Sparse Languages are Not Regular! [HARD Pumping Lemma!]

Here we look at a hard pumping lemma proof, in showing that the language of strings that are "sparse" (i.e., have 1s that are far apart) is not regular. There is a lot of subtlety in generating the string to pick, as well as how the decompositions are formed. However, the structure of the proof is very similar to every other pumping lemma proof.

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▶ADDITIONAL QUESTIONS◀

1. How would you adapt the proof for 1s being at least log_3 |w| characters apart?
2. Is this language context-free?

▶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-sta ... https://www.youtube.com/watch?v=_0xgneWuTnk