Talk:Super-recursive algorithm/Archive1

message: This article seems to be based mainly on the work of Mark Burgin, UCLA, which appeared as a Springer book. While this may look like a clear case of verified science, where only notability could be a concern, the sensational claim made in this article, and the unusual fact that a refutation of Burgin's book appeared in the Bulletin of Symbolic Logic, strongly suggest the opposite; i.e. we need to verify that the claims are accepted by the scientific community before presenting them as facts. (See WP:NPOV.) I currently do not have a strong opinion on this topic, but I will make myself acquainted with it.
In any case I would advise the new editor who has contributed this article and linked to it from various places (I reverted all links but one), to familiarise themselves with our policies and guidelines such as WP:FRINGE and WP:COI, to see if they might apply here, and, as the case may be, to prepare for arguing the opposite. --Hans Adler (talk) 18:00, 7 February 2008 (UTC)
replies: text: I'd recommend attributing the claim to a specific person and citing a relevant, reliable source for the existence of the claim. Michael Slone (talk) 18:26, 7 February 2008 (UTC)

text: See my comments on the Talk page of the Church Turing thesis. Nobody seriously regards Burgin's work as even coming close to a refutation of the thesis. In any case, this article should be merged to hypercomputation and the only fair way to present it is as a fringe theory with no serious backing in the scientific community. Pascal.Tesson (talk) 19:39, 7 February 2008 (UTC)

text: My first thought was also to merge with hypercomputer, but after reflection I think that is not the best solution. My impression is that this is a particular type of hypercomputer that will take a full article to explain. It is important, though, to use due weight in describing philosophical claims about the Church-Turing thesis.  Martin Davis has published a review of Burgin's book and a separate paper on Burgin's book and other notions of hypercomputing; these would be good sources to use to document the position held by the vast majority of computability theorists. — Carl (CBM · talk) 19:57, 7 February 2008 (UTC)

text: Yes, I agree. This means someone needs to pull Burgin's book from the library. Kizeral - can you add the explanation to the article? — Carl (CBM · talk) 21:06, 7 February 2008 (UTC)
thread_title: Are the claims made in this article accepted by the scientific community?

message: There was a claim that the algorithm article discusses an algorithm to tell whether there are more 0s than 1s in an infinite binary sequence. There is no such algorithm (in the article or not, it simply doesn't exist). So I removed that claim from this article. — Carl (CBM · talk)`  —Preceding comment was added at 23:27, 28 February 2008 (UTC)
replies: text: There are no turing machine algorithms that can do this for arbitrary sequences. However, there are things like quantum computers and variants thereof, which are "programmable", and can thus have have "algorithms", and are not isomorphic to turing machines. Some of these can make decisions about infinite sequences ... no surprise, as they are performing calculations with infinite-precision real numbers ...  but I have no clue if this has any bearing on what this article is supposed to be about. linas (talk) 05:21, 29 February 2008 (UTC)
replies: text: The ordinary formulations of "quantum computer" don't compute anything more than Turing machines, they only compute faster. Even accepting that "has more 0s than 1s" can be given any specific meaning, which seems unlikely, I don't believe that there is any algorithm that could decide it.  Like many people in computability theory, I have a specific meaning for the word algorithm: it's a deterministic, finitistic, discrete process that terminates after a finite number of steps with the correct answer. We don't call any possible process an algorithm; at least one necessary property is that a human being could simulate the algorithm given nothing but pencil and paper. — Carl (CBM · talk) 14:52, 29 February 2008 (UTC)

text: It turns out this was actually added to the algorithm article at some point; see talk:algorithm for that. — Carl (CBM · talk) 15:30, 29 February 2008 (UTC)
thread_title:  More 0s than 1s 

message: Having read Davis' articles on Burgin's work, I don't find the following text accurate, so I am moving it here for further discussion. 
:To support this claim, Davis gives two arguments. First what bothers Martin is that there is no bound of time necessary to obtain the result. Second problem that he can see is that in order for a computational result to be useful one must be able to at least recognize that it is indeed the result sought. The first argument is also a problem for Turing machines as for them there is also no bound of time necessary to obtain the result. The second argument involves not the problem of what is computable but what is useful. However, different super-recursive algorithms are successfully used in practice.
— Carl (CBM · talk) 23:30, 28 February 2008 (UTC)
thread_title:  Claims about Davis' arguments 

message: I find the second sentence deeply disturbing: The term was introduced by Mark Burgin to more adequately describe what contemporary computers are doing. In the strict, technical sense, modern computers are just plain Turing machines, so that claim that modern computers are more than just "just plain Turing machines" is non-sense. Yet, in a hand-wavy kind of way, I can perhaps see some inkling of truth: for example, quantum computers, or generalizations thereof (topological/geometric computers) can compute things that turing machines cannot (because they can sum infinite sequences, with turing machines cannot). Insofar as we use floating-point on ordinary computers to approximate these things, we are "using contemporary computers to  do these fancier things" ... but this is a subtle point. Trotting it out, raw, naked, in the second sentence is asking for trouble.  linas (talk) 05:12, 29 February 2008 (UTC)
replies: text: Perhaps "The term was introduced by Mark Burgin, who believes it more adequately describes what contemporary computers are doing."? But I agree that contemporary computers are not doing super-recursive algorithms, and the distinction between a program the repeatedly asks for input and produces an answer, versus a program that asks for input but never produces an answer, is significant. — Carl (CBM · talk) 15:10, 29 February 2008 (UTC)
thread_title:  Strange remarks 

message: Does anyone have a reference for the "computational schemas"? They aren't in the index of Burgin 2005. — Carl (CBM · talk) 16:47, 29 February 2008 (UTC)
thread_title:  ref for computational schemas  

message: In the lead sentence, "...more powerful than Turing machines, that is, computes more..." should be rewritten, as I'll try and explain. The examples of "super-recursive" have no links or particulars; however, one example, a limit of sequences of turing machine compuatations, seems sufficiently well defined by itself. However, it is unclear in what sense that a limit of an infinite sequence of computations, is a computation; since there is a minimum amount of time cost to any computation, the time cost of an infinite sequence would not converge; so in this example it would seem unreasonable to say that the limit process computes more than than a Turing Machine. The lead sentence should not suggest the possibility of constructing a working machine by some design that is better than current machines; I doubt that is the purport of Burgin's work. Better would be to talk about maps to solution spaces (as opposed to computable maps). There is no reason for a very theoretical process to be couched in lay terms. I doubt that currently it can be, without misleading. Pete St.John (talk) 18:13, 4 March 2008 (UTC)
replies: text: I just want to note this online reference, a fairly short paper that seems to address this topic. Hasn't clarified anything for me, though. Pete St.John (talk) 18:28, 4 March 2008 (UTC)
replies: text: The issue is that Burgin does claim that those limiting processes are a sort of computation - and he expands his definition of the term algorithm to accommodate them. He claims that an inductive Turing machine can "solve" the halting problem, and uses this as an argument against the Church-Turing thesis; see that section in this article. — Carl (CBM · talk) 19:43, 4 March 2008 (UTC)
replies: text:  This is sorta like Assuming Good Faith (on the part of Burgin); the article, and maybe even Burgin's original writing, does seem lame to me for the evident reasons. However, I'm sure that something makes a great deal of sense; something I would phrase like "A functor in Burgin Space (of which Algorithms are a subspace closed under Composition) maps problems into Burgin Space Two (a subspace of which is solution space)". Instead of saying that an algorithm (with a new defintion of "algorithm") is computable (with a new definition of "computable") I'd merely be more specific. I had this problem at the Wolfram (2,3) article, because the PR language from Wolfram Research was too general, leading everyone to "refute" the paper (which may not have been worded great, either, but which extends all the defintions and may contain true theorems). Of course we don't mind "generalizing so that Church-Turing does not apply" but "refuting Church-Turing" is wrong. If we can't find the former type of exposition then we should probably AfD this article, you think? But I expect that there is good work underlying Burgin (and maybe some language barrier too) Pete St.John (talk) 19:58, 4 March 2008 (UTC)
thread_title:  lead sentence 

message: I don't think this sentence strikes the right tone:
:Thus, a better way to present new advancements in computability theory is to say that Burgin and other researcher extended the definition of the term algorithm to a broader domain where the Church-Turing thesis is not true.
If someone has said this in print, we can attribute it to them; but to simply claim as fact that this is a "better" way seems somewhat biased. If included, it would also require more explanation, because the "Church-Turing thesis" being described is not the actual Church-Turing thesis, but some other claim. The actual thesis isn't ordinarily considered to be domain dependent. — Carl (CBM · talk) 12:07, 6 March 2008 (UTC)
thread_title:  removed paragraph about "better way" 

message: As our discussion continues, it brings good news and bad news. I would like to start with good news. 
First, our discussion is becoming more scientific because, at least, one participant (Carl) looked into the book that is discussed and another participant (CRGreathouse) correctly wrote (29 February 2008) that before discussing concepts, it's necessary to precisely define these concepts.
All this helps us to make our discussion more scientific. As methodology of science teaches and as scientists know from their own experience, science has developed its own criteria estimating what people say or write. Namely, all statements or claims have to be based on facts and sound logic. What contradicts to facts or/and logic is not scientific. To prove or disprove something, it's necessary to give valid arguments based on facts. Unfortunately, often these scientific criteria are violated in many discussions that took place in science. For example, mocking an opponent is acceptable for a lawyer or politician, but not for a scientist. It is known that the first version of the periodic table of chemical elements was discovered by a British chemist Newlands. However, when he presented his discovery to the Royal Society, he was so severely mocked that he working in this area. 
Second, the article "Super-recursive Algorithm" is becoming more organized, better structured and gives much more information to the reader, and there is reasonable suggestion of Pete St.John (4 March 2008) to change the statement "refuting Church-Turing" to the statement "generalizing so that Church-Turing does not apply".
Third, Hans Adler (29 February 2008) acknowledged that the word "refutation" cannot be used to characterise Martin Davis' review because refutation damands arguments based on facts that prove that something is not true, while the review provides no arguments, but gives only a reflection of some chosen parts of the book "Super-recursive Algorithms." 
Bad news are: the participants of the discussion still have a lot of misconceptions and misunderstanding; the topic of the discussion is still too fuzzy; and many claims made by participants lack grounded arguments.  
For instance, participants of this discussion confuse mathematical theories presented in the book "Super-recursive Algorithms" and practical implications based on some of these theories and on known empirical facts about computers. 
In addition, it's necessary to understand that theory of super-recursive algorithms consists of many subtheories: the theory of inductive Turing machines and inductive computation, theory of limiting recursive functions, theory of analogue computations, theory of neural networks that work with all real numbers, and theory of Turing machines with oracles, to mention but a few. Even if one of these theories is criticized, it does not mean that the whole theory of super-recursive algorithms is criticized. To think so is the same as to think that if some inconsistent mathematical theory is criticized, then the whole mathematics is criticized or to think that if some ungrounded physical theory is criticized, then the whole physics is criticized.
For instance, Carl (29 February 2008) mentions a paper by Soare  (esp. p. 13, second para, "Some have cast doubt..."). Let us take the whole statement of Soare. He writes:
"Some have cast doubt on Turing's Thesis on the grounds that there might be physical or biological processes which may produce, say, the characteristic function of the halting problem. It is possible that these may exist (although there is presently no evidence) but if so, this will have absolutely no effect on Turing's Thesis because they will not be algorithmic or mechanical procedures ...".
This argument of Soare cannot be applied to what is written in the book "Super-recursive Algorithms" because the book gives an algorithmic or mechanical procedure in the form of an inductive Turing machine. Even Davis (in his article "The Church–Turing Thesis: Consensus and opposition") does not argue that these machines model (perform) computations when he writes (Davis 2006, p. 128): "in order for a computational result to be useful one must be able to at least recognize that it is indeed the result sought." 
What worries Davis is that computations controlled by such algorithms will never be useful. It is possible to give valid arguments that such computations will be useful, but even if we agree with his assertion, as scientists we have to build mathematical model for such objects because they exist. Distant stars are not useful in the pragmatic sense but they exist and physics includes them in models of the universe, while physicists are studying these stars by means of theoretical models. In a similar way, computer scientists have to model and to study all phenomena that exist in the field of computing. That is why it's not a good situation that, as Carl writes (29 February 2008),  there is no SERIOUS discussion within the computability community that the Church-Turing thesis has been (or even could be) overturned. 
Such mathematical theories as the theory of inductive Turing machines and inductive computation, theory of limiting recursive functions, and theory of Turing machines with oracles have never been criticized for their mathematical content. What irritates many of researchers who based their works on the CTT is the connected to some of these theories implications that CTT is not true.
Actually everything in these statements about validity of CTT is based on definitions of algorithm that are used. We define that algorithm is the same as Turing machine and then it becomes impossible to disprove CTT. However, if we consider algorithm as some technical or even natural phenomenon, than CTT becomes a scientific hypothesis, which can be proved or disproved. 
Hans Adler (18:00, 7 February 2008) made the most extended contribution to our discussion. Let's analyze what he wrote and what arguments he gives to support his claims.
He correctly observes that the sentence "This is a truly extraordinary book" is ambivalent. To eliminate this ambivalence it's necessary to analyze what is written after this sentence. If we read the review, we see that after this sentence the reviewer explains what he saw in the book he is reviewing. For instance, the next sentence states that the author of the book notes the well-known fact that the majority of program properties related to their output are undecidable by modern computers when they work in the recursive mode. There is nothing extraordinary in this. Then the reviewer states that the author of the book assures that superrecursive algorithms can solve many of such problems for ordinary programs and contemporary computers. Is true or not? As it is written in the book, limiting recursive and limiting partial recursive functions introduced by the mathematician M. Gold are particular cases of super-recursive algorithms. It was proved long ago (more than forty years ago) that these functions can compute and decide sets that are not computable by Turing machines. Since it was done, nobody disproved these results. So, according to mathematical criteria, these results are true. Consequently, the citation from the book is also true. By the way, later the reviewer writes that it is known that not all sets accepted by limiting partial recursive functions of Gold and the “trial and error” predicates of Putnam are accepted by Turing machines. 
Hans Adler is concerned with expressions like:
But he assures us that
Among the author’s astonishing predictions is the following:
Burgin finds a key to what he regards as
The first expression simply shows that the reviewer wants to express the opinion of the author. The second expression explains why the reviewer thinks that the book is extraordinary. To better understand the third expression, it's necessary either to consider the whole sentence of which this expression is only a part or to read the book. In any case, the question is to really understand what contemporary computers are doing, we need to into consideration that there are non-terminating computations that nevetheless give results. Thus, it is necessary to buid mathematical models how to rigorously define results for non-terminating computations. There are several solutions to this problem. One of these solutions was embodied in (represented by) such models as limiting recursive functions and limiting partial recursive functions of Gold, “trial and error” predicates of Putnam, inductive Turing machines of Burgin, and general Turing machines of Schmidhuber. So, actually the opinion of the reviewer that "Burgin finds a key to" this problem is inexact because Gold and Putnam found this "key" much earlier. What the reviewer correctly writes and what Hans Adler reiterates (as if it can be something negative) is that "Burgin’s inductive Turing machines generalize this notion extending their reach through the entire arithmetic hierarchy." Is this statement in favor of the book or contains some negative implications? We know that the approach of Gold and Putnam gave birth to such a respected direction in computer science and artificial intelligence as inductive inference. Many mathematicians and computer scientists, including such well-known researchers as Leonard Adleman, Manuel Blum, Lenora Blum and Carl Smith, contributed to this area. Thus, to say that a book is a further development of a respected direction in computer science and artificial intelligence definitely is a positive statement with respect to this book.
Hans Adler gives one more citation from the review of Davis. Namely, that "mathematicians can eagerly await the proofs of the Goldbach Conjecture and the Riemann Hypothesis that this amazing technology promises!" This is really a strange statement because, as we know, now mathematicians prove theorems and not machines. Even if computers are heavily used for some proofs, such as the Four Color Theorem, people are doing these proofs as they are writing programs for those computers. I took the book "Super-recursive Algorithms" and tried to find a prediction that computers will prove theorems and mathematicians will wait for their results but was not able to do this.
However, in general, the review of Davis describes the content of the book. For instance, when he writes that “neural nets” with arbitrary real numbers as weights "not only accept uncomputable sets, in fact they accept all sets of strings on a given alphabet", he simply reiterates Corollary 4.2.9 (Burgin 2005, p. 151), which can be easily derived from Lemma 4.2.1 (Burgin 2005, p. 149) proved in the book.
When Davis writes:  
"Burgin's inductive Turing machines generalize this notion extending their reach through the entire arithmetic hierarchy. It is a pity that he doesn't stop to tell us how he imagines this power could be realized in real-world physical devices."
It only states that the book "Super-recursive Algorithms" is a theoretical monograph and not an Instruction Manual for some device. It might be very useful if we find in the book how to design useful super-recursive algorithms, but theory is not about "know how" but about "know what." May be "know how" will be a topic of another book in this area.
However, some statements from the review can be misinterpreted without further explanation. For instance, the Davis writes, "Burgin suggests the example of a PC being used as a terminal to access a supercomputer with the supercomputer playing the role of oracle." To understand this sentence, we need to know what is an oracle. I have not found what is written in the book "Super-recursive Algorithms" about oracles. I will look more, but for now I would like to share with other participants of this discussion what methodology and history of science teaches us. The lesson that we can take is that for science (and mathematics), it's more efficient and productive to ask not "What is some concept, say A?" but  "How to understand the concept A?" Here is one example from the history of mathematics. 
When mathematicians started to solve more and more complex algebraic equations and found some formulas for their solutions, they encountered a strange phenomenon. Expressions that had negative numbers under the square root appeared in the process of solving equations. At that time, it made no sense. All mathematicians knew that the square of any number could not be negative. So, at first, mathematicians forbade utilization of these unusual expressions. (In a similar way, now some people try to forbid utilization of super-recursive algorithms.) Then mathematicians started using these unusual expressions only as formal constructions that have no real meaning in mathematics. Only when some mathematicians asked the question "What can be a number?" imaginary and complex were invented. As we know, now complex numbers are very useful in mathematics and beyond in spite that some of them are called imaginary. Those who don't know mathematics can interpret the expression "imaginary numbers" as irony, but we know that it is only a mathematical term. So, to discern irony from a literal statement of some fact, it's necessary to have enough knowledge.
Now let's look what can be an oracle and try not to ascribe to an oracle some extraordinary features. We come to the conclusion that any system B can be an oracle to another system A when B has or can get more knowledge than A.
For instance, now the Internet is an oracle for many people. When I read the "talk" of Hans Adler and saw acronyms WP:NPOV, WP:FRINGE and WP:COI, which I did not know, I went to the WIKIPEDIA as an oracle and found their meaning. When we want to know more about super-recursive algorithms, we go to such oracles as the book "Super-recursive Algorithms" and reviews on this book. 
The example from the book cited in the review of Davis does not mean that the supercomputer is a super-recursive oracle. A supercomputer can be an oracle for a PC even if both of these machines are equivalent in their potential computing power to some Turing machines. From the point of view of recursion theory, such oracle makes no sense as it does not make this PC super-recursive. However, from the point of view of real life in general and computational practice in particular, such an oracle can be very useful to the user of that PC.
This shows that recursion theory that studies classes of algorithms rather than individual algorithms is too coarse and can make wrong conclusions in many practical situations. As S writes, theory of computation eventually was substituted by recursion theory. In recursion theory, the Church-Turing Thesis is an implicit axiom. However, computers science is not a pure mathematics but a technical science and all laws of science are only well-grounded hypotheses. The Church-Turing Thesis is a law of contemporary computer science. However, according to Popper to be a scientific law, a statement needs to allow empirical refutation. Thus, the Church-Turing Thesis can’t be refuted in recursion theory where it is an axiom. However, in computability theory, which is a part of computer science this Thesis allows empirical refutation.  
Hans Adler also writes (18:00, 7 February 2008) about philosophical arguments with respect to the current status of the Church-Turing Thesis. As I can see from the new version of the article "Super-recursive Algorithm", people who support assertions from the book "Super-recursive Algorithms" are all computer scientists and not philosophers. Besides, the book itself contains mathematical proofs and not only general reasoning. 
I would like also to explain to linas who writes (29 February 2008) that “ In the strict, technical sense, modern computers are just plain Turing machines …” and many other people who believe in this that this is a misconception. As is correctly written in the WIKIPEDIA article “Turing Machine,” “… any particular program running on a particular machine is in fact nothing but a deterministic finite automaton, since the machine it runs on can only be in finitely many configurations. Turing machines would actually only be equivalent to a machine that had an unlimited amount of storage space.”
However, Turing machine was one of the best (if not the best) model of computers. However, times are changing. Information technology is developing and we need new models. We can find a variety of such models in computer science. We have only to understand: what models are better in what situations, what models are irrelevant now and have to wait for future computers, what models can pave the way to future computers, and what models are only mathematical abstractions that provide for a good mathematical theory.
With respect to all discussion participants, Multipundit (talk) 00:14, 7 March 2008 (UTC)
replies: text: Multipundit, that was an impressive effort: some good points, some good prose. But frankly I didn't read all of it and I conjecture that most of us will not. Wiki is more like modern business email: if more than one point is made both will be ignored. That kills me every time, constraining myself to compounded sententences on merely finitely many points is unnatural to me, so I'm sympathetic. Pete St.John (talk) 16:50, 7 March 2008 (UTC)

text: Multipundit writes: "Now let's look what can be an oracle and try not to ascribe to an oracle some extraordinary features."  Actually, ascribing extraordinary features to an oracle is the whole point of oracles.  See Oracle machine if you're unacquainted with this sense of the term.  (You should be acquainted with it, if you've studied computing theory formally.  I haven't so much as cracked a book on computing theory in over 25 years, but I did take an undergrad course in it when I was a CS major at UC Berkeley, and then graded homework for that same course, for a few semesters.) Yakushima (talk) 05:30, 24 May 2008 (UTC)
thread_title:  Finding the true meaning 

message: I asked about this subject at Prof Pratt's userpage (we have some prior history on account of the Wolfram (2,3) "Turing Machine" controversy) and he replied at length there. It's quite a severe, but informative, critique of Burgin's work and I commend it to interested editors. Pete St.John (talk) 17:40, 7 March 2008 (UTC)
thread_title:  critique by Prof Pratt 

message: PeterStJohn (8 March 2008) asks to pick one specific point related to what Pratt wrote on his web site. The most important point is scientific validity of Pratt’s claims. Here we are discussing an area in computer science. Thus, the author of some claims must give scientific argument that support these claims. Let’s look what scientific arguments related to the work of Burgin are given by Pratt. The only scientific argument is:
“As best I can tell it seems to be Burgin's made-up name [super-recursive algorithm] for computability classes above Turing degree 0 such as the arithmetic and analytic hierarchies, which have been studied for many decades.”
This statement is completely incorrect because in the article (and I am not speaking about the book in question) there are various examples of super-recursive algorithms that have nothing to do with the arithmetical and analytic hierarchies. In addition, these algorithms (from the article "Super-recursive algorithm") have not been studied for many decades in contrast to what is written.
: First, it would help if there were (are?) citations to use of the term "super-recursive algorithm" from elsewhere. Has anyone else adopted the term, and if so, what formal definition do they give for it? Second, Pratt didn't say "exclusively artithmetic and analytic" but "such as.." instead. I'm sure we'd all be interested in something specific which is new. Pratt's impression seems, to me, to be that anything new and correct is lost among what is old or incorrect. Pete St.John (talk) 03:41, 10 March 2008 (UTC)
replies: text: Yes he corrected that himself. He read some of Burgin's papers and has not read this particular book. Judging from his apparent impression of the papers, I would not expect him to read the book. Pete St.John (talk) 03:41, 10 March 2008 (UTC)

text: It isn't so easy to extrude single, definite, isolatable points from such a complex topic; I appreciate your trying. My replies are embedded above. Pete St.John (talk) 03:41, 10 March 2008 (UTC)
replies: text:  It's not easy to respond to a long, complex arguement; but easier to address single specifics, one at a time. Not that I'm any paragon of "Accurate, Brief, and Concise" myself. Pete St.John (talk) 02:50, 14 March 2008 (UTC)
thread_title:  Grounded scientific arguments 

message: Carl (29 February 2008) asked for a reference for the "computational schemas." I have searched the book "Super-recursive Algorithms" and found terms algorithmic scheme (p. 113 and 115) and computational scheme  (p. 121). It looks they have the same meaning. As I can understand, the term algorithmic scheme is much more general than the term super-recursive algorithm and Burgin argues (on p. 115) that it's necessary to make a clear distinction between super-recursive algorithms and algorithmic schemes that are not algorithms. That's why I changed the term computational schemata to the term algorithmic scheme in the article "Super-recursive algorithm".
With respect to all discussion participants, Multipundit (talk) 00:49, 13 March 2008 (UTC)
thread_title:  References for computational schemas  

message: The vast majority of the references don't use the term "super-recursive algorithm" at all. In fact, it seems to be an idiosyncratic term used exclusively by Mark Burgin, and the claim that what these other authors are doing amounts to something that should be described with that term is attributable solely to Burgin. So there is basically one source for this article, despite the lengthy list of references. Most of the rest more properly describe hypercomputation. --Delirium (talk) 05:49, 11 April 2008 (UTC)
replies: text: By all means, feel free to prune the references to those that actually mention "super-recursive algorithms" or (possibly) that Burgin explicitly mentions. — Carl (CBM · talk) 11:20, 11 April 2008 (UTC)

text: I can agree that for those who don't understand what are super-recursive algorithms and what is hypercomputation the references section is questionable. However, anybody with sufficient knowledge and brains can check that all works listed in the references section are about some kind of super-recursive algorithms. What concerns hypercomputation, it is necessary to understand how super-recursive algorithms are related to hypercomputattion before making any assertions. To conclude, I don't know what Burgin explicitly mentions, but it's sufficient (for a learned and intelligent person) to check definitions to understand the true situation. With surprise, Multipundit (talk) 22:42, 15 April 2008 (UTC)
thread_title:  references section is questionable 

message: It would be useful if somebody explains what relations between hypercomputation and super-recursive algorithms exist. To write that these concepts are closely related is not enough. It would be helpful to describe what these concepts have in common and what are differences between them. Such a description might enhance both articles "Hypercomputation" and "Super-recursive algorithms".
With respect, Multipundit (talk) 01:17, 24 April 2008 (UTC)
replies: text: Yes, that would be helpful. Do you know if there is any published source that gives that analysis? Unfortunately, Burgin doesn't seem to give enough information. Based on Burgin, def 4.1.2 (p. 108), it appears that the procedure I describe above is a "hypercomputation". — Carl (CBM · talk) 01:37, 24 April 2008 (UTC)

text: I searched for relations between hypercomputation and super-recursive algorithms the book "Super-recursive algorithms" and three survey articles in this and related areas (Burgin, M. and Klinger, A. Three Aspects of Super-recursive Algorithms and Hypercomputation or Finding Black Swans, Theoretical Computer Science, v. 317; Toby Ord. Hypercomputation: Computing more than the Turing machine can compute, arXiv.org; and Bruno Loff And Felix Costa, Five views of hypercomputation). However, in spite of the optimistic assertion of DavidD4scnrt, I was not able to find a grounded explanation of such relations. We can try to elaborate such an explanation ourselves, but it looks, that there are no published sources that give it. With respect, Multipundit (talk) 00:56, 1 May 2008 (UTC)
thread_title:  What are relations between hypercomputation and super-recursive algorithms? 

message: "Inductive Turing machines" are cited in the article as capable of super-recursive algorithm behavior.  In the course of trying to find out what (if anything) they are, I ran across some very disturbing things.
Burgin argues (with a co-author) that a very simple piece of "hardware" added to Turing machines (yielding an "inductive Turing machine") makes them more powerful than ordinary Turing machines.  The hardware modification is equivalent to having an LED on the front panel of some hardware realization, telling you whether the machine has halted yet.  This is from Burgin and Klinger's "Experience, generations, and limits in machine learning", Theoretical Computer Science 217 (2004) 71-91.
 "To show that inductive Turing machines are more powerful than ordinary Turing
machines, we need to find a problem that no ordinary Turing machine can solve and to
explain how some inductive Turing machine solves this problem. To do this, consider
the halting problem for an arbitrary Turing machine. It was proved unsolvable for
Turing machines. Today it is one of the most popular unsolvable problems in the
theory of algorithms.
 "However, there is an inductive Turing machine M that solves this problem. This
machine M contains a universal Turing machine U as a subroutine. Given a word u
and description D(T) of a Turing machine T, machine M uses machine U to simulate
T with the input u. While U simulates T, machine M produces 0 on the output tape. If
machine U stops, and this means that T halts being applied to u, machine M produces
1 on the output tape. According to the de4nition, the result of M is equal to 1 when T
halts and the result of M is equal to 0 when T never halts. In such a way, M solves
the halting problem.
 "So, even the simplest inductive Turing machines are more powerful than conventional
Turing machines. At the same time, the development of their structure allowed inductive
Turing machines to achieve much higher computing power than have the simplest
inductive Turing machines described above. This contrasts to such a property of a
conventional Turing machine that by changing its structure, we cannot get greater
computing power."
Yeah, guys, but ... you don't have to "develop" the structure of a conventional Turing machine to get what you're talking about.  You just watch the proposed Turing machine to see if it has halted yet.  Do you object that this implies at least sporadic observation, to see if there's an answer yet?  So what?  So do regular Turing machines.  You haven't "solved the halting problem", because the that problem is defined as specifying an algorithm that itself always halts with an answer.  "Inductive Turing machines" are architecturally no different from regular ones, for all purposes practical and theoretical.  Therefore, what you can compute with them is also no different.  (In this same paper, Burgin and Klinger prove that all Turing machines can be modeled with "inductive Turing machines"; they don't ask the obvious next question "can all inductive Turing machines be modeled with regular ones?"  The answer would have been rather embarrassing for them, I suppose.)
How did this flagrantly silly argument ever make it into a peer-reviewed journal?  Well, a big part of the answer is: it didn't.  The guest editors of this issue of Theoretical Computer Science were also the co-authors of this paper.  This is not what is usually meant by "peer review".
This Wikipedia article should cite independent, reliable sources to support notability.  "Independent" means, at the very least, "not written or co-authored by the progenitor of the topic".  "Reliable", in a scientific context, means "peer reviewed in a mainstream (not fringe) journal."  I'm starting to think that we may never turn up any work that meets these basic criteria when it comes to this supposed field of "super-recursive algorithms", not to mention "inductive Turing machines".  (Now, inductive inference machines, that's a real topic.)  Burgin wrote a monograph?  It got published by Springer?  Almost meaningless: it wasn't peer reviewed.  Parts of it don't even seem to be copy-edited.
While I'm here, I think it's worth noting something else I find inexpressably sad: Burgin's current (2008-05-26) CV mentions that he's a "Member of the Research Board of Advisors, American Biographical Institute.  Yes, please click on that link.  Pretty shocking, eh?
It all makes me wish that somebody would research, write and publish an exposé in a reliable source, so that this can all be pointed out on Wikipedia as a notable case of pseudo-science emerging in the mainstream -- after all, that kind of thing does happen now and again.  Then at least we could be sure there's really a topic here. Yakushima (talk) 09:32, 26 May 2008 (UTC)
thread_title:  This is inexpressably sad 

message: Yes, this not only sad but also disastrous if appealing to false information, spreading disinformation, and using such logical fallacies as "red herring", ignorance impose its low level of knowledge and understanding on WIKIPEDIA. Yes, ignorance does not need scientific criteria and arguments, but attacks using ungrounded claims and accusations.
With a surprise, Multipundit (talk) 21:39, 28 May 2008 (UTC)
replies: text: Produce one unambiguously independent, unambiguously peer-reviewed article that treats the subject in any significant detail. Yakushima (talk) 14:05, 31 May 2008 (UTC)

text: Be specific. What was inaccurate? Yakushima (talk) 14:05, 31 May 2008 (UTC)
thread_title:  This not only sad but also disastrous 

message: To my surprise, I have found that some people, such as Pratt, cannot understand difference between finite and infinite sets.
Let’s look what implies the following claim of Pratt. He writes (27 May 2008), “I don't see what if anything Burgin's notion of inductive Turing machine adds to Gold's notion of language identification in the limit.” At the same time, Gold (1965-7) proved that language identification in the limit cannot generate more than 3 levels of the Arithmetic Hierarchy, while there is a proof in the book “Super-recursive algorithms” that inductive Turing machines can generate all levels of the Arithmetic Hierarchy (for those who work in other fields, the Arithmetic Hierarchy has infinitely many levels). Thus, this statement of Pratt implies that he cannot see the difference between finite and infinite sets.
It’s possible to understand why it might happen if we take a man from a primitive tribe in which people can count only 1, 2, 3 (or many). Such man will not understand what it means to count from 1 through 10. For this man, the proficiency in counting from 1 to 3 and the proficiency in counting from 1 to 1,000,000 will be the same. He will claim that counting from1 to 1,000,000 adds nothing to what people of his tribe can already do, i.e., to counting from 1 to 3.
With respect, Multipundit (talk) 21:10, 30 May 2008 (UTC)
replies: text: It is true that not any person can always understand difference between finite and infinite.  It is opening of mind that creates capability to understand such difference, and it is such opening that is absent in the Pratt and others including the vandal of this article.  Those who comprehend will bury the Pratts by make glorious the citation index, as by article like Three levels of the symbolosphere by which one can inductively increase citation index to infinity, and also more efficiently than with crippled, limited, ordinary blind peer review process.  Instruction pamphlet available from American Biographical Institute.  Yakushima (talk) 16:46, 31 May 2008 (UTC)
thread_title:  Can all people understand difference between finite and infinite? 

message: The entire entrance to the article is obviously written by a figure opposed to the ideas brought forward, and uses primarily antagonistic language, and personal attacks to actively discourage any reader from proceeding beyond the introduction.
The introduction should be rewritten or removed in it's entirety.
--136.159.219.206 (talk) 20:21, 5 June 2008 (UTC)
thread_title:  The opening is incredibly biased and non-encyclopedic 

message: This discussion shows that the science development accelerates. Before each important discovery passed three stages:
1 stage: It’s incorrect (It cannot be true).
2 stage: It’s correct.
3 stage: It has been known before.
Now, in some cases, the first and third stages blended, while the second stage is smashed between them. At the same time, many people don’t understand that the third stage implies the second one. We have this situation in the case of super-recursive algorithms. Indeed, some researchers (e.g., Davis) claim that super-recursive algorithms don’t exist (the first stage), while others (e.g., Pratt) assert that super-recursive algorithms have been used long before their theory was presented in the book “super-recursive algorithms” (the third stage). So, if they have existed before, they continue to exist now.
With respect, Multipundit (talk) 20:26, 23 June 2008 (UTC)
thread_title:  Acceleration of science development 

message: I think reverting to a refutatory lead section is not the answer. I agree with those who say this article should be seriously trimmed down and put into the hypercomputation article. The basic points:
* A super-recursive algorithm is a possibly non-terminating algorithm for incrementally and provisionally producing elements of a sequences; we can consider a part of that sequence a final answer when the algorithm says so OR when we have some external proof that the algorithm will never revoke that part. 
* This is a limited form of hypercomputation which can already be implemented, if one grants the premise that this liberal interpretation of "answer" serves any purpose.
* Burgin considers this to be a refutation of the Church-Turing thesis, but most everyone else seems to think otherwise. --Unzerlegbarkeit (talk) 15:40, 30 July 2008 (UTC)
replies: text: Okay, that would probably be an improvement on the status quo. Haukur (talk) 17:50, 30 July 2008 (UTC)

text: He didn't say hypercomputation can already be implemented, he said a limited form of it can already be implemented if one grants a very generous premise which most people are not willing to grant. Enormous difference there. Haukur (talk) 10:37, 1 August 2008 (UTC)

text: Possibly, but as the article is currently written, it seems that inductive TMs really do reduce to general TMs. Nor or any of the other examples discussed, they are just mentioned. In fact, I don't even know what the first level you are referring to is. Perhaps you could give an example of a higher-level inductive TM which does not reduce to a general TM. --Unzerlegbarkeit (talk) 16:02, 8 August 2008 (UTC)

text: I hadn't carefully read your latest addition. If there are levels of inductive TMs that compute the whole arithmetical hierarchy, then you are correct, they cannot reduce to general TMs. But at the same time, doesn't this contradict the statement that an inductive TM "produces output from time to time and once this output stops changing, it is considered the result of the computation"? And what's more it is no longer implementable on any hardware (even given a liberal interpretation of "answer"). Because it seems to me nothing of this sort will compute the whole arithmetical hierarchy either. --Unzerlegbarkeit (talk) 17:44, 8 August 2008 (UTC)

text: I'm afraid I'm still not getting it. The article claims that simple inductive TMs are "thoroughly grounded in physical machines". Are higher-level inductive TMs still supposed to be physically grounded? If so, how can they possibly calculate the whole arithmetical hierarchy? If not, then what qualifies them as super-recursive algorithms, but not things like neural networks based on real numbers? Thanks for your patience, --Unzerlegbarkeit (talk) 16:41, 13 August 2008 (UTC)

text: Oh, come on Burgin, give up this pretense that Multipundit is someone other than you.  It's becoming too much effort to play along with it with a straight face.  --Vaughan Pratt (talk) 01:35, 13 October 2008 (UTC)
thread_title:  Merge proposal