| olcott <NoOne@NoWhere.com>: Mar 02 06:45PM -0600 On 3/2/2021 6:20 PM, Richard Damon wrote: > function use as the program parameter, thus getting the Halt Decider > function to perform the conditional execution instruction cloaked behind > your false wall of separation. OK, great. That was an important critique. It caused me to focus on examining to input to functions as a key element of halt deciding. Assuming that what Malcolm said also applies to indirect recursion: On 2/26/2021 2:41 PM, Malcolm McLean wrote: > originally invoked with, then it must be infinitely recursive, > even if there are conditional branches. It always takes the > same branches. Then detecting infinite recursion is made even simpler. Noticing that the H_Hat() invocation of Halts() is infinitely recursive for every halt decider that bases it halt deciding decision on examining its own simulation of its input overcame the "impossible" aspect of refuting the halting problem proofs. (Copyright Pete Olcott 2016, 2017). The feedback that you and Malcolm provided was important to implementing this halt decider. You corrected a key short-coming with the initial halt decider design and Malcolm eliminated an unnecessary step. void H_Hat(u32 P) { u32 Input_Halts = Halts(P, P); if (Input_Halts) HERE: goto HERE; return; } int main() { u32 Input_Would_Halt = Halts((u32)H_Hat, (u32)H_Hat); Output("Input_Would_Halt = ", Input_Would_Halt); } -- Copyright 2021 Pete Olcott "Great spirits have always encountered violent opposition from mediocre minds." Einstein |
| gazelle@shell.xmission.com (Kenny McCormack): Mar 03 02:17AM In article <874khtjmtv.fsf@nosuchdomain.example.com>, >Is your newsreader buggy, or are you deliberately making >inappropriate cross-posts, or is something else going on? >I've cross-posted this article and set followups to comp.theory only. Wow. I'm so impressed. -- People who want to share their religious views with you almost never want you to share yours with them. -- Dave Barry |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 07:23AM > I'm a bit late to the discussion, but this sounds a lot like you're > trying to overturn Turing's proof that that a general algorithm to solve > the halting problem for all possible program-input pairs cannot exist. If the halting problem could be solved, it would be great news for mathematics. Pretty much all unsolved problems in mathematics, at least those that can be computed with an algorithm, would be solved in one fell swoop. The Riemann hypothesis? Simply create a program that checks every single non-trivial zero of the zeta function and stops when it finds one that's not on the critical line. Then simply use the marvelous halting problem solution above to see if it ever terminates, and you'll have proven the Riemann hypothesis as true or false. Do odd perfect numbers exist? Simply create a program that goes through every odd number and checks if it's perfect, and terminates when it finds one. Use the halting problem proof to check if it will ever terminate and you'll have solved the problem. Collatz conjecture? Same thing. And so on and so forth. |
| David Brown <david.brown@hesbynett.no>: Mar 03 08:58AM +0100 On 03/03/2021 08:23, Juha Nieminen wrote: > when it finds one. Use the halting problem proof to check if it will > ever terminate and you'll have solved the problem. > Collatz conjecture? Same thing. And so on and so forth. I think Pete Olcott already said that he was not a mathematician. So that means he is not constrained by such simple mathematical logic, and can solve the impossible in an way that mathematicians cannot. You know, in the same way that non-physicists are not constrained by physical laws that they don't understand, and can thus make moon rockets in their garage powered by gravity-repulsive paint. |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 08:50AM > You know, in the same way that non-physicists are not constrained by > physical laws that they don't understand, and can thus make moon rockets > in their garage powered by gravity-repulsive paint. I haven't followed what he has written and said about his approach, but if I were to guess, he's probably doing the same thing as so many other pseudomathematicians do with such things, when trying to contradict well-established corroborated mathematical proofs: Simply *redefine* the original problem so that it (possibly) fits his "proof". A bit like trying to redefine the value of pi so that a proof of squaring the circle becomes correct. (This is an actual real-life example.) |
| mickspud@potatofield.co.uk: Mar 03 09:11AM On Wed, 3 Mar 2021 08:50:22 +0000 (UTC) >A bit like trying to redefine the value of pi so that a proof of >squaring the circle becomes correct. (This is an actual real-life >example.) Strictly speaking pi is an irrational number so doesn't actually have a definable value in the first place :) |
| David Brown <david.brown@hesbynett.no>: Mar 03 10:23AM +0100 On 03/03/2021 09:50, Juha Nieminen wrote: > pseudomathematicians do with such things, when trying to contradict > well-established corroborated mathematical proofs: Simply *redefine* > the original problem so that it (possibly) fits his "proof". I don't think anyone, including Olcott himself, really understands what he is trying to do - though I haven't bothered following the details either. He seems to propose some kind of simulator (for x86 assembly, bizarrely, rather than the usual Turing machine or similar) with all programs reduced to a "u32" type. Programs are split into a "decidable" bit and an "undecidable" bit, and he avoids the usual never-ending execution of the halting decider by only simulating the "decidable" bit. > A bit like trying to redefine the value of pi so that a proof of > squaring the circle becomes correct. (This is an actual real-life > example.) The difference is that you can often learn quite a bit of maths, and quite a bit about what works and doesn't work, by trying to solve that kind of problem. And for those that really think they have got the answer, they can learn about peer review and academic criticism. (There are some people, of course, who never listen and never learn - Olcott seems to be one of them.) Perhaps Olcott could try a political solution, as Edward Goodwin did - when the laws of mathematics stopped him from squaring the circle, he tried to get the "Indiana Pi Bill" passed to trump the mathematical laws. |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 09:39AM >>example.) > Strictly speaking pi is an irrational number so doesn't actually have > a definable value in the first place :) Of course it does. Pi is a computable number, so it can perfectly well be defined, and computed to an arbitrary accuracy, using a finite description. Irrationality in itself has nothing to do with the squaring the circle problem. The square root of 2 is irrational, but if the square root could be used to define the area of a circle then it could be "squared" as per the original problem description. (The reason why the circle cannot be squared (using the tools in the original problem) is because pi is not an algebraic number.) |
| Paavo Helde <myfirstname@osa.pri.ee>: Mar 03 11:53AM +0200 > Strictly speaking pi is an irrational number so doesn't actually have > a definable value in the first place :) You are confusing "definable" with "exactly representable in my favorite notation". The value of pi can be defined with no problems, and it can be also represented exactly. The most common exact representation makes use of a Greek letter. |
| mickspud@potatofield.co.uk: Mar 03 09:56AM On Wed, 3 Mar 2021 09:39:03 +0000 (UTC) >Of course it does. Pi is a computable number, so it can perfectly well >be defined, and computed to an arbitrary accuracy, using a finite >description. You really are an aspie arn't you. You can't even spot a light heated remark when its signposted with a smiley. But if you want to be pedantic irrational numbers cannot be represented by the ratio of 2 integers therefor their true value is unknown and always will be. |
| mickspud@potatofield.co.uk: Mar 03 10:12AM On Wed, 3 Mar 2021 11:53:57 +0200 >> a definable value in the first place :) >You are confusing "definable" with "exactly representable in my favorite >notation". It would appear you are confusing definable with representable. Infinity can be represented, that doesn't mean it can be defined especially given there are numerous different types. >The value of pi can be defined with no problems, and it can be also >represented exactly. The most common exact representation makes use of a >Greek letter. I don't think you understand the meaning of an irrational number. |
| "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>: Mar 03 02:24AM -0800 >> represented exactly. The most common exact representation makes use of a >> Greek letter. > I don't think you understand the meaning of an irrational number. pi is pi, however, when I need to actually use it, I define a certain precision. Say, I need at least 301 digits of pi to accurately work with a certain fractal formula in a very deep zoom that uses an arbitrary floating number package. Anything less than 301 digits of precision will tend to start to creating "artifacts" that with ruin the rendering. |
| David Brown <david.brown@hesbynett.no>: Mar 03 11:48AM +0100 >> description. > You really are an aspie arn't you. You can't even spot a light heated remark > when its signposted with a smiley. A smiley works to indicate a joke like this when it is clear that the poster knows that what they are writing is obviously and completely wrong. > But if you want to be pedantic irrational numbers cannot be represented by > the ratio of 2 integers therefor their true value is unknown and always will > be. However, when you write this is is clear that you are not very familiar with the mathematics at this level. (Nothing wrong with that, of course - few people know or care what in means to be "computable".) The true value of π is known - it is π. It can't be written as a finite decimal expansion, but that has nothing to do with its "true value" - it's just a representation. (As an interesting aside, if you write π using a hexadecimal expansion, you can calculate any given digit without having to calculate the preceding digits. Weird and fascinating, IMHO.) And "pedantic" is a compliment to a mathematician! |
| David Brown <david.brown@hesbynett.no>: Mar 03 12:07PM +0100 >> You are confusing "definable" with "exactly representable in my favorite >> notation". > It would appear you are confusing definable with representable. No, he is not. > Infinity > can be represented, that doesn't mean it can be defined especially given > there are numerous different types. You can define the infinities you want, by specifying characteristics of interest. Infinities are not computable numbers, but they are well-defined. >> represented exactly. The most common exact representation makes use of a >> Greek letter. > I don't think you understand the meaning of an irrational number. Why are you so concerned with irrational numbers here? I am confident that the others in this branch of the thread know exactly what the term means. The irrationality of pi is not relevant as to why its square root is not constructable with ruler-and-compass geometry. √2 is irrational, constructable, algebraic, and computable. π is irrational, non-constructable, non-algebraic (transcendental) and computable - as is √π. |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 11:16AM >>description. > You really are an aspie arn't you. You can't even spot a light heated remark > when its signposted with a smiley. There is no joke in your statement. It's an assertion with no punchline or anything. And it's an incorrect assertion. > But if you want to be pedantic irrational numbers cannot be represented by > the ratio of 2 integers therefor their true value is unknown and always will > be. You said "definable value", which is rather different from "it has a non-recurring decimal expansion". Pi is a computable number, and therefore it's perfectly well definable. |
| mickspud@potatofield.co.uk: Mar 03 11:27AM On Wed, 3 Mar 2021 11:48:12 +0100 >> when its signposted with a smiley. >A smiley works to indicate a joke like this when it is clear that the >poster knows that what they are writing is obviously and completely wrong. Ok, define the value of Pi then. And I don't mean just C/D. >> be. >However, when you write this is is clear that you are not very familiar >with the mathematics at this level. (Nothing wrong with that, of course So you are, good, then see above. Off you go... >- few people know or care what in means to be "computable".) The true >value of π is known - it is π. It can't be written as a finite decimal It can't be written as a finite expansion in any number base and a symbol is not a value - its a symbol. |
| mickspud@potatofield.co.uk: Mar 03 11:29AM On Wed, 3 Mar 2021 11:16:47 +0000 (UTC) >> when its signposted with a smiley. >There is no joke in your statement. It's an assertion with no punchline >or anything. And it's an incorrect assertion. Thank you for proving my point so emphatically and also proving that you don't know what "light hearted" means. But then English is a 2nd or 3rd language for you so fair enough. >Pi is a computable number, and therefore it's perfectly well definable. Define it then. |
| "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>: Mar 03 03:50AM -0800 >> A smiley works to indicate a joke like this when it is clear that the >> poster knows that what they are writing is obviously and completely wrong. > Ok, define the value of Pi then. And I don't mean just C/D. [...] atan(1) * 4 ? pi is pi. Infinite precision... ;^) |
| David Brown <david.brown@hesbynett.no>: Mar 03 01:32PM +0100 >> A smiley works to indicate a joke like this when it is clear that the >> poster knows that what they are writing is obviously and completely wrong. > Ok, define the value of Pi then. And I don't mean just C/D. Defining pi as the ratio of circumference to diameter of any circle is a perfectly good definition. But you can have lots of other definitions, all of which are equivalent, including Chris' suggestion : π = 4.tan⁻¹(1) or, if you prefer, the limit of π = 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... You can define it as the first positive solution "x" to e ^ ix = -1 All of these uniquely specify a single number - that makes them definitions. >> However, when you write this is is clear that you are not very familiar >> with the mathematics at this level. (Nothing wrong with that, of course > So you are, good, then see above. Off you go... See above. So, what do /you/ mean when you say "true value", or that the "true value" of a number is known? Do you just mean that when written down in decimal, the representation is either finite or ends in an infinite repetition of a finite sequence? That's a useful classification - so useful, that we give it a name: "the rational numbers". (It is an equivalent way to define them.) But mathematics does not pretend that only such numbers have a "true value". >> value of π is known - it is π. It can't be written as a finite decimal > It can't be written as a finite expansion in any number base and a symbol > is not a value - its a symbol. I assume you are restricting your number bases to those you are familiar with - using a positive integer greater than 1 as the base. It is perfectly valid, though somewhat obscure, to use other types of number base. For example, π can be represented as 10 in base π. You are mixing up representations and values here. A representation is just how you write a value, and mathematicians use many different representations for different purposes. Sometimes a given value can be written finitely in a particular representation, sometimes not. A good example would be the right angle. This is a well-defined concept, with a well-defined value. Represented in degrees, it is 90°, which in turn has a finite decimal representation. Represented in radians, it is π/2, which does not have a finite decimal representation. But the radian representation in different symbols is clear and finite. Equally, 1 radian is easily and finitely represented in decimal, while the degrees representation of the same angle is 180/π, which does not have a finite representation in decimal. In a desperate and feeble attempt to get on-topic for the group, the value of a C++ object is not dependent on how you choose to display it using an ostream. |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 12:37PM > It would appear you are confusing definable with representable. Infinity > can be represented, that doesn't mean it can be defined especially given > there are numerous different types. Maybe you should specify what you mean by "definable". The number pi can certainly be defined in a completely unambiguous way with a finite expression. Just because it may have an infinite non-recurring decimal expansion doesn't mean it can't be defined. https://en.wikipedia.org/wiki/Definable_real_number There exist real numbers that cannot be defined (with a finite expression), and in fact almost all real numbers are like that, but pi is not one of them. |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 12:41PM > Thank you for proving my point so emphatically and also proving that you > don't know what "light hearted" means. But then English is a 2nd or 3rd > language for you so fair enough. And assholery seems to be your first language. >>Pi is a computable number, and therefore it's perfectly well definable. > Define it then. It's the ratio between the circumference and the diameter of a circle. Maybe you don't understand what a "definable number" is. https://en.wikipedia.org/wiki/Definable_real_number |
| Paavo Helde <myfirstname@osa.pri.ee>: Mar 03 02:46PM +0200 > Juha Nieminen <nospam@thanks.invalid> wrote: >> Pi is a computable number, and therefore it's perfectly well definable. > Define it then. See e.g. Weierstrass definition from 1841: [math] \pi = \int_{-\infty}^\infty {dx \over 1+x^2} [/math] https://en.wikipedia.org/wiki/Pi#cite_note-16 (Remmert, Reinhold (2012). "Ch. 5 What is π?". In Heinz-Dieter Ebbinghaus; Hans Hermes; Friedrich Hirzebruch; Max Koecher; Klaus Mainzer; Jürgen Neukirch; Alexander Prestel; Reinhold Remmert (eds.). Numbers. Springer. ISBN 978-1-4612-1005-4.) |
| mickspud@potatofield.co.uk: Mar 03 02:43PM On Wed, 3 Mar 2021 12:41:33 +0000 (UTC) >> don't know what "light hearted" means. But then English is a 2nd or 3rd >> language for you so fair enough. >And assholery seems to be your first language. Its spelt arsehole in England - the clue is in the name of the language. If I wanted to learn proper Finnish I wouldn't go to Sweden. >It's the ratio between the circumference and the diameter of a circle. >Maybe you don't understand what a "definable number" is. >https://en.wikipedia.org/wiki/Definable_real_number I guess its a case of defining define. |
| Juha Nieminen <nospam@thanks.invalid>: Mar 03 05:05PM >>And assholery seems to be your first language. > Its spelt arsehole in England - the clue is in the name of the language. If I > wanted to learn proper Finnish I wouldn't go to Sweden. And then you call me an "aspie". >>Maybe you don't understand what a "definable number" is. >>https://en.wikipedia.org/wiki/Definable_real_number > I guess its a case of defining define. Let me ask it this way: Why do you consider a fraction, like 1/3, to be a definable number? |
| mickspud@potatofield.co.uk: Mar 03 05:11PM On Wed, 3 Mar 2021 17:05:55 +0000 (UTC) >> Its spelt arsehole in England - the clue is in the name of the language. If I >> wanted to learn proper Finnish I wouldn't go to Sweden. >And then you call me an "aspie". Just being helpful :) >> I guess its a case of defining define. >Let me ask it this way: Why do you consider a fraction, like 1/3, >to be a definable number? It has a quantifiable value. |
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