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I am confused about Turing Machines that are able to decide languages that contain infinite words.

  1. Are languages with an infinite amount of only finite strings always decidable?

  2. How can a Turing Machine halt on an infinite input string?

  3. Can Turing Machines loop on finite strings?

  4. What is the difference between an infinite input string that a Turing Machines can halt on and an infinite input string that a Turing Machine cannot halt on (decidable vs recognizable)?

It's a lot of questions but they are related, please help me.

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    $\begingroup$ I have never heard of Turing machines operating on infinite words. Can you supply a reference? $\endgroup$ – Yuval Filmus Aug 10 at 7:24
  • $\begingroup$ The example question I got was: Give a high level description of a Turing Machine that decides the language L = {w | count(a) == 2 * count(b)}. A regular expression may be infinite as well, for example a*. Given that it is a regular expression it is also a decidable language, however I don't quite understand how a Turing Machine can actually decide on such a language. $\endgroup$ – BHK Aug 10 at 8:56
  • $\begingroup$ Your language consists of finite words. The regular expression $a^*$ denotes a language of finite words. $\endgroup$ – Yuval Filmus Aug 10 at 9:05
  • $\begingroup$ @YuvalFilmus yes, but it could also contain a word consisting of infinite a's $\endgroup$ – BHK Aug 10 at 10:37
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    $\begingroup$ No it couldn’t. By definition, a language only contains finite words. $\endgroup$ – Yuval Filmus Aug 10 at 15:31
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There's is a significant difference between "arbitrarily long" and "infinite".

As a simple example, an integer can have an arbitrarily great magnitude; formally, for every integer, there exists a larger integer (its successor), which in turn has a successor, and so on. But all integers are finite; $\omega \notin \mathbb{N}$. (Or, if you prefer, $\infty \notin \mathbb{N}$.)

Similarly, the Kleene star operator $A^*$ represents the concatenation of an arbitrarily large number of elements from $A$, but not an infinite number of elements of $A$.

There is an interesting part of formal language theory which deals with sets of infinitely-long strings (ω-strings), but such strings cannot be produced by any regular expression. (You can use ω-regular expressions, which include the infinite repetition operator $A^\omega$.) But you might want to master the material you're currently studying before venturing onto Buchi automata, interesting though they may be.

To answer your questions:

  1. No, there are undecidable languages consisting only of finite strings. Indeed, since such languages are undecidable, it is not necessarily even decidable whether they are finite sets. A classic example of an undecidable language is the set of (descriptions of) Turing machines which halt on every input. Note that every Turing machine (like every integer) has a finitely-long description, so the language is a set (an infinite set, in this case) of finite strings.

  2. A Turing machine is under no obligation to read its entire input. Consider the language of ω-strings which start with an $a$. This is clearly a set of infinitely-long strings, but only a single character needs to be examined to determine whether a string belongs to the language.

  3. Sure, why not. For example, the Turing machine trying to determine whether its input describes a Turing machine which halts on every input could simulate the described machine using every possible input, but that is going to go on forever even if the language describe a Turing machine which always halts.

  4. See, for example, deterministic Buchi automata. Basically, an ω-language $L$ can be deterministic if there is a language $L' \subset Pref(L)$ of finite prefixes which can deterministically predict inclusion in $L$.

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