Relation between semi decidable, undecidable and countable sets

Question

I know that

decidable problem: has both counting (bijection with $\mathbb N $) and membership algorithm (TM halts for both member and non member strings )
semidecidable problem: has counting algorithm and TM halts for member strings
countable problem: has only counting algorithm

Now I have come across undecidable language set which does not have membership algorithm (as semi decidable languages at least have TM which halts for member strings, which is not the case for undecidable problems), that is it does have TM that halt neither for member string nor for non member string.

I have doubt about the relationship between these language classes.

I know "decidable $\subset$ semi decidable $\subset$ countable".

Where does "undecidable" fit here?

Is it like this?:

(Match the color of text with that of the border. Region color is lighter shades of corresponding enclosed texts.)

Q. Is above diagram correct? Also I want to add "countable" to above. I feel, for finite alphabet languages, countable will be proper superset of "undecidable". Is it right?

PS: I feel, earlier I based my understanding of "undecidable" on incorrect definition given in this question. So I draw wrong diagram earlier. I believe its correct now.

I think your diagram is wrong, since it assumes that undecidable problems are always semi-decidable. Some semi-decidable problems are undecidable (like the Halting Problem) and all decidable problems are semi-decidable. — Sagnik, Jan 22 '19 at 19:30
Also decidable problems aren't a subset of undecidable problems. The sets are complements of each other. — Sagnik, Jan 22 '19 at 19:31
This is a good answer related to this topic:
https://cs.stackexchange.com/questions/83815/if-a-problem-is-not-semi-decidable-and-not-decidable-can-we-say-it-is-undec — Sagnik, Jan 22 '19 at 19:33
I'm not sure what you mean by a "counting algorithm". To me, a counting algorithm is one that returns the number of elements of some set. That doesn't make much sense for infinite sets. Indeed, for most infinite sets, there isn't even any way of presenting that set as the input of an algorithm. — David Richerby, Jan 22 '19 at 19:39
@Raphael 1) just modified the diagram. Originally I didnt meant to say decidable is subset of undecidable earlier. I hope this diagram correctly represents what I mean. 2) I might be completely wrong about countable part. Need to check. — RajS, Jan 22 '19 at 19:46
@DavidRicherby wikipedia defines them as: "a set is countably infinite if it has one-to-one correspondence with the natural number set, N." Wikipedia further says "Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a unique natural number." So I believe term "countable" or counting algorithm makes sense for infinite sets. Please, Let me know if I am wrong here. — RajS, Jan 22 '19 at 19:50
@anir "countable" certainly makes sense; it's "counting" that I'm disputing. Wikipedia seems to be describing an enumeration algorithm. — David Richerby, Jan 22 '19 at 19:51
what about that containment diagram I prepared for my understanding? Are those relationships wrong? — RajS, Jan 22 '19 at 19:52
@Raphael I also know that countable are not comutable as I said, they just have counting algorithm, no membership algorithm. Is diagram depicting something wrong? — RajS, Jan 22 '19 at 20:05
can you suggest better/correct graphic floating online which covers all these terms together? Or any help in any form correcting those several mistakes? — RajS, Jan 22 '19 at 20:23
Isnt this answer states recursively enumerable languages are countable? — RajS, Jan 22 '19 at 20:31
Is the diagram correct, if we are to consider only finite alphabet? — RajS, Jan 22 '19 at 21:42
@Raphael I feel, earlier I based my understanding of "undecidable" on incorrect definition given in this question. So I draw wrong diagram earlier. I believe its correct now. Can you check it now, please. — RajS, Jan 23 '19 at 09:41

David Richerby · Answer 1 · 2019-01-23T14:26:45.973

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Undecidable is simply the complement of decidable, as the name suggests: anything that is not decidable is undecidable. So the whole pink area of your diagram consists of undecidable languages.

All languages over finite alphabets are countable. For example, every string over alphabet $\{0,1\}$ is a natural number written in binary.¹ Everything in your diagram is countable. We only ever consider finite alphabets; infinite alphabets aren't physically realizable because we can't store infinitely many distinguishable possibilities in a finite space.

^{1. A small problem is that $0$, $00$, $000$, etc. are all the number zero written in binary. So, instead, associate the binary string $xyz$ with the natural number $1xyz$ and now, every binary string represents a different natural number.}

edited Jan 23 '19 at 14:26

answered Jan 23 '19 at 12:43

David Richerby

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hey David, I have updated the question along with the diagram three hours back. There is no pink color now in the diagram. I believe you mean to say that the whole square is now undecidable and countable? Also the whole square is "countable" only if we are talking about finite alphabet languages, right? (It wont be if we start talking about infinite alphabet languages.) – RajS Jan 23 '19 at 13:35
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It's pink on my screen. And please don't edit your question in ways that invalidates answers. – David Richerby Jan 23 '19 at 13:39
Sorry I edited the question quite a lot, reflecting what I learnt after I first asked the question. But I edited it when there was no answer... – RajS Jan 23 '19 at 14:23
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@anir OK, so your question hasn't changed since I posted my answer, and your diagram still contains a pink region. I'll add a few words about finiteness of alphabets. – David Richerby Jan 23 '19 at 14:25

John L. · Accepted Answer · 2019-01-23T14:12:09.123

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A language is defined as a set of strings over an alphabet. We will assume the usual situation where the alphabet is a finite set. Then the set of all strings are countably infinite. Why is it countable? Because we can list all strings of length 0, all strings of length 1, all strings of length 2, all strings of length 3 and so on.

The correct diagram should look like the following (where we assume P!= NP), where the enclosing unlabeled plane represents languages that are countable. Note all languages that sit outside of the region of "TM decidable" is "TM undecidable". Note that we usually say "decidable", "recognizable", "co-recognizable" and "undecidable" simply without the prefix "TM".

$CS21 Decidability and Tractability Published by Shanna Hood https:\slideplayer\com-slide\5289813$

For completeness, we have the following diagram that includes chomsky hierarchy.

$CS 208 Computing Theory by Dr. Brahim Hnich https:\slideplayer.com\slide\8093532$

edited Jan 23 '19 at 14:12

answered Jan 23 '19 at 14:02

John L.

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My confusion was primarily with decidable, recognizable, undecidable and countable languages. You have shown other possible language classes too. Q1. And I believe diagram that I have put in the question adheres with your diagrams. Right? Q2. How the scenario will be if the languages are of infinite alphabet? – RajS Jan 23 '19 at 14:33
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Q1. I would say yes, assuming that "undecidable" in your diagram means all regions outside of region "decidable / co-decidable". – John L. Jan 23 '19 at 14:42
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Q2. The case with infinite alphabet is an entirely different matter. Let a language over an infinite alphabet consists of an infinite subset of all strings of length one, i.e., the alphabet itself. Even for such a "simple" language, we may need to redefine Turing machine to allow infinite states so as to recognize it. That is why we rarely talk about infinite alphabet. There are huge (I mean, infinitely large) differences between finiteness and infinity. – John L. Jan 23 '19 at 14:48

Relation between semi decidable, undecidable and countable sets

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