How to Count to Three, a Tutorial

Here we look at the language THREE_DFA, which is the set of all DFAs that accept at most three strings. This is related to the language INFINITE_DFA in that it helps us solve the problem. If the DFA does not accept infinitely many strings, then we can brute force through all possible remaining strings (at most the number of states - 1 in length), and then count the number of accepted strings. If you can solve this a different way, I would love to know about it!

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▶ADDITIONAL QUESTIONS◀

1. What is the time complexity of this method in solving THREE_DFA?
2. If instead we asked ONE_DFA (i.e., all DFAs accepting at most one string), is there any structure within the DFA we can assume or try to find?

▶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 ... https://www.youtube.com/watch?v=HinUZ2f64bg