DEV Community: Meu57

Transition function?

Meu57 — Sun, 07 Apr 2024 17:14:36 +0000

[Bobs Explains the transitive function]

Bob: The transition function defines the rules for moving. We can use our light bulb switch as an example of finite automata. Suppose in the off state, if the input is ON, then the state of the light bulb will shift towards ON, right? And in the ON state, if the input is OFF, then the state of the light bulb will shift towards OFF. So if we denote OFF input by 0 and ON input by 1, now we can show this using the transition function δ(ON, 0) = OFF or δ(OFF, 1) = ON. I hope you are getting my point, Alice. This means that here, delta is showing what the input will be to move towards a particular state.

Alice: Okay, so does the delta or transition function show the rule that determines what input will cause a move to a particular state, right?

Bob: Exactly, Alice. The transition function, often denoted by ( δ ), specifies the conditions under which a transition occurs from one state to another in a finite automaton. It’s a formal way to express the rules that govern the state changes. So, in the context of the light bulb example:

When the light bulb is OFF and the input is 1 (representing the ON switch), the transition function (δ(OFF, 1)) will result in the light bulb state changing to ON.
Conversely, when the light bulb is ON and the input is 0 (representing the OFF switch), the transition function (δ(ON, 0)) will result in the light bulb state changing to OFF.

Alice: Now, I got you.

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Bob: You are now a tech thing.

Meu57 — Sun, 07 Apr 2024 11:15:10 +0000

Bob: You’ve transformed yourself into a computer tech. The analogy of finite automata you provided was absolutely fantastic. I’m sure you wouldn’t mind if we delved deeper into finite automata, right?

Alice: go ahead, please.

Bob: Anything starts with its name, right? The first introduction of anything is its name, right?

Alice: I agree, sir. My name is Alice, and I am human.

Bob: Correct, name decides the definition, so first we have to understand the definition. A formal definition of finite automata, right?

Alice: Totally right. Then, what is the formal definition of finite automata?

Bob: So, the definition of finite automata is divided into chunks, or we can say that it divides into sets.

Alice: Means, as we merge them, we get the finite automata, right?

Bob: Totally, let me clarify for you. Finite automata have rules for going from one state to another, so we have a set of rules. These rules depend on the input. So, we have set of inputs. For example, we can use the example of a light switch again. The rule of the light bulb turning off and on depends on the switch, and the switch state, decides the state of light bulb or we can say switch current state, decides the state of the light bulb, right?

Alice: Right, Input is deciding the state and state is binding the rule means if when the switch will turn off the light bulb will be off its rule.

(Then Alice concludes)
Alice: So, rules, states and inputs are interlinked.

Bob: Absolutely, rules are related to states and these rules depends on the input.

(Then Bob adds an statement by continuing his previous statement)

Bob: This input is part of a set of inputs. Such as this light switch scenario, we have a set of inputs which have two inputs: on and off, right? So, we can say on and off belong to this set. We call this set Alphabet.

Alice: yeah, ON is an input and OFF is an input both making a group of input, and we are calling it a set of input, and this set is Alphabet.

Bob: Correct, and then Finite automata have a start state and a set of end states, which we can say are accept states.

Alice: so, we have states, rules, inputs, starting state and end states which we are calling accept states right.

(Bob says with smile)

Bob: So, what will be the formal definition of finite automata, The formal definition will be a finite automaton is a list of those five objects: set of states, input alphabet, rules for moving, start state, and accept states. In mathematical language, a list of five elements is often called a 5-tuple.

5-tuple (Q, Σ, δ, q0, F)

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

Alice: Okay, these five ‘musketeers’ come together to define finite automata.

Bob: Hahhaha, you can say that.

Alice: But here in δ(Delta) is a transition function would you please explain its function please. How it's working.

Bob: OfCourse!!

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Alice: Finite Automata kind of simple to understand.

Meu57 — Thu, 04 Apr 2024 10:25:28 +0000

[Next day]
Alice: (breathing in the fresh air) Isn't it amazing how quickly the weather can change? Just last night, we were cooped up inside because of the storm.

Bob: (smiling) Absolutely. It's like nature's own version of nondeterminism. Speaking of which, did you finish reading that chapter on finite automata?

Alice: I have read about finite automation and I find finite automata kind of simple to understand.

Bob: I am glad that you find it simple.

Alice: Yeah, made an analogy for that to explain to myself.

Bob: that's interesting I would love to hear that tell me I am excited.

Alice: okay we know what states are, right and we know that states are depends on input right as we see in light switch example. So, you must have played Snakes and Ladders in your childhood.

Bob: Oh man, nostalgia!

Alice: So, we take here three states and two inputs 0 and 1 like we take two inputs in light switch but now here we have a snake-ladder game, and we have a dice too in our dice, there are only two numbers you can get, first is '1' and second is '0', right. Now, suppose your player is at home (starting point) and until you get a 1 on the dice, you will not come out from your house. Now suppose instead of ladders, you assume that there are states which can bring you upward or downward same Snakes and Ladders game.

Bob: Okay!!! go ahead it sounds interesting

Alice: yeah, but there is a catch in the game there are limited chances to roll the dice in this game.

Bob: okay so snake and ladder with limited chances to roll the dice.

So now start in state q1 which is starting point of our player or can say home.

You roll dice and suppose you got 1 (on q1 state if dice get 1), then your player will transition go from q1 (ladder brings your player) to q2 (state).
Now you are on q2 state, and you roll dice and suppose you get 1 again (on q2 state if dice gets 1), then the state will not change your player will remain on the q2 state.
Now you are on q2 state, and you roll dice and suppose you get 0 (still you are on q2 now you roll dice and get 0), then you will move to q2 to q3 state.
Now you are on q3 state, and you roll dice, and you get 1 (now you are on q3 now you roll dice and get 1), then you will come back q3 to q2.

Okay, so in this game we have limited chances to roll the dice that's why if you reach the destination which is the final state or can say in the last chance of rolling dice keeps you on the final state or brings you to the final state which is our destination, then you will win or lose.

Bob: That's something new even I have to say, your analogy is quite creative! It simplifies the concept of finite automata by comparing it to a familiar game. In finite automata, we have a set of states and the system transitions between these states based on input, similar to moving on a game board based on dice rolls.

In your analogy, the dice rolls represent the input symbols (1 and 0), and the states (q1, q2, q3) represent positions on the game board. The rules of moving between these states based on the dice rolls are like the transition functions in finite automata. And just like having limited chances to roll the dice, finite automata have a limited sequence of inputs to reach a final state.

This is a great way to visualize the process of state transitions in finite automata. It’s important to remember that in actual finite automata, the transition functions are predefined, and the system deterministically follows these rules to process a string of inputs and decide whether to accept or reject it.

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Alice: What is the meaning of state in TOC.

Meu57 — Wed, 03 Apr 2024 05:55:47 +0000

Alice asked: but what is "state" meaning here? Is it the same "state" we use as synonym of condition or it's something else?

Bob: "Great question, Alice! In the context of Theory of Computation and finite automata, a 'state' is indeed similar to a 'condition'. It represents the status of a system at a particular point in time. For example, in the case of the automatic door, the 'state' could be 'OPEN' or 'CLOSED', indicating whether the door is open or closed at a given moment. This 'state' changes based on inputs, like someone approaching the door. So, while 'state' and 'condition' can be synonyms in everyday language, 'state' has a specific meaning in the context of computation and automata.

Alice (Exclaimed): oh!! you said 'state' has a specific meaning in the context of computation and automata." I wonder what it would be.

Bob: "Absolutely, Alice! In the context of computation and automata, a 'state' is a specific condition or status that a system or process can be in at a given time. It's a snapshot of the system at a particular moment.
For example, consider a traffic light. It can be in one of three states: 'RED', 'YELLOW', or 'GREEN'. Each state represents a different condition of the traffic light and dictates a different behavior for drivers.
Similarly, in a computational system like a finite automaton, states represent different conditions of the system. Transitions between states occur based on certain inputs or conditions.
In our automatic door example, the states were 'OPEN' and 'CLOSED', and the door transitioned between these states based on whether someone was near the door or not.
So, while 'state' in everyday language often means a condition or circumstance, in computation and automata, it has this more specific meaning related to the status of a system or process."

Alice: So… okay… nice…. If I am getting you right, then the state depends on input, and as input changes, so does the state. It's like a button for turning off and turning on. If you turn on the bulb, it will light up, and if you turn it off, the bulb will be turned off.

Bob: "Exactly, Alice! You've got it. The state of a system, like a light bulb, can indeed change based on input, like flipping a switch. If the switch is turned 'ON', the bulb's state changes to 'ON', and it lights up. If the switch is turned 'OFF', the bulb's state changes to 'OFF', and it goes dark. This is a simple example of how states and inputs work in systems. Well done!"

Alice: Okay, now we are getting somewhere, but you have used some strange words in your explanation: 'Automata,' 'automation,' and 'finite automaton.' I am not aware of these words.

Bob: "Well these are typical technical words which we use in TOC. Let me help you in these words.
Automata is nothing but a mathematical machine for clear understanding you can omit mathematical word for now. Think of it as a machine that can be in different states You can assume automata as traffic light that can be red, yellow, or green, and if these lights are working without any intervention of human it's Automation that's why we use word Automatically and Finite Automata is a type of machine that has a limited number of states. For example, a door that can be either 'open' or 'closed or a traffic light which has 6 stages only."

Alice: "Ah, so these terms are closely related!"

Bob: "Exactly! They are indeed closely related."

[The Snowstorm has been stopped; Alice was ready heading towards his home]

Alice: (gathering her things) Well, Bob, this has been enlightening. I'm going to head home and dive into this chapter on Finite Automata.

Bob: Sounds like a plan. Let me know if you have any questions after you've read it.

Alice: Definitely. (smiling) And thanks for the impromptu lesson today. It's cleared up a lot of confusion for me.

Bob: Anytime, Alice. It's always a pleasure discussing TOC with someone as curious as you.

Alice: (heading towards the gate) Alright then, see you tomorrow! We'll have a lot to talk about, I'm sure.

Bob: (waving goodbye) See you, Alice. Enjoy the reading and the sunshine!

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Alice: How we can understand TOC?

Meu57 — Wed, 03 Apr 2024 05:52:49 +0000

Alice: “How can we understand TOC in our real life? Can you break down how we see Theory of Computation (TOC) in our everyday lives?”

Bob: “Absolutely! Take our recent trip to the supermarket. Have you noticed how there were two doors for exiting and a separate one for entering?

Alice: “Yes!!”

Bob: That’s a practical example of TOC at play. We encounter it all the time, especially in electromechanical devices.”

Alice: “Like what exactly?”

Bob: “Think about those automatic doors at the supermarket entrance. They swing open when you approach, right? That’s controlled by a device at the heart of the mechanism.” *

Alice: “And how does that device work?”

Bob: “Well, it’s pretty clever. There’s a sensor pad in front that detects someone coming. Then, there’s another pad at the back to make sure the door stays open long enough for you to pass through safely.”

Alice: “So, this device has two states, either ‘OPEN’ or ‘CLOSED,’ depending on what?”

Bob: “Exactly! And it responds to four possible inputs: ‘FRONT’, ‘REAR’, ‘BOTH’, or ‘NEITHER’. Based on these inputs, it switches between its states.”

Alice: “Could you give me an example?”

Bob: “Sure thing. Let’s say A person walks up to the door from the front (‘FRONT’ signal). The door opens (‘OPEN’).

The person is now behind the door (‘REAR’ signal). The door stays open (‘OPEN’).
The person walks away, and no one is at the door (‘NEITHER’ signal). The door closes (‘CLOSED’).
Another person walks up to the door from the front (‘FRONT’ signal). The door opens again (‘OPEN’).
Now, there are people at both sides of the door (‘BOTH’ signal). The door stays open (‘OPEN’).
Everyone walks away from the door (‘NEITHER’ signal). The door closes (‘CLOSED’).
A person walks up to the door from the back (‘REAR’ signal). The door stays closed (‘CLOSED’).
The person walks away, and no one is at the door again (‘NEITHER’ signal). The door stays closed (‘CLOSED’).
So, the door’s state changes based on whether someone is near it or not. This is a simple example of how a finite automaton, a concept in Theory of Computation, works in real life.”

Alice: “So, viewing the automatic door controller as a finite automaton helps us understand its behavior better?”

Bob: “Exactly! It’s like looking at a simple computer with just a single bit of memory, capable of remembering whether it’s ‘OPEN’ or ‘CLOSED’. More complex devices like elevator controllers might need more memory bits to track their states. This approach helps us grasp the design of various devices, from household appliances to digital watches, using the concepts of finite automata.”

*Reference Introduction to the theory of Computation by Michael Sipser

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TOC (Theory of Computation)

Meu57 — Tue, 02 Apr 2024 17:21:34 +0000

In a cozy house, where crackling fire warms the room. a man was sitting by the fireplace, lost in his thoughts with sipping tea. Outside, the winter's breath whispered through the trees, painting the landscape in a serene blanket of snow. The air was crisp, the sky adorned with clouds pregnant with more flakes to come. A gentle snowfall had turned the streets into a wonderland of white, muffling the sounds of the city.

Bob glanced out the window, watching the snowflakes dance in the glow of the streetlights. He sipped his tea, reveling in the warmth it brought to his chilled fingers. It was the perfect evening to cozy up by the fireplace.

Suddenly, a knock at the door broke the peaceful ambiance. Bob rose from his armchair and made his way to the entrance. He swung open the door to reveal his friend, Alice, bundled up in a thick coat, snowflakes clinging to his hair.

"Hey, Bob," Alice greeted with a smile, brushing snow off his shoulders. "Sorry for dropping by unannounced. The snowstorm caught me off guard, and my car got stuck a few blocks away. Mind if I crash here for the night?"

"Of course not, come on in," Bob replied, ushering Alice inside. "I was just about to make some tea. Perfect timing."

As they settled by the crackling fire, Alice shook off his coat and took a seat opposite Bob. The warmth of the fire thawed the chill from their bones, and soon they were lost in conversation, catching up on work and life.

Bob leaned back in his chair, taking in the cozy atmosphere. "You know, Alice," he began, "there's something fascinating about the weather tonight. The way the snow falls, each flake unique yet part of a larger pattern. It's like nature's own symphony."

Alice nodded in agreement, taking a sip of his tea. "Absolutely, Bob. Weather is one of those complex systems that constantly evolves and influences our lives. It's a dance of states — from calm and serene to wild and chaotic."

And so, nestled by the fire, with the snowflakes whispering outside, Bob and Alice embark on a journey of exploration, from the intricacies of weather patterns to the mysteries of computation and beyond.

Bob: You know, TOC is like the heartbeat of our digital world. It’s this fascinating blend of Computer Science and Mathematics.

Alice: Ah, yes! The mystical dance of machines and logic. But what exactly is TOC?

Bob: Well, my friend, TOC is all about understanding how machines think. Imagine every input as a little spark it sets the machine in motion, changing its state. Events matter the tiniest shift can lead to a whole new outcome.

Alice: So, it’s like a symphony of ones and zeros, orchestrated by the laws of computation?

Bob: Exactly! Now, let’s break it down. TOC was born to unravel the secrets of discrete systems. It’s like peeking behind the curtain to see how our digital actors perform.

Alice: And why did they invent this mystical theory?

Bob: Well, my friend, they wanted to describe and analyze the dynamic behavior of these systems. Think of it as deciphering the language of binary whispers.

Alice: But wait, how does math fit into this grand tale?

Bob: Ah, math Linear algebra, calculus, eigenvalues, and eigenvectors they’re the tools that power our AI. Vectors, matrices, and tensors they’re like the building blocks of machine learning.

Alice: And what about AI? Is TOC its secret mentor?

Bob: Indeed! TOC whispers to AI. It teaches algorithms to walk perfectly, and limits to respect. Without TOC, our AI would stumble in the dark.

Bob: TOC, the unseen architect of our digital universe! Let’s delve deeper.

Alice: Hear, hear! 🍵🤖