Talk:Pumping lemma for regular languages

message: The second paragraph says "w = xyz where the middle portion y must not be empty", but then immediately lists xz as an example?  This would mean y is the empty string.  The paragraph also goes on and says, "...constructed by repeating y zero or more times..."  This seems a tiny bit misleading.  It might mean repeating it 0 or more times AFTER the first occurrence of 'y', but it seems to imply y itself can be listed 0 or more times, which again, contradicts the definition.  — Preceding unsigned comment added by 73.145.200.192 (talk) 08:24, 19 January 2019 (UTC)
thread_title: Error in examples/statements regarding the lemma?

message: Any thoughts on TeXifying at least the lemma statement? Might look a little nicer. 141.210.102.198 16:49, 20 April 2007 (UTC)
thread_title: TeXify

message: Where is the statement of the lemma?  Shouldn't it be front and center?  
:Good point.  It used to be there.  I haven't found the edit yet that removed it. Fixed. Jrincayc 04:14, 14 November 2006 (UTC)
thread_title: Lemma statement

message: I made the change to y so it would be 'bc' rather than 'bcbc' (below) because [so I thought] |xy| must be less than or equal to p.
The pumping length appears to be 4 (4 states in the FSA), so if x=a and y=bcbc, |xy|=6 and the language would not be regular. The
conditions in Sipser (p.78 2nd ed.) don't seem, IMHO, consistent, but could be read to say y must be at least 'bc'.
"In terms of the pumping lemma, the string abcbcd is broken into a x portion a, a y portion bcbc and a z portion d. (Note, it could be broken in different ways such as a x portion a , a y portion bc and a z portion bcd)"
:Yes, you are correct.  I will fix the page.  Jrincayc 03:24, 6 October 2006 (UTC)
thread_title:  Don't understand how y can be bcbc 

message: The first example says: "If we let |xy|=p and |z|=p, then xy is the first half of w, or all p of the as". I don't think we can do this - don't we have to consider any case where |xy| ≤ p? The important thing is that y contains at least one a and no bs, which does follow directly from |xy| ≤ p. Dcoetzee 22:14, 14 September 2007 (UTC)
thread_title:  Unwarranted assumption in first example? 

message: Why would this be in the philosophy section?  It should be in the computer science / lingustics section.  —Preceding unsigned comment added by 24.108.202.22 (talk) 09:05, 19 November 2007 (UTC)
thread_title:  Philosophy? 

message: IMHO the inner part of the Lemma is always true (edit: which reduces the lemma to \forall L\subseteq \Sigma^* (\mbox{regular}(L) \Rightarrow true)):
\exists x,y,z \in \Sigma ^* (w=xyz \Rightarrow [\text{three conditions}])
There always exists a combination of x,y,z which is unequal to w which makes the whole term true without even having to consider the three conditions. This becomes more clear when trying to negate the lemma (in order to prove a language is not regular). The inner part in this case becomes:
\forall x,y,z \in \Sigma ^* (w = xyz \land \neg[\text{three conditions}])
Obviously w = xyz cannot be true for all x,y,z in \Sigma ^*.
But what the lemma should say is that there is a concatenation xyz = w which also satisfies the three conditions. So imho the implication should be replaced by a conjunction:
\exists x,y,z \in \Sigma ^* (w=xyz \land [\text{three conditions}])
Please correct me if I'm wrong. Otherwise I'll change the lemma in a few days.
Tcommbee (talk) 00:57, 22 June 2009 (UTC)
replies: text: This is correct, a small oversight on the part of the original author. Dcoetzee 22:38, 22 June 2009 (UTC)
thread_title:  Lemma Statement too weak? 

message: Let Σ = {a, b}, is L = {ab} (a language that consists of only one word) a formal language? If so, is it a regular language? [My guess is that it is, since it can be constructed by the grammar ({A, S}, Σ, {S -> aA, A -> b}, S)] If so, does the pumping lemma hold for it? (It doesn't seem to...) Wyverald (talk) 02:37, 8 November 2010 (UTC)
replies: text: Oh wait. Does the pumping lemma hold true for all limited-length languages because you can just let p be greater than the length of the longest word in the language? Silly me. On second thought, shouldn't we clarify this in the article? Wyverald (talk) 02:49, 8 November 2010 (UTC)
replies: text: It is there: "Note that finite languages satisfy the pumping lemma trivially by having p equal to the maximum string length in L plus one."  It could probably be clearer though.  Jrincayc (talk) 13:33, 8 November 2010 (UTC)
thread_title:  Confused 

message: Pumping lemma holds true for a language of balanced parentheses (which is still non regular):
It is always possible to find a substring of balanced parentheses inside any string of balanced parenthesis. Such substring can be safely removed or repeated any number of times without ruining the balance.
For example there are two possible strings of length 4: (()) and ()(). We can define y to be () in both cases and see that any number of adding or removing () from those strings will not break the balance.
WhitestOwl (talk) 01:10, 27 February 2011 (UTC)
thread_title:  Removed example of balanced parenthesis 

message: ((Wikipedia email from , received 28 Sep 2015:))
Hello, Jochen!
Thank you for providing an example for the general version of the pumping lemma for
regular languages.
But one can solve the problem you've mentioned ({ ambncn : m≥1 and n≥1 }) using a homomorphism from the alphabet to a set of strings. We can just erase "a" and the rest is easy.
Could you provide an example where a homomorphism is of no use but the general version of the pumping lemma shows in its full glory?
I'm sorry if I'm asking too much or I've chosen a wrong way to communicate (I'm still new to the whole Wikipedia stuff).
Thanks in advance
:Hello , I prefer to answer this way, so other editors may help us with the issue. 
:You are right with your objection to the example from Hopcroft and Ullman, but I don't have a full-glory example at hand. I'll try to think about one; possibly { cmbncn : m≥1 and n≥1 } would work? However, someone might ask for a citation, anyway. - Jochen Burghardt (talk) 16:20, 28 September 2015 (UTC)
replies: text: Oops, I think you are right. In fact, Hopcroft and Ullman didn't claim that { ambncn : m≥1 and n≥1 } satisfies the non-generalised Pumping Lemma; they just asked to prove its non-regularity by the generalized version. Also, I agree that there is no obvious way to fix the example. For now, I'll just withdraw the wrong claim from the article. You don't happen to know a good example language that (1) satistifes the non-generalised Pumping Lemma, (2) is nevertheless non-regular, and (3) can easily be proven so by the generalized Pumping Lemma? Or maybe even a language that in addition (4) can't be proven to be non-regular by some of the regular languages' closure properties, such as closure w.r.t. homomorphism mentioned above by ? - Jochen Burghardt (talk) 12:23, 16 May 2016 (UTC)
thread_title:  General version of the pumping lemma 

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I have just modified  on Pumping lemma for regular languages. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
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thread_title:  External links modified 

message: Headings are usually complete subjects, however "Converse not true" is not. "Problems with the converse" would be more accurate, or "Invalidity of the converse". Even just "The Converse" would be better, although I suspect it was changed from that specifically to make clear that the converse is untrue. Bearlyalivent (talk) 08:42, 11 February 2024 (UTC)
replies: text: You have a point here. (As I'm not a native English speaker, I confine myself to some mathematical considerations.) Of your suggestions, I dislike "", for the same reason I dislike e.g. "" - complaining about math facts is of no use.  "" is ok, while "" might suggest at first glance that we have to say something important about it, beyond that it is just false (by analogy, we wouldn't make a heading "" only to explain that this is a false proposition). I also thought about using "", and adding in the section body a sentence like ". - Jochen Burghardt (talk) 18:12, 11 February 2024 (UTC)
thread_title:  "Converse not true" is grammatically incomplete