GATE 2020 Question 36 (Which language is undecidable?)

Here we look at four languages: the set of TMs with empty language, the set of TMs that reach a certain state on a certain input in exactly 100 steps, the set of TMs with an undecidable language, and the set of TMs that accept at least 21 strings. We use Rice's theorem for three of the languages to show that they are undecidable. Not sure which ones? Watch the video to find out!

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▶ADDITIONAL QUESTIONS◀

1. What if the first language was all TMs with language not being the empty set?
2. What if the second language was all TMs that reach state q at any point in the computation on input w?

▶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 ab ... https://www.youtube.com/watch?v=b2Gy6AArupo