Why is the Halting Problem Undecidable?

Here we concern ourselves with the Halting Problem (called HALT_TM) which asks whether a given Turing Machine halts (accepts or rejects) a given input w. We show that this problem is undecidable (i.e., no algorithm exists that runs in a finite amount of time for it). The idea is to suppose that a decider exists for HALT_TM, and then to create a decider for A_TM (which we know to be undecidable) based on that supposition, which gives us a contradiction.

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▶ADDITIONAL QUESTIONS◀

1. Can you prove that HALT_TM is undecidable without the knowledge of any other undecidable problem?
2. Can you prove it is undecidable using a different undecidable problem than A_TM?
3. Can you prove it undecidable in a completely different way?
4. Is HALT_TM recognizable?

▶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 aspe ... https://www.youtube.com/watch?v=4tbCR10QGTI