The halting problem revisited

Discussion in 'Java' started by Roedy Green, Mar 26, 2011.

  1. Roedy Green

    Roedy Green Guest

    The proof that the halting problem is insoluble has discouraged any
    sort work on static analysing what computer programs will do.

    What if the problem were reformulated like this:

    The input to the detector is a Java program, syntactically valid. The
    output is :

    1. definitely halts
    2. definitely does not halt
    3. can't tell

    Obviously such a program CAN be written. The question is how high
    quality is such a detector?

    What counts is the percentage of real-life programs that fall in
    category 1 or 2.

    It does not really matter how it fairs with pathological constructions
    favoured of mathematicians.

    There is some research in this area, going on in the name of code
    optimisation, where you do analysis on a fairly local basis.

    You might imagine a compiler warning you of parts of your program
    where you have created an endless loop or endless recursion under some
    unusual run time conditions. It would not necessarily catch
    everything, but it might catch something.


    --
    Roedy Green Canadian Mind Products
    http://mindprod.com
    There are only two industries that refer to their customers as "users".
    ~ Edward Tufte
     
    Roedy Green, Mar 26, 2011
    #1
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  2. In message <>, Roedy Green wrote:

    > The proof that the halting problem is insoluble has discouraged any
    > sort work on static analysing what computer programs will do.


    If that were true, we wouldn’t have optimizing compilers.

    > What if the problem were reformulated like this:
    >
    > The input to the detector is a Java program, syntactically valid. The
    > output is :
    >
    > 1. definitely halts
    > 2. definitely does not halt
    > 3. can't tell
    >
    > Obviously such a program CAN be written.


    Trivially always answer 3.

    > The question is how high quality is such a detector?


    Can’t tell.
     
    Lawrence D'Oliveiro, Mar 26, 2011
    #2
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  3. Roedy Green

    Alex Mentis Guest

    Lawrence D'Oliveiro wrote:

    > > The question is how high quality is such a detector?

    >
    > Can’t tell.


    hehe
     
    Alex Mentis, Mar 26, 2011
    #3
  4. Roedy Green

    Lew Guest

    On 03/26/2011 02:45 AM, Roedy Green wrote:
    > The proof that the halting problem is insoluble has discouraged any
    > sort work on static analysing what computer programs will do.


    "I'm ... well ... ummm ..., just not - er - sure how I ... um, ... feel about
    Turing," Tom said haltingly.

    --
    Lew
    Honi soit qui mal y pense.
    http://upload.wikimedia.org/wikipedia/commons/c/cf/Friz.jpg
     
    Lew, Mar 26, 2011
    #4
  5. On 03/26/2011 02:45 AM, Roedy Green wrote:
    > The proof that the halting problem is insoluble has discouraged any
    > sort work on static analysing what computer programs will do.


    No it hasn't. Static analysis is a very hot topic of research in
    compilers, despite the fact that is provably impossible. In fact, I
    would wager that computer science is the only field where we try to
    solve problems we already know to be impossible (well, there's also
    government, but that doesn't count).

    > What if the problem were reformulated like this:
    >
    > The input to the detector is a Java program, syntactically valid. The
    > output is :
    >
    > 1. definitely halts
    > 2. definitely does not halt
    > 3. can't tell
    >
    > Obviously such a program CAN be written. The question is how high
    > quality is such a detector?


    No, the question is how useful is the definite yes/no compared to the
    amount of time it takes to run. Also, it turns out that the halting
    problem is actually rather boring for static analysis; perhaps the most
    important question in general would either be the equivalency question
    (are these two expressions equivalent) or pointer aliasing (can these
    pointers point to the same value). Or, if you are looking at
    machine/assembly code, the most important question is code/data separation.

    > You might imagine a compiler warning you of parts of your program
    > where you have created an endless loop or endless recursion under some
    > unusual run time conditions. It would not necessarily catch
    > everything, but it might catch something.


    Sometimes, though, you want an infinite loop. Think of the event loop in
    a GUI toolkit.

    --
    Beware of bugs in the above code; I have only proved it correct, not
    tried it. -- Donald E. Knuth
     
    Joshua Cranmer, Mar 26, 2011
    #5
  6. Roedy Green

    Lew Guest

    Roedy Green wrote:
    > The input to the detector is a Java program, syntactically valid. The
    > output is :
    >
    > 1. definitely halts
    > 2. definitely does not halt
    > 3. can't tell
    >
    > Obviously such a program CAN be written. The question is how high
    > quality is such a detector?


    That is very far from obvious. You assume that the decider will halt.

    Now if #3 were "can't tell within period /x/", that'd be different.

    Joshua Cranmer wrote:
    > Roedy Green wrote:
    >> The proof that the halting problem is insoluble has discouraged any
    >> sort work on static analysing what computer programs will do.


    > No it hasn't. Static analysis is a very hot topic of research in compilers,
    > despite the fact that is provably impossible. In fact, I would wager that
    > computer science is the only field where we try to solve problems we already
    > know to be impossible (well, there's also government, but that doesn't count).


    False analysis. The government doesn't try to solve problems, irrespective of
    whether they are possible.

    Seriously, though, just because a problem isn't ultimately solvable doesn't
    mean that research into heuristics or more efficient approaches or bounding
    problems or probabilistic approaches won't be fruitful.

    For example, suppose you could prove that a decider has a 99.5% probability of
    proving haltability or the impossibility thereof within /x/ seconds. You can
    prove that the decider will halt by adding the constraint that it must end
    after /x/ seconds. You then call the answer "can't tell" with an epsilon
    probability of being wrong.

    I should think the impossibility of the theoretical would in no wise
    discourage the search for the profitable.

    --
    Lew
    Honi soit qui mal y pense.
    http://upload.wikimedia.org/wikipedia/commons/c/cf/Friz.jpg
     
    Lew, Mar 26, 2011
    #6
  7. On 03/26/2011 02:46 PM, Lew wrote:
    > Roedy Green wrote:
    >> The input to the detector is a Java program, syntactically valid. The
    >> output is :
    >>
    >> 1. definitely halts
    >> 2. definitely does not halt
    >> 3. can't tell
    >>
    >> Obviously such a program CAN be written. The question is how high
    >> quality is such a detector?

    >
    > That is very far from obvious. You assume that the decider will halt.


    Program A:
    Always returns "can't tell" for any input.

    Program B:
    Looks for a for/while/do-while loop in the code. If one is present,
    output "can't tell". Otherwise, check a method callgraph for the
    presence of a cycle. If one is present, output "can't tell", otherwise,
    output "definitely does not halt"

    Program A clearly satisfies the requirements of the detector, in that it
    is never wrong. Obviously, it always terminates, so there is an obvious
    program (at least to me) that satisfies the requirements.

    Program B is a program which is moderately more useful but still
    satisfies the requirements.

    So it seems quite obvious that it exists, at least to me.

    --
    Beware of bugs in the above code; I have only proved it correct, not
    tried it. -- Donald E. Knuth
     
    Joshua Cranmer, Mar 26, 2011
    #7
  8. Roedy Green

    Lew Guest

    Joshua Cranmer wrote:
    > Lew wrote:
    >> Roedy Green wrote:
    >>> The input to the detector is a Java program, syntactically valid. The
    >>> output is :
    >>>
    >>> 1. definitely halts
    >>> 2. definitely does not halt
    >>> 3. can't tell
    >>>
    >>> Obviously such a program CAN be written. The question is how high
    >>> quality is such a detector?

    >>
    >> That is very far from obvious. You assume that the decider will halt.

    >
    > Program A:
    > Always returns "can't tell" for any input.
    >
    > Program B:
    > Looks for a for/while/do-while loop in the code. If one is present, output
    > "can't tell". Otherwise, check a method callgraph for the presence of a cycle.
    > If one is present, output "can't tell", otherwise, output "definitely does not
    > halt"
    >
    > Program A clearly satisfies the requirements of the detector, in that it is
    > never wrong. Obviously, it always terminates, so there is an obvious program
    > (at least to me) that satisfies the requirements.
    >
    > Program B is a program which is moderately more useful but still satisfies the
    > requirements.
    >
    > So it seems quite obvious that it exists, at least to me.


    Silly me. I assumed we wanted it to produce a *correct* answer based on an
    actual effort to determine the answer.

    Now that I look again, I see that attempts at correct results were not
    explicitly specified. What was I /thinking/?

    Will program B always succeed? Now that I hear your answer it's no longer not
    obvious.

    --
    Lew
    Honi soit qui mal y pense.
    http://upload.wikimedia.org/wikipedia/commons/c/cf/Friz.jpg
     
    Lew, Mar 26, 2011
    #8
  9. On 26/03/2011 07:07, Lawrence D'Oliveiro wrote:
    > In message<>, Roedy Green wrote:
    >
    >> The proof that the halting problem is insoluble has discouraged any
    >> sort work on static analysing what computer programs will do.

    >
    > If that were true, we wouldn’t have optimizing compilers.
    >
    >> What if the problem were reformulated like this:
    >>
    >> The input to the detector is a Java program, syntactically valid. The
    >> output is :
    >>
    >> 1. definitely halts
    >> 2. definitely does not halt
    >> 3. can't tell
    >>
    >> Obviously such a program CAN be written.

    >
    > Trivially always answer 3.


    No.
    What cannot be done is to create an algorithm that will tell in all
    cases whether a program will halt.

    It is trivial to detect whether most small programs will do in a finite
    time. Otherwise nobody could ever read somebody else's code and spot
    halting/loop bugs. The Human brain is also subject to the Turing
    limitation (as far as is known)

    --
    Dirk

    http://www.neopax.com/technomage/ - My new book - Magick and Technology
     
    Dirk Bruere at NeoPax, Mar 26, 2011
    #9
  10. On 03/26/2011 04:53 PM, Lew wrote:
    > Will program B always succeed? Now that I hear your answer it's no
    > longer not obvious.


    If you make the modification that I had (I mixed up "definitely halts"
    with "definitely does not halt"), yes, with the added caveat that if you
    cannot construct the callgraph, you print out "can't tell." The reasons
    for such might include native methods, effects of reflection, violation
    of the closed world assumption, or multithreaded pitfalls (viewing the
    callgraph at an actual instruction level, so context switches become
    extra edges--I'm viewing this as its correlation to a Turing machine).

    --
    Beware of bugs in the above code; I have only proved it correct, not
    tried it. -- Donald E. Knuth
     
    Joshua Cranmer, Mar 26, 2011
    #10
  11. On 26.03.2011 21:56, Dirk Bruere at NeoPax wrote:
    > The Human brain is also subject to the Turing
    > limitation (as far as is known)


    If what you're saying is true, then how can the human brain prove things
    like, say, Fermat's last theorem?
     
    Screamin Lord Byron, Mar 27, 2011
    #11
  12. On 27.03.2011 21:18, Patricia Shanahan wrote:
    > On 3/27/2011 9:10 AM, Screamin Lord Byron wrote:
    >> On 26.03.2011 21:56, Dirk Bruere at NeoPax wrote:
    >>> The Human brain is also subject to the Turing
    >>> limitation (as far as is known)

    >>
    >> If what you're saying is true, then how can the human brain prove things
    >> like, say, Fermat's last theorem?

    >
    > How would you deduce inability to prove Fermat's last theorem from the
    > Turing limitation?


    There can not be an algorithmic proof of Fermat's last theorem because
    of the halting problem. And yet, humans can produce the proof.
    Therefore, human brain is not subject to Turing limitation.
     
    Screamin Lord Byron, Mar 27, 2011
    #12
  13. On 27.03.2011 22:30, Patricia Shanahan wrote:
    > On 3/27/2011 1:19 PM, Screamin Lord Byron wrote:
    >> On 27.03.2011 21:18, Patricia Shanahan wrote:
    >>> On 3/27/2011 9:10 AM, Screamin Lord Byron wrote:
    >>>> On 26.03.2011 21:56, Dirk Bruere at NeoPax wrote:
    >>>>> The Human brain is also subject to the Turing
    >>>>> limitation (as far as is known)
    >>>>
    >>>> If what you're saying is true, then how can the human brain prove
    >>>> things
    >>>> like, say, Fermat's last theorem?
    >>>
    >>> How would you deduce inability to prove Fermat's last theorem from the
    >>> Turing limitation?

    >>
    >> There can not be an algorithmic proof of Fermat's last theorem because
    >> of the halting problem....

    >
    > Why does the halting problem imply anything at all about the existence
    > or otherwise of a proof for Fermat's last theorem?
    > Can you give a proof, or point to one I can read?


    Yes. Sir Roger Penrose's book "Emperor's New Mind". I have a copy which
    is translated to my language, so I can't give you exact page numbers
    where he talks about that specifically (should be within first 100
    pages), but the book is absolutely worth a read in its entirety.
     
    Screamin Lord Byron, Mar 27, 2011
    #13
  14. On 27.03.2011 23:03, Screamin Lord Byron wrote:
    > On 27.03.2011 22:30, Patricia Shanahan wrote:
    >> On 3/27/2011 1:19 PM, Screamin Lord Byron wrote:
    >>> On 27.03.2011 21:18, Patricia Shanahan wrote:
    >>>> On 3/27/2011 9:10 AM, Screamin Lord Byron wrote:
    >>>>> On 26.03.2011 21:56, Dirk Bruere at NeoPax wrote:
    >>>>>> The Human brain is also subject to the Turing
    >>>>>> limitation (as far as is known)
    >>>>>
    >>>>> If what you're saying is true, then how can the human brain prove
    >>>>> things
    >>>>> like, say, Fermat's last theorem?
    >>>>
    >>>> How would you deduce inability to prove Fermat's last theorem from the
    >>>> Turing limitation?
    >>>
    >>> There can not be an algorithmic proof of Fermat's last theorem because
    >>> of the halting problem....

    >>
    >> Why does the halting problem imply anything at all about the existence
    >> or otherwise of a proof for Fermat's last theorem?
    >> Can you give a proof, or point to one I can read?

    >
    > Yes. Sir Roger Penrose's book "Emperor's New Mind". I have a copy which
    > is translated to my language, so I can't give you exact page numbers
    > where he talks about that specifically (should be within first 100
    > pages), but the book is absolutely worth a read in its entirety.


    Just to be clear, in the time of writing of that book Fermat's last
    theorem wasn't yet proven nor Penrose provides the proof. That was done
    later of course.
    http://en.wikipedia.org/wiki/Wiles'_proof_of_Fermat's_Last_Theorem

    In his book, Penrose argues that such problems cannot be solved
    algorithmically.
     
    Screamin Lord Byron, Mar 27, 2011
    #14
  15. On 27.03.2011 23:18, Patricia Shanahan wrote:
    > On 3/27/2011 2:03 PM, Screamin Lord Byron wrote:
    >> On 27.03.2011 22:30, Patricia Shanahan wrote:
    >>> On 3/27/2011 1:19 PM, Screamin Lord Byron wrote:
    >>>> On 27.03.2011 21:18, Patricia Shanahan wrote:
    >>>>> On 3/27/2011 9:10 AM, Screamin Lord Byron wrote:
    >>>>>> On 26.03.2011 21:56, Dirk Bruere at NeoPax wrote:
    >>>>>>> The Human brain is also subject to the Turing
    >>>>>>> limitation (as far as is known)
    >>>>>>
    >>>>>> If what you're saying is true, then how can the human brain prove
    >>>>>> things
    >>>>>> like, say, Fermat's last theorem?
    >>>>>
    >>>>> How would you deduce inability to prove Fermat's last theorem from the
    >>>>> Turing limitation?
    >>>>
    >>>> There can not be an algorithmic proof of Fermat's last theorem because
    >>>> of the halting problem....
    >>>
    >>> Why does the halting problem imply anything at all about the existence
    >>> or otherwise of a proof for Fermat's last theorem?
    >>> Can you give a proof, or point to one I can read?

    >>
    >> Yes. Sir Roger Penrose's book "Emperor's New Mind". I have a copy which
    >> is translated to my language, so I can't give you exact page numbers
    >> where he talks about that specifically (should be within first 100
    >> pages), but the book is absolutely worth a read in its entirety.
    >>

    >
    > I don't have a copy, so that is not currently one I can read. Perhaps
    > you can restate the proof in your own words, or give an on-line reference?


    In fact I have. I didn't have much hope to find it online, but I got
    lucky I guess. :)

    http://bit.ly/gCPoot

    Long link:
    <http://books.google.hr/books?id=oI0grArWHUMC&pg=PA75&lpg=PA75&dq=emperor's+new+mind+hilbert+problem&source=bl&ots=04Ljj-YNVy&sig=GNJwD-YkfZ1R5u2oFHD2vjrEqns&hl=hr&ei=T6yPTYnyHIrysgbi7Y2NCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBgQ6AEwAA#v=onepage&q&f=false>
     
    Screamin Lord Byron, Mar 27, 2011
    #15
  16. Roedy Green

    Lew Guest

    Dirk Bruere at NeoPax wrote:
    > It is trivial to detect whether most small programs will do in a finite time.
    > Otherwise nobody could ever read somebody else's code and spot halting/loop
    > bugs. The Human brain is also subject to the Turing limitation (as far as is
    > known)


    Fortunately the human mind is not limited to the limitations of the human brain.

    --
    Lew
    Honi soit qui mal y pense.
    http://upload.wikimedia.org/wikipedia/commons/c/cf/Friz.jpg
     
    Lew, Mar 28, 2011
    #16
  17. In article <>,
    Patricia Shanahan <> wrote:

    > On 3/27/2011 2:03 PM, Screamin Lord Byron wrote:
    > > On 27.03.2011 22:30, Patricia Shanahan wrote:
    > >> On 3/27/2011 1:19 PM, Screamin Lord Byron wrote:
    > >>> On 27.03.2011 21:18, Patricia Shanahan wrote:
    > >>>> On 3/27/2011 9:10 AM, Screamin Lord Byron wrote:
    > >>>>> On 26.03.2011 21:56, Dirk Bruere at NeoPax wrote:
    > >>>>>> The Human brain is also subject to the Turing
    > >>>>>> limitation (as far as is known)
    > >>>>>
    > >>>>> If what you're saying is true, then how can the human brain
    > >>>>> prove things like, say, Fermat's last theorem?
    > >>>>
    > >>>> How would you deduce inability to prove Fermat's last theorem
    > >>>> from the Turing limitation?
    > >>>
    > >>> There can not be an algorithmic proof of Fermat's last theorem
    > >>> because of the halting problem....
    > >>
    > >> Why does the halting problem imply anything at all about the
    > >> existence or otherwise of a proof for Fermat's last theorem? Can
    > >> you give a proof, or point to one I can read?

    > >
    > > Yes. Sir Roger Penrose's book "Emperor's New Mind". I have a copy
    > > which is translated to my language, so I can't give you exact page
    > > numbers where he talks about that specifically (should be within
    > > first 100 pages), but the book is absolutely worth a read in its
    > > entirety.
    > >

    >
    > I don't have a copy, so that is not currently one I can read. Perhaps
    > you can restate the proof in your own words, or give an on-line
    > reference?


    Several related arguments, including Penrose's, are summarized here:

    <http://en.wikipedia.org/wiki/Mechanism_(philosophy)#G.C3.B6delian_arguments>

    --
    John B. Matthews
    trashgod at gmail dot com
    <http://sites.google.com/site/drjohnbmatthews>
     
    John B. Matthews, Mar 28, 2011
    #17
  18. On 27/03/2011 10:20 PM, Lew wrote:
    > Dirk Bruere at NeoPax wrote:
    >> It is trivial to detect whether most small programs will do in a
    >> finite time.
    >> Otherwise nobody could ever read somebody else's code and spot
    >> halting/loop
    >> bugs. The Human brain is also subject to the Turing limitation (as far
    >> as is
    >> known)

    >
    > Fortunately the human mind is not limited to the limitations of the
    > human brain.


    Non sequitur. In fact, self-contradicting. The human mind is implemented
    on the human brain*. Your statement is analogous to saying that some
    computer program you ran on your dual-core x86-64 machine was not
    limited to the limitations of your dual-core x86-64 machine, or that if
    you put a fancy enough GPS unit in your Chevy Tahoe you could drive it
    from New York to Sydney, Australia in under two hours, regardless of the
    Tahoe's engine speed, and without having to drive it onto a plane or a ship.

    Obvious nonsense.

    * And to forestall the inevitable objections from dualists, every single
    faculty of the human mind has been found to be able to be specifically
    impaired by particular brain injuries, including memory, face
    recognition, conscious awareness, language (and particular subskills of
    language), and more. If the brain was just the antenna by which some
    external mind-stuff connected to pilot the body, it seems unlikely that
    this would be the case. Indeed, other than specific low-level sensory or
    motor impairments, and perhaps epilepsy, any sensible dualist theory
    would predict a near-total absence of mental disorders resulting from
    identifiable physical trauma or chemical states of the brain, or
    correctable by same. But instead it's possible to get whacked upside the
    head and become unable to *think certain thoughts* anymore.

    Read "The Man Who Mistook His Wife For A Hat" and try to reconcile *any*
    of it with *any* flavor of dualism. On the other hand, the broad
    spectrum of often very quirky bug reports in there are pretty much
    exactly what you'd expect if the mind was complex software that ran as
    specialized daemons distributed over a complex computer-network inside
    the skull, and then various parts of that network became "host unreachable".

    I'll add that they've mostly already reverse engineered auditory
    perception, have made big strides in reverse engineering visual
    perception, and will probably have decompiled and begun to analyze the
    very algorithms underpinning conscious awareness itself in another
    decade or three. MP3 encoding is a lossy compression whose lossiness was
    optimized partly based on understanding the algorithms used in the human
    brain to process incoming sound data. With every passing year, that much
    less of what goes on in there remains unexplained in terms of simple
    mechanics, electrochemistry, and various algorithms, and there's no
    reason to believe that process will stop short of explaining *all* of it
    in such terms.

    --
    public final class JSnarker
    extends JComponent
    A JSnarker is an NNTP-aware component that asynchronously provides
    snarky output when the Ego.needsPuncturing() event is fired in cljp.
     
    javax.swing.JSnarker, Mar 28, 2011
    #18
  19. "Patricia Shanahan" <> wrote in message
    news:...
    > On 3/27/2011 8:26 PM, John B. Matthews wrote:
    >> In article<>,
    >> Patricia Shanahan<> wrote:
    >>
    >>> On 3/27/2011 2:03 PM, Screamin Lord Byron wrote:

    > ...
    >>>> Yes. Sir Roger Penrose's book "Emperor's New Mind". I have a copy
    >>>> which is translated to my language, so I can't give you exact page
    >>>> numbers where he talks about that specifically (should be within
    >>>> first 100 pages), but the book is absolutely worth a read in its
    >>>> entirety.
    >>>>
    >>>
    >>> I don't have a copy, so that is not currently one I can read. Perhaps
    >>> you can restate the proof in your own words, or give an on-line
    >>> reference?

    >>
    >> Several related arguments, including Penrose's, are summarized here:
    >>
    >> <http://en.wikipedia.org/wiki/Mechanism_(philosophy)#G.C3.B6delian_arguments>
    >>

    >
    > Thanks. I happen to agree with the view that human reasoning is not
    > always consistent. It has evolved to produce decisions as needed, even
    > if there is limited data, which puts completeness ahead of consistency.



    In particular, I think, the human mind sees patterns everywhere, and is
    ready to reason from them without knowing for sure whether they're real or
    spurious. Ramanujan was well-known for deriving amazing results from this
    kind of reasoning by analogy which other, sometimes lesser, mathematicians
    would later prove rigorously.
     
    Mike Schilling, Mar 28, 2011
    #19
  20. In message <imp859$n5i$>, Mike Schilling wrote:

    > In particular, I think, the human mind sees patterns everywhere, and is
    > ready to reason from them without knowing for sure whether they're real or
    > spurious.


    And the part of the mind that believes in these patterns is different from
    the part that finds out whether they’re spurious.

    And so you have the odd situation of scientists who continue to believe in
    religion, for example. Or Johnny Carson’s famed debunking of Uri Geller, the
    effect of which on his popularity was ... zero.
     
    Lawrence D'Oliveiro, Mar 28, 2011
    #20
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