0" }

* q0 is the start state of the FSA. The first symbol of the input string is processed on this state.
* q0 must be a member of Q
* Type: string
const start = "q1";

* F is the set of accept states, which determine whether a provided input is "accepted" by the FSA or "rejected".
* This determination is made after the last symbol has been read by checking if the final state is in the accepting set
* F must be a subset of Q - it can also be an empty set (signifies an FSA that always rejects)
* For simulation purposes, F is passed in as a string array that cannot contain duplicate states
* Type: string[]
const accepts = ["q1"]; // Note here that the start state can also be an accept state

// createFSA() takes the 5 objects defined above, according to the definition, and returns a custom FSA object as seen in FSA.js
const fsa = createFSA(states, alphabet, tfunc, start, accepts);

There are two simulation options available:

simulateFSA(w: string[], fsa: FSA, logging: boolean = false): string <>

Runs the entire input w through the finite state automaton fsa. w is passed in as an array of strings and must only contain symbols from the alphabet defined in fsa. This function returns a string representing the final state, which can be compared with your accepts array to determine if input w is accepted by this FSA.

stepOnceFSA(w: string, qin: string, fsa: FSA, logging: boolean = false): string <>

Returns the destination state, based on input symbol w and input state qin, as defined by δ of fsa. w is passed in as a string and must match a symbol from the alphabet defined in fsa. qin must be a state in Q. Use this function if you'd like to iterate through a full input string step-by-step.

Note: In both functions above, a third logging parameter is available (defaults to false) which will print useful messages to the console as the simulator processes the input string. This can be used for debugging purposes or server-side logs. It is recommended to leave it defaulted to false for browser applications.

* Both simulators require an FSA input created with createFSA(). Here we will use the FSA created in the above block.
* fsa is a machine which accepts strings that end with 0
simulateFSA("0", fsa); // returns "q1"
simulateFSA("01", fsa); // returns "q2"
simulateFSA("010", fsa); // returns "q1"
simulateFSA("", fsa); // returns "q1"

stepOnceFSA("0", "q1", fsa); // returns "q1"
stepOnceFSA("1", "q1", fsa); // returns "q2"
stepOnceFSA("0", "q2", fsa); // returns "q1"
stepOnceFSA("1", "q2", fsa); // returns "q2"

// Check for acceptance
const finalState = simulateFSA("010", fsa);
if(accepts.indexOf(finalState) !== -1)

// Step through an input string w
const w = "01101";
let inputState = start; // The first input state is q0
for (const symbol of w) {
    inputState = stepOnceFSA(symbol, inputState, fsa);
// Now, check for acceptance...


This library provides an engine that creates and simulates an FSA. However, it does not include any visualization components. The demo below provides a UI that utilizes this engine and visualizes the FSA as it's being processed. It's a great way to learn about FSAs and experiment with your own FSA creations! The UI and graph visualizations were built using preact, d3.js, and d3-graphviz.

Demo on ObservableHQ (Learn more about ObservableHQ here)


This library is distributed under the GPL 3.0 license found in the LICENSE file.