Deciding If a DFA's Language Is Infinite: A Clear & Simple Explanation 🤖

Here we show that the problem of checking whether a DFA's language is infinite is decidable. The idea is to appeal to the pumping lemma for regular languages, and then make a certain DFA with the property that its language is empty if and only if the original DFA's language was finite (or infinite). Then the question boils down to asking whether the constructed DFA's language is empty (which is decidable).

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▶ADDITIONAL QUESTIONS◀
1. Can we do a similar question with PDAs/CFGs?
2. What about with Turing Machines?

▶SEND ME THEORY QUESTIONS◀
ryan.e.dougherty@icloud.com

▶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.