3.2: Undecidability

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Sep 12, 2021
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- 3.1.10: Summary
- 3.2.1: Introduction

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3.2.1: Introduction
It is one thing to not be able to imagine how one would compute complicated functions, and quite another to actually prove that they are uncomputable.
3.2.2: Enumerating Turing Machines
We can show that the set of all Turing machines is countable. This follows from the fact that each Turing machine can be finitely described.
3.2.3: The Halting Problem
The halting problem is one example of a larger class of problems of the form “can $X$ be accomplished using Turing machines.” (The answer is no: the halting problem is unsolvable.)
3.2.4: The Decision Problem
We say that first-order logic is decidable iff there is an effective method for determining whether or not a given sentence is valid. As it turns out, there is no such method: the problem of deciding validity of first-order sentences is unsolvable.
3.2.5: Representing Turing Machines
In order to represent Turing machines and their behavior by a sentence of first-order logic, we have to define a suitable language. The language consists of two parts: predicate symbols for describing configurations of the machine, and expressions for numbering execution steps (“moments”) and positions on the tape.
3.2.6: Verifying the Representation
In order to verify that our representation works, we have to prove two things. First, we have to show that if $M$ halts on input $w$ , then $T(M, w) \rightarrow E(M, w)$ is valid. Then, we have to show the converse, i.e., that if $T(M, w) \rightarrow E(M, w)$ is valid, then $M$ does in fact eventually halt when run on input $w$ .
3.2.7: The Decision Problem is Unsolvable
If the decision problem were solvable, we could use it to solve the halting problem.
3.2.8: Summary

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