Talk:General recursive function

This page was a REDIRECT to Talk:Recursive function. Let us have a real talk page instead. JRSpriggs 05:54, 11 August 2006 (UTC)

This article or section may contain too much repetition.

Well DUH its about recursion.

Definition of search operation mu

I brought back the old definition of the search operation because the new one introduced by CMummert is not actually computable. JRSpriggs 06:15, 11 August 2006 (UTC)

My previous definition wasn't very clear; I was using the word function to mean total function. Of course minimization of partial functions is not computable. I saw last week that you had fixed it. Thanks. CMummert 19:15, 13 August 2006 (UTC)

---

χ ψ Ψ ψ φ τ Χ μ ≤ ≥ ≠ → ← Ø ≡ ℛ

Am confused how the 0 as the least number suddenly jumped in here. My question has to do with "the function R" inside the mu-operator -- shouldn't it actually be a representing function of a predicate such that it evaluates to 0 if true and 1 if false?

I figured it out. So I will cut this junk below and save some trees. wvbaileyWvbailey 22:37, 8 November 2006 (UTC)

Defines the same thing as Computable function

To me, the title suggests that the purpose of the article is to define a certain class of functions. However, the same class is already defined in computable function. Wouldn't it make more sense for the title of the article to be "Mu-recursion", since what is unique to this article is that particular definition of "computable function"? Also, is there any reason the article should not be merged with Mu-operator? —The preceding unsigned comment was added by Quux0r (talk • contribs) 2007-04-11T02:14:22.

The idea is that computable function is written so that it is independent of the specific model of computation, and this page is about one particular model of computation. The computable function article does not define this class of functions specifically, it just links here. You could merge this with mu operator but both this page and that page are quite long so it would take a lot of trimming to merge them. I for one would not object to the trimming. You might want to put up merge tags and wait a few days, and/or write the merged version in a sandbox. CMummert · talk 02:20, 11 April 2007 (UTC)

Cleanup needed

This needs some cleanup to be readable. A large part of the article reads like a collection out-of-context snippets from Kleene's book. Pcap ping 05:07, 21 August 2009 (UTC)

Get to work and do it. BillWvbailey (talk) 14:33, 21 August 2009 (UTC)

Minimisation operator and partial functions

I belive that the minimisation operator can only accept total functions, otherwise the recursive functions wouldnt be closed under the minimisation operator.

An example is:

f(x, y) = 0 <==> (y = 0 and x ∈ A) or y = 1

where A is recursively enumerable (but not recursive)

f is partial and is M-recursive.

g (x) = μ y. (f(x,y)=0)

g is total (if it's not 0 then its 1).

g cant be M-recursive, otherwise A would be recursive, since g would be its indicator function (absurd).

I'm going to modify the article, if i'm wrong please state the reason here. — Preceding unsigned comment added by X-Overload-X (talk • contribs) 04:31, 27 June 2011 (UTC)

If A isn't recursive, then f(x,0) is undefined for x not in A. μ operator is also undefined if checked function is partial (as stated here). Then, function g will also be partial and then it can be indicator for non-recursive set. Maybe I am wrong, 'cause I'm not very familiar with μ operator 79.184.100.6 (talk) 17:20, 30 March 2012 (UTC)

Plagiarized Intro?

Sorry if this has been discussed before, but the introduction is identical to https://www.princeton.edu/~achaney/tmve/wiki100k/docs/%CE%9C-recursive_function.html

Which came first?

Also, a cosmetic note: The last line of the intro is very confusing and should be omitted in my opinion, so it's not clear to someone unfamiliar with the material what "recursive function" means in this case. It might easily be thought that "recurisive" is being used here as a synonym for "mu-recursive". This line also seems a bit irrelevant. — Preceding unsigned comment added by Luv2run (talk • contribs) 21:33, 26 October 2012 (UTC)

In a nutshell: the lead paragraph has been stable for 10 years. A bit of research in the history indicates that in May 2002 Axelboldt introduced a significant change in the first two or three sentences, then someone else came along and added the Ackermann function, and etc. You can ask Axelboldt, although I suggest that before spending any time on this and pestering Axelboldt, you should ask yourself the question: who cares? And given the answer: "I really care", then go through "the seven whys" ("Why do you care?" . . . "Because it's academically dishonest [etc]" . . . Why is it academically dishonest? . . . [etc]). BillWvbailey (talk) 23:13, 26 October 2012 (UTC)

The Princeton page says (at the bottom) "The article content of this page came from Wikipedia and is governed by CC-BY-SA." Tijfo098 (talk) 13:18, 12 November 2012 (UTC)

Is There A Contradiction In The Introduction?

The introduction states that all recursive functions are a part of the R class of functions. However; the mu-operator described in this very page is not expected to halt in some conditions, therefore clearly not in R. Am I missing something basic here? - 8 June 2019‎ 185.3.147.227

No, it says the set of all total recursive functions (with values in {0,1}) is R. The restriction "total" means all mu-operators involved in a function definition must eventually halt. - Jochen Burghardt (talk) 16:42, 8 June 2019 (UTC)

For the record, when the question was asked, the last sentence of the lead was wrongly stated, and there was therefore a contradiction between it and the remainder of the lead (thanks to the IP for pointing it). I have fixed this contradiction before Jochen reply, which uses the new formulation. I wrote a reply explaining this, but apparently, I forgot to save it. D.Lazard (talk) 08:31, 9 June 2019 (UTC)

Minimisation operator and partial functions again

It is not required to demand that function f in the definition of the μ-operator is total. The μ-operator can be implemented in any programming language by some code similar to

    μ(f)(x*) <= i := 0; while f(i, x*) > 0 do i := i + 1 end; return(i)

As f is a total and μ-recursive, f(i, x*) > 0 is decidable and consequently μ(f)(x*) is undefined iff f(i, x*) > 0 for each i (as the while-loop then never terminates). Now let

    μ'(f)(x*) <= i := 0; while f(i, x*) > 0 do i := i + 1 end; return(i)

for each (i.e. not necessarily total) function f. μ'(f)(x*) is undefined iff f(i, x*) > 0 for each i (as before) or f(i, x*) is undefined and f(j, x*) > 0 for each j < i (as the test in the while-statement then does not terminate). It is immediately obvious from the definition with the while-loop, that μ'(f) is a computable function, and that μ(f) = μ'(f) if f is total.

Example 1: If f(i, x) computes the residue class of i+1 mod x if x > 0 and is undefined for x = 0, then μ cannot be applied to f. μ'(f)(0) is undefined and μ'(f)(n+1) = n.

Example 2: If f(i) := 42 - i, if i ≤ 42, and f(i) := f(i+1) otherwise, then μ cannot be applied to f. μ'(f) is a total function as μ'(f) = 42.

The definition of μ-recursive functions gives a notation (syntax) for the definition of computable functions, as the syntax of any programming language L does. However, it is decidable whether a certain text defines a program of L (each compiler for L provides a decision procedure). But it is undecible, whether a certain text defines a μ-recursive function because it cannot be decided whether a computable function is total (this is even not semi-decidable in fact). By allowing the minimization of non-total functions, the notation of μ-recursive functions becomes decidable. Jack Rusell (talk) 13:04, 21 November 2019 (UTC)

You are right. This is the reason why, some time ago, I have added (the domain of a function defined by an operator is the set of the values of the arguments such that every function application that must be done during the computation provides a well-defined result) before the definition of the operators (see above thread). But I forgot to remove "total" in the definition of the μ operator. Now, this is done, and presently the article defines your function μ'. The formulation can certainly be improved, but, getting a formally correct result would needs a much deeper edit. D.Lazard (talk) 15:18, 21 November 2019 (UTC)

Thanks for the update. I have added a note as it may confuse readers when finding the demand for total functions in a textbook (and may also minimize traffic on the talk page). Btw. I do not understand your (the domain of a function ... well-defined result). Can you reformulate to make it understandable (at least for people like me). Jack Rusell (talk) 16:48, 21 November 2019 (UTC)

I have added an explanation in each case. Maybe the quoted sentence may be edited and removed. By the way, there was an edit conflict with your added note. I have tagged it because if some textbooks use the definition with "total", at list one must be cited. Also, as the general recursive functions have been originally defined by Gödel (this must be said in the article), the original definition must be checked. In any case, the assertion that the two definitions lead to the same class of function seems dubious, and, if true, it proof is certainly not easy. D.Lazard (talk) 17:42, 21 November 2019 (UTC)

 Comment: As for the {{citation needed}} after "the class of μ-recursive functions remains the same", I remember the reason: when a Turing machine is converted into an equivalent mu-recursive function, the minimalization needs to be use only a single time, viz. for the outermost loop, informally: while (not in a stop state) do (perform a single machine step) done. The loop body itself can be implemented by a primitive recursive function that looks up the program, writes the new symbol, moves the head, and changes the state. Maybe this helps to find a textbook reference (I just tried, but failed). - Jochen Burghardt (talk) 20:45, 21 November 2019 (UTC)

I just saw that this is mentioned in section μ-recursive function#Normal form theorem; a reference to there should suffice. However, the Kleene theorem there needs a citation in turn. - Jochen Burghardt (talk) 20:47, 21 November 2019 (UTC)

I have provided the required references. However, I was unable to cite without creating a duplicate entry in the reference list (I searched for a while). Maybe you (or someone else) can replace my non-standard citations with the correct command. I also provided references for the "same class" statement. It is obvious that μ-computable entails μ'-computable. I think a direct proof for the opposite needs some work. It is easier to use equivalence proofs, i.e. μ'-computable implies Turing-computable implies μ-computable. I checked my course notes where I proved equivalence of μ'-recursion with a simple while-language. When proving while-computable implies μ'-computable, the μ'-operator is applied to a total function in the proof (the function which evaluates the while-condition), hence my proof works fore the μ-operator as well. As I cannot give a reference to this proof here, I used Kleene's Normalform Theorem as an evidence. (PS: This comment was written before the former comment was, but sended later.) Jack Rusell (talk) 21:12, 21 November 2019 (UTC)

Historical section needed, and other things to do

Thanks to Jochen Burghardt and Jack Rusell for Having fixed my approximative knowledge of the subject. It appears from the preceding discussion that the historical context is completely lacking, although it is fundamental to really understand the subject. It could be provided by a history section where should appear (at least)

• Who introduced the concept, and for which purpose (AFIK, this is Gödel for the purpose of decidability theory, which is now a part of computability theory).
• Equivalence with lambda calculus and Turing machines (AFIK, this is due to Kleene), relationship with Church–Turing thesis.
• History of the terminology (AFIK, "μ-recursive function" is relatively recent, and Kleene used "general recursive function"). I ignore whether Gödel used "general recursive function" or simply "recursive function"). See also next thread.

Such an historical section seems the best way for explaining the choice of not imposing "total" for the minimization operator.

By the way there are many other things that deserve to be added to this article. In particular the section "Equivalence with other models of computability" need to be rewritten and expanded; the assertion that μ-recursive functions and Turing machines are "parallel" theories must be corrected.

Also, examples of μ-recursive functions that are not primitive recursive must be given (Ackermann function). D.Lazard (talk) 15:06, 22 November 2019 (UTC)

For the equivalence, an overview can be found in the Church–Turing thesis in footnote 6. Some snippets of that article's history information may also be useful here. - Jochen Burghardt (talk) 21:48, 22 November 2019 (UTC)

Proposed move

I suggest renaming this article General recursive function. If there is a consensus for that, I have the rights needed for doing the move. The reason for such a move are

• This is a much more common name: Scholar Goofle provides 1230 hits for "General recursive function" (with quotes), while mu-recursive and μ-recursive (without "function") provide 64 and 356 hits, respectively.
• This is the name used by the founders of the theory
• This would make the search easier (this is the reason for which I have created some time ago the redirect General recursive function, after a rather difficult search of the article about this concept).

Clearly a redirect must be kept after moving.

Opinions and comments? D.Lazard (talk) 15:30, 22 November 2019 (UTC)

The theory of computation considers different models of computation, such as Turing machines, μ-recursive functions, the λ-calculus, RAM-machines and more. Common to all models (which are completely different in their definitions) is that they all compute the same functions, i.e. they are equivalent in their computing power. This observation (different notions yields identic results) lead to Church's Thesis. One defines the class of general recursive functions as the class of functions which can be computed by either of these models (of computation). Therefore the title "μ-recursive functions" fits precisely what the article is all about. A title "General recursive functions" would be misleading because the article completely ignores the other models, Turing machines and the λ-calculus at least. For this reason, I suggest not to change the title. Jack Rusell (talk) 17:02, 22 November 2019 (UTC)

My view is that this article should be merged with computable function, because Soare has effectively succeeded in making it standard for "computable" to mean this class of functions, formally definable in a number of provably equivalent ways.
The downside is that it's (a little) harder to explain what Church's thesis actually is, because if computable function means a certain informally defined, intuitively comprehensible class of functions, then you can just say that Church's thesis is the assertion that all computable functions are recursive. If all computable functions are recursive by definition, that doesn't seem to say anything.
But as unfortunate as that might be, that ship has sailed; this is what "computable" means, now. When explaining Church's thesis, we just have to find other language. --Trovatore (talk) 06:56, 23 February 2020 (UTC)

Before Church's thesis, the term "computable function" was informally defined. It is for elaborating a formal definition that general recursive functions, Turing machines and lambda calculus have been introduced. The fundmental theorem that supports Church's thesis is that these three models of computation compute the same functions. So, the modern definition of a "computable function" is a function than can be computed by any of these models of computation (and many others). As each of these models of computation are rather technical to define, many authors focuse on one of them. As far as I understand, general recursive functions are preferred by pure mathematicians and philosophers, because they are closer to their way of thinking. In computational complexity theory, computability is usually defined by Turing machines. In high level programming and semantics (computer science), lambda-calculus is generally preferred, as making easier self referencing computations (universal machines, and functions that define other functions and use them for further computations).

So, I oppose the proposed merge with computable function. This article is about the equivalence between the main models of computation, while General recursive function is about one of them. Merging would make computable function too technical for many readers, who do not need to learn the technical definitions, and, overall would breaks WP:NPOV policy by privileging one theory over the others. D.Lazard (talk) 09:13, 23 February 2020 (UTC)

By the way: Total recursive function currently redirects to Computable function. By D.Lazard's argument, it should better redirect to General recursive function. Any objections? - Jochen Burghardt (talk) 15:29, 23 February 2020 (UTC)

Fine for me. D.Lazard (talk) 16:25, 23 February 2020 (UTC)

 Done Per apparent consensus. - Jochen Burghardt (talk) 22:02, 23 February 2020 (UTC)

I don't actually agree with that, as I mention below (though I made the comment after your change). I think someone linking or searching "total recursive function" is more likely interested in the class of functions, rather than the "implementation-specific" details of how the class is defined. --Trovatore (talk) 22:35, 23 February 2020 (UTC)

It seems to me that, if the article is really about the model of computation rather than the class of functions, then it's misnamed. I'm not sure what the name should be, but it's misleading to name it after a class of functions if it's really about a model of computation.

Also, in that case, this article is much less important than computable function; it's logically just a sub-section of that one. This should be indicated somehow in the article. --Trovatore (talk) 21:16, 23 February 2020 (UTC)

I agree with D.Lazard that General recursive function and computable function should not be merged; it seems that the latter article is long enough by its own, containing only the information common to all models of computation. And I agree with Trovatore that the logical sub-section relation should be made clear. As for the article name, I don't know a better one that is somewhat common in the literature. As a kind of justification, one might say that general recursive functions have domain and range ℕ, while lambda functions operate on lambda terms, and Turing machines operate on strings (or even "tapes"?). The equivalence proofs underlying the Chruch-Turing thesis construct isomorphisms from one function class to another, this is (slightly) more than just establishing mutual subset properties. - Jochen Burghardt (talk) 22:02, 23 February 2020 (UTC)

My concern is that the article appears to be about a class of functions but apparently is not. This is confusing to the reader. It could be moved to a more descriptive title, perhaps definition of computable functions by μ operator or some such. I think total recursive function and general recursive function should point to computable function, as these are synonyms (they didn't use to be, pre-Soare, but they are now). --Trovatore (talk) 22:09, 23 February 2020 (UTC) Oh — "synonyms" except for the "total" bit, of course; that was my goof. --Trovatore (talk) 22:26, 23 February 2020 (UTC)

I see your point, but I guess nobody would enter "definition of computable functions by μ operator" in a search field. Using "What links here?", I found that the following pages redirect to General recursive function - maybe one of these would be better suited as original (rather than redirected) title: General recursive - General-recursive - M-recursive function - Mu recursive function - Mu-recursive - Mu-recursive function - Partial recursive function - Recursive function (computability) - Recursive function theory - Total recursive function - Μ recursion - Μ-recursive function. Following your argument, names ending in "recursion" are particularly interesting; e.g. what about Recursion (computability)?

As a related issue, following your objection to my hasty re-targeting, names ending in "function" should better link to Computable function. I'm not convinced about that: looking for e.g. "M-recursive function" would end up at Computable function, while looking for "Μ recursion" would end up in General recursive function; this would be likely to lead to confusion. On the other hand, once the logical sub-section relation is made clear, a user ending up in General recursive function will easily find the way to the more general Computable function is it's that (s)he was looking for. - Jochen Burghardt (talk) 08:03, 24 February 2020 (UTC)

Two basic comments: firstly, if the common terminology is not formally correct, we must not change it to a teminology that is more correct. Secondly, the theory of computability is independent of modern set theory ZFC. It is often manipulated in other logical frameworks, such as intuitionistic logic. Therefore, this article must avoid to be too much dependent on set theory. This is for this reason that I have edited the lead for removing the word "class" from the first sentence. It is also a reason for opposing to the merge, as Computable function is about a set or class of functions, while this article is about individual functions.

Also, there is some ambiguity in the literature whether a (general) recursive funtion is a function that can be defined by a recursive process or is the pair of a function and a recursive process. Clearly it is the second interpretation that must be preferred, as otherwise the following theorem would be nonsensical: "There are general recursive functions such that it is computationally undecidable whether they equal the constant function zero." As this ambiguity is common in the literature, it is not our job to avoid it in the article title, although the content of the article could deserve to be edited for clarifying this point, or, at least, avoiding to make the ambiguity confusing.

As the concept described in the article is known either as "general recursive function" or "μ-recursive function" but has no commonly used other names, the article title must be one of these two names, and changing it would be WP:OR. My move from the latter title to the former one could be discussed, but the former name seems more common, specially when used outside the theory of recursive functions. Also, the latter title appears in the search engine as "M-recursive function", which is confusing, as it may be difficult for the reader to understand that "M" means "μ". D.Lazard (talk) 11:59, 24 February 2020 (UTC)

This is a bit of a multithreaded discussion with a lot of facets, which is always hard to keep track of. Let me try a bulleted list and see if that helps.

• Thanks Jochen Burghardt for pointing out the redirect μ recursion. I think that's a better title for this article than any so far proposed (though there might be still better options; e.g. I would probably prefer μ-recursion with the hyphen). The reason is that we seem to be agreed that this article is about a model of computation, and μ-recursion is a model of computation.
• I agree with D.Lazard that we can't make up our own names for things; if a name is standard in the literature then we use it. However I am skeptical that "general recursive function" truly standardly refers to the model of computation we're discussing, as opposed to the functions thereby computed. I also think that anywhere "general recursive function" is likely to be linked from, or most of the searches for it, will be more usefully served by finding an article about the functions rather than the model of computation.
• Therefore general recursive function, total recursive function, and all similar phrases with "function" as the head, should point to computable function.
• The fact that the μ gets capitalized in lists is unfortunate, but the damage is limited by the fact that this article is very rarely the first one a reader should be searching for. Most times, the reader will be better served by reading computable function first. Only if he/she truly wants to know specifically about this model of computation is this the right article, and in that case hopefully he/she can find it via a link from another article.
• I agree that we aren't talking about set theory — for "class of functions" substitute "kind of function", if you like. The kind of function computed by the various models are all the same, and are treated at computable function. The differences in model of computation are "implementation details", like different programming languages if you will.
• The point about including the computation as part of the function in some sense is generally well-taken, but doesn't really change anything in this discussion. The computation that computes a given function can be effectively translated between different models of computation in a completely stereotyped way, and you can prove using very mild assumptions (and I'm pretty sure, even in intuitionistic logic) that they compute the same function. Therefore we can still speak of whether a computable function is provably total (for example) without worrying about which model of computation is used.

Well, that's a lot, better stop here. The big takeaway is we should move this content to a name that clearly names a model of computation, for example μ-recursion, and all the "function" links should point at computable function. --Trovatore (talk) 21:18, 24 February 2020 (UTC)

Could "partial recursive function" please be clarified?

I'm removing this edit request because I looked it up in a textbook. I'll make a small clarification. TricksterWolf (talk) 13:16, 14 February 2021 (UTC)

Total Recursive Function redirects here.

Total Recursive Function redirects here, and it should not. Total recursive functions are functions that always halt, and this article is about general functions (or partial functions) that may or may not halt. The concept of 'total functions' is very interesting in itself and deserves its own article, so if 'total recursive function' means 'total function' then 'total recursive function' should redirect there. If it doesn't mean that then maybe it should get its own article. Comiscuous (talk) 06:02, 7 April 2021 (UTC)

I think the redirect is OK, but it (currently) may be confusing.

In Wikipedia, there are many redirects of subtopics to more general topics. The most recent example I remember is Harvard minimizing chart redirecting to Logic optimization, although the former is one of the lesser-known logic optimization methods.

To reduce the amount of confusion, we should add a section "Total recursive function" and link Total recursive function to General recursive function#Total recursive function. This would make clear that the redirect concerns a special case. - Jochen Burghardt (talk) 09:44, 7 April 2021 (UTC)

 Done I started a section as suggested, and re-targeted Total recursive function to there. Please help me in fleshing-out the new section. - Jochen Burghardt (talk) 09:59, 7 April 2021 (UTC)

I want to renew my point above that this article is misnamed for the content. The content is about a model of computation, whereas the name appears to be about a kind of function. I still think the best title for this content is μ-recursion (yes, I'm aware it looks like M-recursion in lists; that's unfortunate but I think it's still the right name). --Trovatore (talk) 19:09, 7 April 2021 (UTC)

The title of the article is the name given by the inventor(s) of the concept, and used by many textbooks. Wikipedia cannot and must not change this. Moreover, all the content is about a class of functions. The article is thus not misnamed, as named with a common name of these functions. D.Lazard (talk) 20:17, 7 April 2021 (UTC)

No, the content is not about a class of functions. The class of functions is the computable functions, which are defined in several different ways, all provably equivalent and mechanically inter-translatable. This article is about one such way, not about a different class of functions. --Trovatore (talk) 20:22, 7 April 2021 (UTC)

No, no, no. The article defines "general recursive functions" or "mu-recursive functions", and the first section is devoted to their definition. This defines a class of functions, which is the main subject of the article. This is a remarkable result (supporting Church–Turing thesis) that every tentative for an acceptable definition of computable functions leads to the same class of functions. This does not change the fact that this is a class of functions that is defined here. D.Lazard (talk) 20:54, 7 April 2021 (UTC)

The class of functions should have a single article, no matter which model of computation is used. The natural (post-Soare) name for that class is "computable function", and that is where the content regarding the class of functions (as opposed to this specific definition/model) should reside. --Trovatore (talk) 20:57, 7 April 2021 (UTC)

(By the way, the Church–Turing thesis is not really relevant to this discussion. In the post-Soare era, the term "computable function" no longer means "informally computable function". It means "function obtained in one of these precise and provably equivalent ways". The Soare redefinition of "computable" makes it a little harder to state the Church–Turing thesis, because there's no longer a convenient phrase for "informally computable function". In my view that's a little unfortunate, but there's not much we can do about it; this is the nomenclature as it has evolved.) --Trovatore (talk) 21:03, 7 April 2021 (UTC)

I now see that I had misunderstood D.Lazard's position a bit. I thought he wanted the article to be about functions using this specific model of computation, but give it a name that makes it look like a class of functions. Instead it appears (unless I've misunderstood again, which is certainly possible) that he wants this to be the general article about the class of functions, and he just wants to pick one definition.
I see two problems with that. First, I don't see any reason in that case to privilege the definition by μ-recursion. It's the most traditional, perhaps the oldest, but it's not really the most perspicuous.
The other main problem is that it ignores the Soare-renaming, which I am given to understand has now become almost entirely standard.
So my view is still what I said above, with the refinement that the computable function article needs to be somewhat revised to be about the precise class of functions, and not about "informally computable functions". --Trovatore (talk) 22:58, 7 April 2021 (UTC)

I'm not sure I understood all subtleties of your above discussion. Anyway, a look at Regular language, Regular grammar, Regular expression, Nondeterministic finite automaton, Deterministic finite automaton might be inspiring for the problem we have here. The first article is about the language class, each of the remaining ones is about one particular mechanism to describe class members (all mechanisms lead to the the same class). Of course, e.g. Deterministic finite automaton speaks about the mechanism (automaton definition) and the result of using it (languages accepted by a DFA); however, general properties of the latter are outsourced to the Regular language article. I think the situation is quite similar here. Following that scheme, a title "μ-recursion" would be preferred about "μ-recursive function", but the article would speak about both. - Jochen Burghardt (talk) 14:26, 8 April 2021 (UTC)

What I'm saying is, the analogue of the regular language article is or should be called computable function, not "general recursive function". That's the effect of the linguistic shift led by Soare in, I think, the nineties? or the aughts at latest. It is quite standard by now. --Trovatore (talk) 16:42, 8 April 2021 (UTC)

@Trovatore: Thanks, that helped me to understand your above discussion with D.Lazard. I cannot comment on a linguistic shift, since I'm not a native English speaker, and, even worse, were forced to work as an applied computer scientist who had no professional contact with computability theory. However, after re-reading the above discussion, I'd be in favor of following the "Regular language" scheme: having a "main" article named after the function class ("Computable function" or "General recursive function" or similar; note that the "regular" in "regular" language originated from one of its defining mechanisms, see the last paragraph of Regular language#Equivalent formalisms), having one article for each of the equivalent defining mechanisms, named after the mechanism (e.g. "Lambda calculus", rather than "Lambda-definable function"), moving as much common stuff to the "main" article (including brief discussion of the Church-Turing thesis). My impression is that the editing effort to achieve such a structure would be minimized if we start from the current Computable function (possibily renamed) as "main", and the articles linked under Computable function#Definition as "subordinate"; this implies my favor for renaminig General recursive function to (e.g.) μ-recursion, to refer to the defining mechanism, not the function class. - Jochen Burghardt (talk) 18:46, 8 April 2021 (UTC)

Yes, I agree. Frankly computable function is in a bit better state than this article at the current time. This article is mostly the definition and some syntax for the definition. There are three brief sections that are about the functions; they could be moved to computable function. At the computable function article, we should make sure there's content about normal forms, closure properties, where the functions sit between generalizations (computability relative to oracles, for example) and finer analysis (for example the polynomial hierarchy).

And my current thought is that μ-recursion is indeed the best name for "this" article, meaning the article that discusses this particular definition/model of computation. I'm open to other suggestions as long as they don't have the word "function" as the head. --Trovatore (talk) 18:08, 9 April 2021 (UTC)

Related discussion

Note that there is a related discussion at Wikipedia talk:WikiProject Mathematics#Proposal: change terminology from "recursive" to "computable". --Trovatore (talk) 05:14, 23 April 2021 (UTC)