Regular Languages Closed Under Homomorphism

Here we show that the regular languages are closed under homomorphism, which is a function from Sigma* to itself that has a nice "splitting" property. We use this property in the proof by constructing an NFA for the desired "homomorphism" language, by breaking up every transition into many smaller transitions.

#easytheory #nfa #dfa #gate #gateconcept #theoryofcomputing #turingmachine #nfatoregex #cfg #pda #undecidable #ricestheorem

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▶ADDITIONAL QUESTIONS◀
1. Construct an NFA for the example homomorphism given in the video applied to a "starter" DFA.
2. Can it be possible for a finite language L to have h(L) be infinite?
3. Can it be possible for an infinite language L to have h(L) be finite?

▶SEND ME THEORY QUESTIONS◀

▶ABOUT ME◀
I am a professor of Computer Science, and am passionate about CS theory. I have taught over 12 courses at Arizona State University, as well as Colgate University, including several sections of undergraduate theory.

▶ABOUT THIS CHANNEL◀
The theory of computation is perhaps the fundamental theory of computer science. It sets out to define, mathematically, what exactly computation is, what is feasible to solve using a computer, and also what is not possible to solve using a computer. The main objective is to define a computer mathematically, without the reliance on real-world computers, hardware or software, or the plethora of programming languages we have in use today. The notion of a Turing machine serves this purpose and defines what we believe is the crux of all computable functions.

This channel is also about weaker forms of computation, concentrating on two classes: regular languages and context-free languages. These two models help understand what we can do with restricted means of computation, and offer a rich theory using which you can hone your mathematical skills in reasoning with simple machines and the languages they define.

However, they are not simply there as a weak form of computation--the most attractive aspect of them is that problems formulated on them are tractable, i.e. we can build efficient algorithms to reason with objects such as finite automata, context-free grammars and pushdown automata. For example, we can model a piece of hardware (a circuit) as a finite-state system and solve whether the circuit satisfies a property (like whether it performs addition of 16-bit registers correctly). We can model the syntax of a programming language using a grammar, and build algorithms that check if a string parses according to this grammar.

On the other hand, most problems that ask properties about Turing machines
are undecidable. This Youtube channel will help you see and prove that several tasks involving Turing machines are unsolvable---i.e., no computer, no software, can solve it. For example, you will see that there is no software that can check whether a
C program will halt on a particular input. To prove something is possible is, of course, challenging. But to show something is impossible is rare in computer
science, and very humbling.

[denikhalikov]

Hey dude dont stop these please, we're sharing you on Reddit to get more exposure, these are REALLY helpful!

[naydernn]

You explain this stuff much better than my teacher ever could! Thank you!!

[marekvargovcik]

commenting so you get more recognition. introduction to theory of computation is a life saver! thanks

[christiansinger]

Really helped, since I was able to use the same idea to show that regular languages are closed under the application of a finite-state transducer.

[deveshnandan]

Thanks a lot man, your videos are really helpfull for aspirants preparing for gate exam :)

[trevormartin]

Thank you for uploading this video, I really appreciate it.

[ibadiix]

thanks bro ♥, you deserve more views !

[turkirob]

Fantastic, really very good explanation

[attafriski]

Thanks a lot, helps me understand this subject

[benasabu]

Sir what is the exact application of this homomorphism like I dont understand where it is usefull

[bemsorcaabay]

Good explanation... Better if there is a concrete example.

[bionictran]

Please do decision properties of CFL's. Otherwise, great video!