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Should this be merged with Recursive function? -- Saforrest 00:27, 27 January 2006 (UTC)
Well imho you should make it clear already on the rec.func. page that a "recursive function" is not the thing that most (unmathematical) people would first think it is, i.e. a function that references itself. --Sigmundur
Isn't this definition of a computable function:
A partial function is called computable if the graph of is a recursively enumerable set.
circular with the definition of a recursively enumerable set:
A countable set S is called recursively enumerable if there exists a partial computable function such that is the range of ? -- Michael Stone 00:24, 11 March 2006 (UTC)
The circularity is resolved in practice by defining either computable function or c.e. set directly in terms of Turing machines or some other class of computing devices. The wikipedia artcles that use the word recursive seem to do this.
--
CMummert
02:14, 20 March 2006 (UTC)
This article is redundant with recursive function and they should be merged. I don't really care whether the merged article is here or at recursive function, but definitely it's wrong to have two articles, since by the definitions given in each article, the two sets of functions are provably coextensive.
It would be theoretically possible to have an article at computable function for functions that are informally computable, and to put the information about the formally-defined class of functions at recursive function, with a note that Church's thesis is the (unformalizable, and therefore unprovable) claim that the two classes of functions are coextensive. But I don't think that would be a good idea, given the trend, started by Robert I. Soare, to use "computable" for the formal notion. -- Trovatore 00:47, 11 March 2006 (UTC)
I have problems finding any info (on Wikipedia) about functions which are not computable. Now, it could be there exists an article about such a topic and that I should just look harder, but if it is so, the I think it should be linked from "See also" section of this article. If not, what a shame, lets write one :-) -- Dijxtra 09:18, 7 April 2006 (UTC)
I rewrote and significantly extended the article today. I tried to keep it balanced between Recursion theory and Computability theory (computer science). I also tried to leave in anything that was already there, to the extent possible. I made the examples of uncomputable functions more explicit, per the comment higher on the talk page. CMummert 23:33, 16 July 2006 (UTC)
First, I need to emphasize that I am a new learner in this field and I am reading this article from a pure mathematical point of view.
As I finish the article, I still don't get a clue as to what the precised mathematical definition is for computable function. Is it that any function that maps an n-tuple of natural numbers into a natural number is considered a computable function?
In the following, I want to quote some statements from the article and give my comments on them.
"Computable functions can be used to discuss computability without refering to any concrete model of computation such as Turing machines or register machines. Their definition, however, must make some reference to some specific model of computation."
My comments: It appears from the second statement that you cannot define computability without first knowing what model you are talking about. If this is the case, why shouldn't we be talking about Turing Machines computable or Register Machines computable instead of talking about computable functions in general. Why doesn't the author simply make it clear that there is no such thing as computable function in general?
"Equivalently, this thesis states that any function which has an algorithm is computable."
My comments: What is the mathematical definition for algorithm? Does this statement imply that a function that is not computable today may be computable some day when an algorithm is found for it? (Note: You can define algorithm as a finite procedure but then how would you define finite procedure mathematically?).
Finally, I believe that we'd better have a precised mathematical definition for any object that we have to deal with if we are to be able to resolve many of the important issues in this field (e.g. P vs NP).
CBKAtTopsails 11/29/2006
I went back to read the article again and I still couldn't find a definition based on which mathematical work can be performed. The basic question is do we or do we not have a precise mathematical definition for computable function on this web? If so, where is it? I must stress again that I am pretty new in this field and my interest in this stuff was solely motivated by the curiosity about a mathematical solution to the P vs NP problem. I do not know why things are treated the way they are and whether or not there are any clearly defined objectives to accomplish and therefore I don't quite understand why so much time was spent on debating the separating and merging of topics instead of providing more precise definitions for terminologies and more rigorous treatment of the concepts. I do feel that things are very loosely tied in this field. CBKAtTopsails 11/30/2006
First, thank you for your courtesy of responding. It is now my understanding based on your comment that computable function = mu recursive function. On that basis, I went into the article of Mu-recursive function. In this article, Mu-recursive function is defined as follows:
It is a function that maps a finite tuple of natural numbers into a single natural number and it is a member of a set of functions ( Let's call it C )that satifies the following conditions:
(i) The set contains the initial functions (1), (2), (3) and
(ii) The set is closed under operations (4), (5) & (6).
I would appreciate it if you can clarify the following for me:
(a) With respect to operation (4) (Composition of Operator), what are the functions gi's? Are they simply arbitrary elements from C? Apparently, given a set of gi's there are infinitely many ways of defining h. So, my second question is : Is h also an arbitrary function taken from C? Does closeness here means that if h and the gi's belong to C, then f belongs to C. Finally, I notice a little contradiction about the function h. The article first writes h(x1,....,xk) and then h(g1,.....,gm). This will result in conflicting definitions for h if m is not equal to k.
(b) With respect to operation (5) (Primitive Recursion Operator), what are g and h? Are they also arbitrary functions taken from C? Why does the article write g(x1,....,xk) and then g(x2,...,xk)? Does it mean that the domain of g contain both k-tuples and (k-1)-tuples? If so, g cannot belong to C. (Same question for h). Finally, does "closed under primitive recursion operator" in this case means if g and h belong to C, then f belongs to C. If so, how can f have 0 as the first argument (0 is not a natural number). Furthermore, f is not well defined because we don't know how to compute f(y,x2,....,xk) given only g(x1,....,xk) and h(y,z,x1,....,xk).
(c) With respect to operation (6) (Mu Operator), first, a little inconsistency about the function Muyf. Muyf(y, x1,.....,xk) clearly indicates that the function has (k+1) arguments but the article says "whose arguments are x1,....,xk. It is not clear to me what the precise definition is for Mu. I can only take a guess (from what I read) and my guess is the following (Correct me if I am wrong):
Given a function f from C whose domain consists of (k+1)-tuples and given a k-tuple, (x1,.....,xk), if there exists a y such that f(0, x1,....,xk), f(1, x1,......,xk),............f(y,x1,.....xk) are defined and that f(y,x1,.....,xk)=0 then Muyf(x1,....xk)=y. However, under this definition, "f(0,x1,.....,xk) is defined." immediately disqualifies f to be a member of C because 0 is not a natural number. Furthermore, how can I prove that if f is from C, then Muyf belongs to C?
CBKAtTopsails (11/30/2006)
When I typed my comments in the edit box, I did have separate paragraphs for each comment. I don't know why it came out the way it looks after I saved the page. Maybe you can offer some help on this. Do you have written instructions on how to use this edit box? CBKAtTopsails (12/1/2006).
One other comment. Presumably, I can go back to the Mu Recursive Function article to do more reading about the article and the discussion about that article but my original plan was to do some research on computable functions. I still have some doubts in my mind whether computability should be equated to recursiveness. If this topic is to be made a separate topic from the Mu Recursive Function, shouldn't it have its own definition? Why do we have two different names? I am interested in knowing whether there is a consensus in this field on how computability should be defined. CBKAtTopsails (12/1/2006).
The problem with equating the computable functions with the mu-recursive or Turing-computable functions is that the former (and probably the latter, depending upon whether you take the output of a Turing machine to be a numeral or a number) can only have sets of natural numbers as their range; so if we equate computability with recursive function-hood or Turing-computability, goedel numbering functions (or their inverses), for example, cannot be computable (since the range of one of them will be a set of strings of some alphabet), though they clearly are computable in the intuitive sense.
I propose that we not equate computable functions with any well-defined class of functions of natural numbers anywhere in the article. We should leave the discussion, warts and all, in terms of "algorithms," "finite procedures," etc., and e.g. note explicitly that the Church-Turing thesis is plausible only when restricted to functions f from subsets of NN to N.
That being said, cleaning up all the articles conceptually related to this one would be a pain in the ass. But if there are no objections I'd volunteer to do this one, leaving intact as much as possible. -- Futonchild 19:34, 1 March 2007 (UTC)
- In this way, we can list out all theorems, i.e., exactly all the valid formulas of first-order logic, can be listed out by a simple mechanical procedure. More precisely, the set of valid formulas is the range of a computable function. In modern terminology we say that the set of valid formulas of first-order logic is recursively enumerable (r.e.).
- This, then, is the "working hypothesis" that, in effect, Church proposed:
- Church's thesis:
- A function of positive integers is effectively calculable only if recursive.
- In this connection, it is important to remember that in the technical literature the word ‘computable’ is often tied by definition to effective calculability. Thus a function is said to be computable if and only if there is an effective procedure for determining its values. Accordingly, a common formulation of the Church-Turing thesis in the technical literature and in textbooks is:
- All computable functions are computable by Turing machine.
I'm not quite sure what a better phrasing should be but
seems awkward or even inaccurate . If I get a chance I'll try to think about something better.
From the description of Church's Thesis.
Later on, an example of a computable function ...
If I am to take the rendition of Church's thesis on face value, this example seems wrong. So, if for a certain n, call it m, there is no sequence of m 5s in the expansion of pi, then (1) the function is defined at m and (2) the procedure will continue ad infinitum without producing a result. This contradicts the second statement, above. Church's thesis, as described here, says nothing about the procedure going on forever when given an input which _is_ in the domain of f. It only speaks, in the third part, of going on forever if given an input which is _not_ in the domain of f. So, I think that the example should be:
The function f whose domain is {n in the Natural numbers | there is a sequence of n consecutive fives in the decimal expansion of pi} and which is defined to be 1 on that domain, is computable.
Could someone please either correct this function of correct Church's thesis? —Preceding unsigned comment added by 72.74.113.222 ( talk) 05:50, 7 September 2007 (UTC)
I think the claim about the busy beaver function: "every finite sequence of Σ values, such as Σ(0), Σ(1), Σ(2), ..., Σ(n) for any given n, is (trivially) computable" is false.
At the very least, it's contradictory to the claim on the busy beaver page that "there is a true-but-unprovable sentence of the form 'Σ(10↑↑10) = n' "
As, if the fist statement were true, one should "trivially" be able to compute the finite sequence Σ(0), Σ(1), Σ(2), ..., Σ(10↑↑10) thus "trivially" computing Σ(10↑↑10).
Also, as a general note, I think articles should avoid claims that something is "(trivially) computable" unless a simple proof is given or referenced.
— Preceding
unsigned comment added by
18.111.124.28 (
talk •
contribs) 18:25, 4 October 2011
Trovatore, the busy beaver numbers are eventually independent of set theory by Chaitin's theorem, aren't they? For example, the authors here: http://www.scottaaronson.com/busybeaver.pdf exhibit a 7918-state Turing machine that ZFC can't prove runs forever, but that a slightly-augmented version of ZFC can, which seems to me to suggest that BB(7918) cannot be defined in ZFC. We might do well to mention that limitation, here, if someone can find appropriate sources, since it means the phrase "the finite sequence {BB(1), BB(2), ..., BB(7918)}" cannot be assigned a definition (in computer science based on ZFC, anyway), and hence cannot be computable, even though any given sequence of 7918 natural numbers is of course computable. — Preceding unsigned comment added by 50.58.96.2 ( talk) 15:48, 21 August 2017 (UTC)
Please note that most standard texts refer to these functions as UNcomputable and not INcomputable. — Preceding unsigned comment added by 132.205.45.13 ( talk) 17:08, 23 August 2012 (UTC)
"Similarly, most subsets of the natural numbers are not computable. The halting problem was the first such set to be constructed. The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements (coded as natural numbers) are true. Turing and Church independently showed in the 1930s that this set of natural numbers is not computable. According to the Church–Turing thesis, there is no effective procedure (with an algorithm) which can perform these computations."
The first sentence is hard enough to interpret (do you mean finite or infinite subsets of the natural numbers?). But then you move on to "the halting problem was the first such set to be constructed". This makes no sense. Firstly, whatever you mean by that statement needs to be unpacked at the bare minimum (if not just thrown out.). Secondly, on it's face, problems are not sets. The halting problem is about computer programs that could be implemented in any turing complete language. A function that determines halting of arbitrary programs for any language in this class of languages would lead to paradox. You in no way make it clear how this relates to any uncomputable subset of the natural numbers. But then you jump to entscheidungsproblem and you're talking about statements encoded as natural numbers. How does the uncomputability of some statements encoded as natural numbers imply the uncomputability of some (finite? infinite?) subsets of natural numbers? This whole paragraph makes no sense and needs to be rewritten (or maybe just thrown out). Comiscuous ( talk) 06:39, 20 January 2022 (UTC)
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Should this be merged with Recursive function? -- Saforrest 00:27, 27 January 2006 (UTC)
Well imho you should make it clear already on the rec.func. page that a "recursive function" is not the thing that most (unmathematical) people would first think it is, i.e. a function that references itself. --Sigmundur
Isn't this definition of a computable function:
A partial function is called computable if the graph of is a recursively enumerable set.
circular with the definition of a recursively enumerable set:
A countable set S is called recursively enumerable if there exists a partial computable function such that is the range of ? -- Michael Stone 00:24, 11 March 2006 (UTC)
The circularity is resolved in practice by defining either computable function or c.e. set directly in terms of Turing machines or some other class of computing devices. The wikipedia artcles that use the word recursive seem to do this.
--
CMummert
02:14, 20 March 2006 (UTC)
This article is redundant with recursive function and they should be merged. I don't really care whether the merged article is here or at recursive function, but definitely it's wrong to have two articles, since by the definitions given in each article, the two sets of functions are provably coextensive.
It would be theoretically possible to have an article at computable function for functions that are informally computable, and to put the information about the formally-defined class of functions at recursive function, with a note that Church's thesis is the (unformalizable, and therefore unprovable) claim that the two classes of functions are coextensive. But I don't think that would be a good idea, given the trend, started by Robert I. Soare, to use "computable" for the formal notion. -- Trovatore 00:47, 11 March 2006 (UTC)
I have problems finding any info (on Wikipedia) about functions which are not computable. Now, it could be there exists an article about such a topic and that I should just look harder, but if it is so, the I think it should be linked from "See also" section of this article. If not, what a shame, lets write one :-) -- Dijxtra 09:18, 7 April 2006 (UTC)
I rewrote and significantly extended the article today. I tried to keep it balanced between Recursion theory and Computability theory (computer science). I also tried to leave in anything that was already there, to the extent possible. I made the examples of uncomputable functions more explicit, per the comment higher on the talk page. CMummert 23:33, 16 July 2006 (UTC)
First, I need to emphasize that I am a new learner in this field and I am reading this article from a pure mathematical point of view.
As I finish the article, I still don't get a clue as to what the precised mathematical definition is for computable function. Is it that any function that maps an n-tuple of natural numbers into a natural number is considered a computable function?
In the following, I want to quote some statements from the article and give my comments on them.
"Computable functions can be used to discuss computability without refering to any concrete model of computation such as Turing machines or register machines. Their definition, however, must make some reference to some specific model of computation."
My comments: It appears from the second statement that you cannot define computability without first knowing what model you are talking about. If this is the case, why shouldn't we be talking about Turing Machines computable or Register Machines computable instead of talking about computable functions in general. Why doesn't the author simply make it clear that there is no such thing as computable function in general?
"Equivalently, this thesis states that any function which has an algorithm is computable."
My comments: What is the mathematical definition for algorithm? Does this statement imply that a function that is not computable today may be computable some day when an algorithm is found for it? (Note: You can define algorithm as a finite procedure but then how would you define finite procedure mathematically?).
Finally, I believe that we'd better have a precised mathematical definition for any object that we have to deal with if we are to be able to resolve many of the important issues in this field (e.g. P vs NP).
CBKAtTopsails 11/29/2006
I went back to read the article again and I still couldn't find a definition based on which mathematical work can be performed. The basic question is do we or do we not have a precise mathematical definition for computable function on this web? If so, where is it? I must stress again that I am pretty new in this field and my interest in this stuff was solely motivated by the curiosity about a mathematical solution to the P vs NP problem. I do not know why things are treated the way they are and whether or not there are any clearly defined objectives to accomplish and therefore I don't quite understand why so much time was spent on debating the separating and merging of topics instead of providing more precise definitions for terminologies and more rigorous treatment of the concepts. I do feel that things are very loosely tied in this field. CBKAtTopsails 11/30/2006
First, thank you for your courtesy of responding. It is now my understanding based on your comment that computable function = mu recursive function. On that basis, I went into the article of Mu-recursive function. In this article, Mu-recursive function is defined as follows:
It is a function that maps a finite tuple of natural numbers into a single natural number and it is a member of a set of functions ( Let's call it C )that satifies the following conditions:
(i) The set contains the initial functions (1), (2), (3) and
(ii) The set is closed under operations (4), (5) & (6).
I would appreciate it if you can clarify the following for me:
(a) With respect to operation (4) (Composition of Operator), what are the functions gi's? Are they simply arbitrary elements from C? Apparently, given a set of gi's there are infinitely many ways of defining h. So, my second question is : Is h also an arbitrary function taken from C? Does closeness here means that if h and the gi's belong to C, then f belongs to C. Finally, I notice a little contradiction about the function h. The article first writes h(x1,....,xk) and then h(g1,.....,gm). This will result in conflicting definitions for h if m is not equal to k.
(b) With respect to operation (5) (Primitive Recursion Operator), what are g and h? Are they also arbitrary functions taken from C? Why does the article write g(x1,....,xk) and then g(x2,...,xk)? Does it mean that the domain of g contain both k-tuples and (k-1)-tuples? If so, g cannot belong to C. (Same question for h). Finally, does "closed under primitive recursion operator" in this case means if g and h belong to C, then f belongs to C. If so, how can f have 0 as the first argument (0 is not a natural number). Furthermore, f is not well defined because we don't know how to compute f(y,x2,....,xk) given only g(x1,....,xk) and h(y,z,x1,....,xk).
(c) With respect to operation (6) (Mu Operator), first, a little inconsistency about the function Muyf. Muyf(y, x1,.....,xk) clearly indicates that the function has (k+1) arguments but the article says "whose arguments are x1,....,xk. It is not clear to me what the precise definition is for Mu. I can only take a guess (from what I read) and my guess is the following (Correct me if I am wrong):
Given a function f from C whose domain consists of (k+1)-tuples and given a k-tuple, (x1,.....,xk), if there exists a y such that f(0, x1,....,xk), f(1, x1,......,xk),............f(y,x1,.....xk) are defined and that f(y,x1,.....,xk)=0 then Muyf(x1,....xk)=y. However, under this definition, "f(0,x1,.....,xk) is defined." immediately disqualifies f to be a member of C because 0 is not a natural number. Furthermore, how can I prove that if f is from C, then Muyf belongs to C?
CBKAtTopsails (11/30/2006)
When I typed my comments in the edit box, I did have separate paragraphs for each comment. I don't know why it came out the way it looks after I saved the page. Maybe you can offer some help on this. Do you have written instructions on how to use this edit box? CBKAtTopsails (12/1/2006).
One other comment. Presumably, I can go back to the Mu Recursive Function article to do more reading about the article and the discussion about that article but my original plan was to do some research on computable functions. I still have some doubts in my mind whether computability should be equated to recursiveness. If this topic is to be made a separate topic from the Mu Recursive Function, shouldn't it have its own definition? Why do we have two different names? I am interested in knowing whether there is a consensus in this field on how computability should be defined. CBKAtTopsails (12/1/2006).
The problem with equating the computable functions with the mu-recursive or Turing-computable functions is that the former (and probably the latter, depending upon whether you take the output of a Turing machine to be a numeral or a number) can only have sets of natural numbers as their range; so if we equate computability with recursive function-hood or Turing-computability, goedel numbering functions (or their inverses), for example, cannot be computable (since the range of one of them will be a set of strings of some alphabet), though they clearly are computable in the intuitive sense.
I propose that we not equate computable functions with any well-defined class of functions of natural numbers anywhere in the article. We should leave the discussion, warts and all, in terms of "algorithms," "finite procedures," etc., and e.g. note explicitly that the Church-Turing thesis is plausible only when restricted to functions f from subsets of NN to N.
That being said, cleaning up all the articles conceptually related to this one would be a pain in the ass. But if there are no objections I'd volunteer to do this one, leaving intact as much as possible. -- Futonchild 19:34, 1 March 2007 (UTC)
- In this way, we can list out all theorems, i.e., exactly all the valid formulas of first-order logic, can be listed out by a simple mechanical procedure. More precisely, the set of valid formulas is the range of a computable function. In modern terminology we say that the set of valid formulas of first-order logic is recursively enumerable (r.e.).
- This, then, is the "working hypothesis" that, in effect, Church proposed:
- Church's thesis:
- A function of positive integers is effectively calculable only if recursive.
- In this connection, it is important to remember that in the technical literature the word ‘computable’ is often tied by definition to effective calculability. Thus a function is said to be computable if and only if there is an effective procedure for determining its values. Accordingly, a common formulation of the Church-Turing thesis in the technical literature and in textbooks is:
- All computable functions are computable by Turing machine.
I'm not quite sure what a better phrasing should be but
seems awkward or even inaccurate . If I get a chance I'll try to think about something better.
From the description of Church's Thesis.
Later on, an example of a computable function ...
If I am to take the rendition of Church's thesis on face value, this example seems wrong. So, if for a certain n, call it m, there is no sequence of m 5s in the expansion of pi, then (1) the function is defined at m and (2) the procedure will continue ad infinitum without producing a result. This contradicts the second statement, above. Church's thesis, as described here, says nothing about the procedure going on forever when given an input which _is_ in the domain of f. It only speaks, in the third part, of going on forever if given an input which is _not_ in the domain of f. So, I think that the example should be:
The function f whose domain is {n in the Natural numbers | there is a sequence of n consecutive fives in the decimal expansion of pi} and which is defined to be 1 on that domain, is computable.
Could someone please either correct this function of correct Church's thesis? —Preceding unsigned comment added by 72.74.113.222 ( talk) 05:50, 7 September 2007 (UTC)
I think the claim about the busy beaver function: "every finite sequence of Σ values, such as Σ(0), Σ(1), Σ(2), ..., Σ(n) for any given n, is (trivially) computable" is false.
At the very least, it's contradictory to the claim on the busy beaver page that "there is a true-but-unprovable sentence of the form 'Σ(10↑↑10) = n' "
As, if the fist statement were true, one should "trivially" be able to compute the finite sequence Σ(0), Σ(1), Σ(2), ..., Σ(10↑↑10) thus "trivially" computing Σ(10↑↑10).
Also, as a general note, I think articles should avoid claims that something is "(trivially) computable" unless a simple proof is given or referenced.
— Preceding
unsigned comment added by
18.111.124.28 (
talk •
contribs) 18:25, 4 October 2011
Trovatore, the busy beaver numbers are eventually independent of set theory by Chaitin's theorem, aren't they? For example, the authors here: http://www.scottaaronson.com/busybeaver.pdf exhibit a 7918-state Turing machine that ZFC can't prove runs forever, but that a slightly-augmented version of ZFC can, which seems to me to suggest that BB(7918) cannot be defined in ZFC. We might do well to mention that limitation, here, if someone can find appropriate sources, since it means the phrase "the finite sequence {BB(1), BB(2), ..., BB(7918)}" cannot be assigned a definition (in computer science based on ZFC, anyway), and hence cannot be computable, even though any given sequence of 7918 natural numbers is of course computable. — Preceding unsigned comment added by 50.58.96.2 ( talk) 15:48, 21 August 2017 (UTC)
Please note that most standard texts refer to these functions as UNcomputable and not INcomputable. — Preceding unsigned comment added by 132.205.45.13 ( talk) 17:08, 23 August 2012 (UTC)
"Similarly, most subsets of the natural numbers are not computable. The halting problem was the first such set to be constructed. The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements (coded as natural numbers) are true. Turing and Church independently showed in the 1930s that this set of natural numbers is not computable. According to the Church–Turing thesis, there is no effective procedure (with an algorithm) which can perform these computations."
The first sentence is hard enough to interpret (do you mean finite or infinite subsets of the natural numbers?). But then you move on to "the halting problem was the first such set to be constructed". This makes no sense. Firstly, whatever you mean by that statement needs to be unpacked at the bare minimum (if not just thrown out.). Secondly, on it's face, problems are not sets. The halting problem is about computer programs that could be implemented in any turing complete language. A function that determines halting of arbitrary programs for any language in this class of languages would lead to paradox. You in no way make it clear how this relates to any uncomputable subset of the natural numbers. But then you jump to entscheidungsproblem and you're talking about statements encoded as natural numbers. How does the uncomputability of some statements encoded as natural numbers imply the uncomputability of some (finite? infinite?) subsets of natural numbers? This whole paragraph makes no sense and needs to be rewritten (or maybe just thrown out). Comiscuous ( talk) 06:39, 20 January 2022 (UTC)