> On 25 Aug 2018, at 01:15, agrayson2...@gmail.com wrote:
>
>
>
> On Friday, August 24, 2018 at 12:25:03 PM UTC, telmo_menezes wrote:
> On 23 August 2018 at 06:31, <agrays...@gmail.com <javascript:>> wrote:
> >
> >
> > On Thursday, August 23, 2018 at 2:01:24 AM UTC, Jason wrote:
> >>
> >>
> >>
> >> On Wed, Aug 22, 2018 at 4:43 PM <agrays...@gmail.com <>> wrote:
> >>>
> >>>
> >>>
> >>> On Tuesday, August 21, 2018 at 3:22:04 PM UTC, Jason wrote:
> >>>>
> >>>>
> >>>>
> >>>> On Tue, Aug 21, 2018 at 1:16 AM <agrays...@gmail.com <>> wrote:
> >>>>>
> >>>>> I've been looking at the Wiki article on this topic. I find that I
> >>>>> really don't understand what it is, or why it's important. Maybe a few
> >>>>> succinct words from the usual suspects can be of help. TIA.
> >>>>>
> >>>>>
> >>>>
> >>>>
> >>>> Bruno provided a great definition and background of the Church-Turing
> >>>> Thesis. I will try to answer why it is important and comes up often in
> >>>> our
> >>>> discussion.
> >>>>
> >>>>
> >>>> The Church-Turing thesis says that anything that is computable is
> >>>> computable by any computer. In other words, there is nothing that the
> >>>> computer in your cell phone can't compute, that your laptop or that a
> >>>> super
> >>>> computer (or even a quantum computer) can. It just comes down to having
> >>>> enough time and memory.
> >>>>
> >>>> This is why you don't need to buy a new phone with new hardware every
> >>>> time you want to install a new app. Regardless of the type of CPU in
> >>>> your
> >>>> phone, it can be extended in its power of what it might compute only
> >>>> given
> >>>> some new software. It is in this sense that computers are "Universal",
> >>>> they
> >>>> are universal in the same sense that of a universal remote, or in the
> >>>> sense
> >>>> that a record player is a universal sound imitating device. A record
> >>>> player
> >>>> might emulate the sounds of an orchestra, Britney Spears, whale songs,
> >>>> etc.,
> >>>> all it needs is the appropriate record and it can produce the sound.
> >>>>
> >>>> In the same sense, all a Turing Machine (computer) needs to imitate (or
> >>>> emulate) the right program or function is the right software. Because
> >>>> of
> >>>> this, anything that can be described in software, be it a brain
> >>>> emulation,
> >>>> an AI, a virtual environment, a virtual machine or operating system, can
> >>>> never know what hardware is running it, because the Church-Turing thesis
> >>>> says that any computer is capable of running it.
> >>>>
> >>>> This is why if consciousness is computable (the computational theory of
> >>>> mind) we cannot know what is computing us (e.g. we could be in a matrix
> >>>> type
> >>>> simulation for all we know). The other implication is that if
> >>>> computations
> >>>> exist in mathematics (and they do), then we exist within mathematics.
> >>>> Mathematics (or at least the part necessary to describe computations)
> >>>> becomes the fundamental science of what we experience and what is
> >>>> possible
> >>>> to experience or what we may predict about our future experiences
> >>>> (physics).
> >>>>
> >>>>
> >>>> Jason
> >>>
> >>>
> >>> If someone digitizes (emulates) the Mona Lisa, is this equivalent to the
> >>> Mona Lisa?
> >>
> >>
> >> If you digitize a person and put the digitized Mona Lisa before them, it
> >> is equivalent to the real Mona Lisa to that person, at least as far as
> >> they
> >> can tell.
> >>
> >>
> >>>
> >>> Can you write a function which is not computable? AG
> >>>
> >>>
> >>
> >> If by not computable you mean it never returns, then this is easy:
> >>
> >> function foo():
> >> while (true)
> >> {
> >> // loop forever
> >> }
> >>
> >> There are also programs for which no one knows if they are computable or
> >> not. If you can prove whether or not this function ever completes, you
> >> will
> >> be world famous, and may even earn a million dollars (though I think the
> >> prize has been retracted, it might be oferred again):
> >>
> >> Step 1: Set X = 4
> >> Step 2: Set R = 0
> >> Step 3: For each Y from 1 to X, if both Y and (X – Y) are prime, set R = 1
> >> Step 4: If R = 1, Set X = X + 2 and go to Step 2
> >> Step 5: If R = 0, print X and halt
> >>
> >> All you have to prove is the computer either never gets to step 5 or that
> >> it does get to step 5. Mathematicians have been working on a related
> >> problem for 300 years, no one has solved it yet.
> >>
> >>
> >> Jason
> >
> >
> > I was asking about a well-defined mathematical function that can be written
> > in closed form, or possibly as an infinite series. I believe that all such
> > functions are computable. I was not discussing subroutines that might never
> > terminate. If all well defined mathematical functions are computable, why
> > did computability become a big deal? AG
>
> It is not true that all well-defined functions are computable. You
> have already been given examples by Jason and John of well-defined
> mathematical functions that are non-computable.
>
> You seem to confuse "well-defined" with "written in closed form". The
> latter is not even well-defined (heheh) because it hangs on the idea
> of a set of "well-known" functions, and people already have different
> ideas on what that set includes. Having well-known representations
> such as sin(x) or e^x, or even x + y does not magically make the
> related computations non-algorithmic. How do you think you learned how
> to add, subtract, multiply and divide in basic school? Those were
> algorithms.
>
> Well-defined just means that there is a non-ambiguous way to know if a
> given value corresponds to a given input of the function. If I tell
> you to consider the function f, such that its value is zero no matter
> the input, then I gave you a well-defined function in plain English.
> There is nothing magical about notation.
>
> Telmo.
>
> So a "function" must have a well defined domain set, finite or infinite, and
> is not limited to closed forms but includes infinite series and algorithms.
> In such case, an infinite loop, even if it has an initial value, is not a
> function and not computable, whereas all closed forms are computable. Agreed?
> AG
Infinite loop can be extended into the computable. The problem are infinite
computations getting more and more complex, and with no loop.
I am not sure what you mean by close form.
No worry, we will see soon, example of well defined functions which are not
computable. Basically all attributes of universal machine are not computable.
Bruno
>
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