The connection between the code by Arie Groeneveld and Louis de Forcrand
can be shown as follows,
NB. Arie Groeneveld...
S=: 1 :0
:
if. 0= y do. 0
else. 'q r'=. (0,m) #: y
( r *(1+m) ^ (0 (m S) x)) + (1+x) m S q
end.
)
G=: (>:@[ $: _1+ 4 : '0 (x S)y')`[@.(0=])
NB. Louis de Forcrand...
hbi=: ([ (] +/ . * >:@[ ^ ($: i.@-@#)) #.^:_1)^:(*@])"0
gs=: , >:@# <:@hbi {:
NB. Connection...
G0=. <:@:(2x&G"0)
G1=. (#@:(gs^:(0~:{:)^:_))"0@:x:
(G0 -: G1) 1 2 3
1
G1 1 2 3
2 4 6
This is the Goodstein function as per Wikipedia:
https://en.wikipedia.org/wiki/Goodstein%27s_theorem#Sequence_length_as_a_function_of_the_starting_value
Reportedly, it is a total function but unprovable in Peano arithmetic. Of
course, neither G0 nor G1 can produce the result corresponding to the
argument 4.
> This bug (is it a bug?) is in j805
I do not know for sure if the crash is a bug; yet, certainly, it is a
change in behavior. It seems that the issue can be shown also using gs,
gs^:(0~:{:)(^:_) 2
|domain error: hbi
| gs^:(0~:{:)(^:_)2
|[-0]
gs^:(0~:{:)(^:4) 2
2 2 1 4.44089e_16 _1
Interestingly,
gs^:(0~:{:)(^:_) 3
3 3 3 2 1 0
and likewise,
2 G 3
7
yet 2 G 2 produces a crash (running on J805 and above).
Furthermore, I put some extra code in S to display y in the comparison 0=
y. For 2 G 3 the display is identical, running on J804 and J806, and shows
only integers. However, for 2 G 2, running on J804 shows, again, only
integers but on J806, in contrast,
2 G 2
2.000000000000000000000000000000000000000
1.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
1.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
2.000000000000000400000000000000000000000
0.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
1.000000000000000400000000000000000000000
0.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
0.000000000000000444089209850062620000000
0.000000000000000000000000000000000000000
0.000000000000000000000000000000000000000
_0.999999999999999560000000000000000000000
_1.000000000000000000000000000000000000000
_1.000000000000000000000000000000000000000
_1.000000000000000000000000000000000000000
the numbers 2.000000000000000400000000000000000000000 and
1.000000000000000400000000000000000000000 show up before the number
mentioned by Arie Groeneveld (0.000000000000000444089209850062620000000).
On Wed, Apr 11, 2018 at 7:44 PM, 'Jon Hough' via Programming <
[email protected]> wrote:
> Must be due to x ^ y returning floating datatypes, and losing precision.
>
> depth=:0
> S=: 1 :0
> :
> y=.<.y NB. <--------------------- floor of y
> if. 0= y do. 0
> else.
> 'q r'=. (0,m) #: y
> ( r *(1+m) ^ (0 (m S) x)) + (1+x) m S q NB. <------------------ ^ returns
> floating
> end.
> )
>
> That will also fix it.
>
> This bug (is it a bug?) is in j805:
> JVERSION
> Engine: j805/j64/darwin
> Release: commercial/2016-12-11T08:17:56
> Library: 8.05.14
> Qt IDE: 1.5.4s/5.6.2
> Platform: Darwin 64
> Installer: J805 install
>
>
>
> --------------------------------------------
> On Thu, 4/12/18, Arie Groeneveld <[email protected]> wrote:
>
> Subject: Re: [Jprogramming] Goodstein Sequences and Hereditary base-n
> notation... CRASH!
> To: [email protected]
> Date: Thursday, April 12, 2018, 6:38 AM
>
> I noticed it has to do with the
> 0=y testing in S. Instead of producing a
> zero at a certain point it gives a small value
> 4.4....e_16 and then
> starts filling the
> stack.
>
> Op 11-04-18 om 22:05
> schreef Jose Mario Quintana:
> > You might
> like to look also at Arie Groeneveld's message,
> >
> > [Jprogramming] Crash
> calculating large hyper operations
> > http://www.jsoftware.com/pipermail/programming/2015-
> November/043351.html
> >
> > Unfortunately,
> running on the latest stable release,
> >
> > JVERSION
> > Engine: j806/j64nonavx/windows
> > Release: commercial/2017-11-06T10:01:33
> > Library: 8.06.09
> > Qt
> IDE: 1.6.2/5.6.3
> > Platform: Win 64
> > Installer: J806 install
> > InstallPath: j:/program files/j
> > Contact: www.jsoftware.com
> >
> > the following,
> >
> > S=: 1 :0
> > :
> > if. 0= y do. 0
> > else. 'q r'=. (0,m) #: y
> > ( r *(1+m) ^ (0 (m S) x)) +
> (1+x) m S q
> > end.
> >
> )
> >
> > G=: (>:@[ $:
> _1+ 4 : '0 (x S)y')`[@.(0=])
> >
> > 2 G"0 i.4
> >
> > now produces a crash ("J has stopped
> working") instead of,
> >
> > 2 3 5 7
> >
> > As far as I can see, the code should run
> on J804 but I do not know if it
> > runs on
> J805 (or on the latest and greatest j807).
> >
> > Anyway, in theory, 2
> G"0 i.5 should include the additional number,
> >
> > _1 + 3 * 2 ^
> 402653211x
> >
> > in
> practice, of course, it cannot (even trying to execute the
> sentence _1 +
> > 3 * 2 ^ 402653211x
> produces a limit error).
> >
> >
> >
> >
>
>
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