How many DFAs accept ANY language L?

Here we look at the question of how many DFAs accept a given language L, which can be anything. This may seem like a silly question but it turns out to be interesting! If L is not regular, then clearly there is no DFA for L. But if L is regular, we can always add an "unreachable" state to an existing DFA for L as many times as we want.

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▶ADDITIONAL QUESTIONS◀

1. How many minimal DFAs are there for a regular language L?
2. Can you modify the construction so that you can always add a state that is always reachable?

▶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 hard ... https://www.youtube.com/watch?v=Zbqq6G5CipU