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Apr 13, 2016 at 21:42 comment added candied_orange @Kevin in mathematics, "well-defined", is a welcome reminder that you're still in mathematics.
Apr 7, 2016 at 13:45 comment added cst1992 The first paragraph made me feel I was back in theory of computation class.
Apr 6, 2016 at 18:31 comment added Thibault D. Haha good job @5gon12eder , I did not expect you to change that much. You do things thoroughfully :)
Apr 6, 2016 at 15:48 comment added 5gon12eder @ThibaultD. Good point, I've added that. It didn't exactly make the answer simpler, though.
Apr 6, 2016 at 15:47 history edited 5gon12eder CC BY-SA 3.0
Added explanation of grammars as suggested by Thibault D.
Apr 6, 2016 at 8:10 comment added Thibault D. The well-formed property or validation is performed by a grammar. This answer had been perfect if you had mentioned that.
Apr 5, 2016 at 0:39 comment added Pharap While your answer is not wrong that is a very scary answer to pose to a layperson with no knowledge of set theory and little experience with fancy mathematical terms like 'concatenate' and 'well-defined'.
Apr 4, 2016 at 5:47 comment added Giorgio @kevin: Exactly: any mathematical object for which there is a proper definition is well-defined, by definition. Normally, you do not say: let's consider a well-defined graph, with a well-defined node, etc. You only say well-defined if there is a chance to have an ill-defined one. So, since in the answer there is no mention of what could be an ill-defined subset of Σ*, I do not see why one should stress that the subset should be well-defined. If I speak of a set of strings, it is obvious that it should be a well-defined one.
Apr 4, 2016 at 5:41 comment added Giorgio @5gon12eder: “The set containing all strings that are not contained in the set” is not a set. Any set is, by definition, well-defined. IMO the term well-defined is redundant here.
Apr 3, 2016 at 21:11 comment added 5gon12eder @Kevin Yes, it was meant as an example of a silly definition of something that doesn't actually exist. Maybe a better (less obvious) one could be given.
Apr 3, 2016 at 21:09 comment added Kevin @5gon12eder: “The set containing all strings that are not contained in the set” does not exist. The term "well-defined" is ironically not well-defined.
Apr 3, 2016 at 21:08 comment added 5gon12eder @Kevin To which set are you referring? The Kolmogorov tuples? I'm pretty sure the language exists, though I have to admit that I've only been taught it informally and didn't work through a formal prove.
Apr 3, 2016 at 21:03 comment added Kevin @5gon12eder: That set does not exist under ZFC (because the axiom schema of separation is not powerful enough to describe it); if you're using some other set theory, you should specify it.
Apr 3, 2016 at 21:02 comment added 5gon12eder … “The set containing all strings that are not contained in the set” is self-contradicting and not useful.
Apr 3, 2016 at 21:02 comment added 5gon12eder @Giorgio With “well-defined” I mean that there is a formal predicate that tells whether an item belongs to the set or not. This predicate will in general not be computable but it has to be clearly specified without contradiction. Otherwise, bad things might happen. “The pairs of strings (w, M) where M is the smallest description of a Turing machine that outputs w and then halts” is a well-defined but non-computable (see Kolmogorov complexity) predicate. …
Apr 3, 2016 at 21:01 comment added Kevin @Giorgio: In mathematics, "well-defined" is largely just an intensifier: everything that mathematically exists is already well-defined.
Apr 3, 2016 at 20:33 comment added Giorgio What do you mean by well-defined subset? And why do you require that a subset of Σ* be well-defined in order for it to be a language? Wouldn't any subset be called a language?
Apr 3, 2016 at 13:40 history edited 5gon12eder CC BY-SA 3.0
added 1002 characters in body
Apr 3, 2016 at 13:35 history edited 5gon12eder CC BY-SA 3.0
added 1002 characters in body
Apr 3, 2016 at 13:25 history answered 5gon12eder CC BY-SA 3.0