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Re: looping

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Leonid Shifrin

Jan 23, 2010, 7:30:50â€¯AM1/23/10
to
Hi Glenn,

You don't need a loop at all, if I understand your goal correctly. What you
need is a bit of Dynamic functionality. Hope this will get you started:

Clear[makeScorePanel, getScores, getCalled, getPrcntp, bpanel];
makeScorePanel[names : {__String}] :=
Module[{text, n = RandomInteger[Length[names] - 1] + 1, studentInfo,
classFlag = False, buttons, class, percent, next,
score , called , prcnt},
Clear[getScores, getCalled, getPrcntp, bpanel];
score = called = prcnt = Table[0, {Length[names]}];
getScores[] := score;
getCalled[] := called;
getPrcntp[] := prcnt;
text := "Was the answer correct?";
next := (called[[n]]++; n = RandomInteger[Length[names] - 1] + 1);
class :=
If[classFlag,
Panel[Grid[
Transpose[{names, score,
IntegerPart[100*score/(called /. (0 -> 1))]/100.}
],
Spacings -> 3, Dividers -> Center]],
""];
buttons :=
Row[{
Button[" Yes ", score[[n]]++; next],
Button["Class", classFlag = ! classFlag],
Button[" No ", next]}];
percent :=
If[called[[n]] == 0, 0,
IntegerPart[100*score[[n]]/called[[n]]]/100.];
studentInfo :=
Panel[Grid[
{{"Name", names[[n]]},
{"Times called", called[[n]]},
Alignment -> Left]];
bpanel[] :=
Dynamic@Panel[Column[{
studentInfo,
text,
buttons,
class
}]]];

The way to use: first call the makeScorePanel[] with your actual list of
names, then call bpanel[]:

makeScorePanel[{"apple", "bob", "cat", "dog", "ear", "frog", "greg",
"hippo", "i9", "joe"}]

bpanel[]

You stop when you feel like it. You can also call getScores[], getCalled[]
, getPrcntp[] at any time to get the current state of these variables (if
you need that for further processing).

Regards,
Leonid

On Thu, Jan 21, 2010 at 12:55 PM, glenn kuhaneck <mcguy...@yahoo.com>wrote:

> this code is supposed to randomly select a student from a list, keep track
> of how many times he/she has been called, and how many answers they have
> gotten correct.
>
> I am having problems getting the following code to repeat on request: i
>
> These are the sample lists I am using for the code below
> name = ( {
> {"apple"},
> {"bob"},
> {"cat"},
> {"dog"},
> {"ear"},
> {"frog"},
> {"greg"},
> {"hippo"},
> {"i9"},
> {"joe"}
> }); score = ({
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0}
> }); called = ({
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0}
> }); prcnt = ( {
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0},
> {0}
> } );
>
>
>
>
> this is the code to manipulate the lists created above
> repeat = "yes";
> c = "wrong";
> l = Length[name];
> n = RandomInteger[l - 1] + 1;
>
>
> Label[begin]
> "Student #"
> n
> name[[n]]
> called[[n]] += 1;
> "# times called"
> called
> "Old Score"
> score
> Input["Was the answer correct?", Button["Yes", c = "Right"]];
> If[c == "Right", score[[n]] += 1]; c
> "New Score"
> score
> score[[n]]
> prcnt[[n]] = score[[n]]/called[[n]]
> "Class %"
> prcnt
> Input["Another ?", Button["No", repeat = "no"]];
> If[repeat == "yes", Goto[begin], Goto[end]];
> Label[end];
> prcnt[[n]]
>
>
>
> thank you for your assistance,
> Mr. Glenn J. Kuhaneck
>
>

DrMajorBob

Jan 23, 2010, 7:33:23â€¯AM1/23/10
to
This is awkward too, but have a look:

initialize[] := (
names = {"apple", "bob", "cat", "dog", "ear", "frog", "greg",
"hippo", "i9", "joe"};
Clear[called, score, percent];
called[_] = 0;
score[_] = 0;
percent[any_] :=
Quiet@AccountingForm[100. score[any]/called[any], 3];
)

initialize[]
repeat = True;
While[repeat,
n = RandomInteger[{1, Length@names}];
name = names[[n]];
called[name]++;

SetterBar[Dynamic[result], {1 -> "correct", 0 -> "incorrect"}]];
score[name] += result;
Input["Continue?",
SetterBar[Dynamic[repeat], {True -> "Go on", False -> "Stop"}]]
]
Grid[
Table[Through[{Identity, called, score, percent}@name], {name,
names}] /. Indeterminate -> "No scores"]

Bobby

On Thu, 21 Jan 2010 03:55:25 -0600, glenn kuhaneck <mcguy...@yahoo.com>
wrote:

DrMajorBob

Jan 24, 2010, 5:41:30â€¯AM1/24/10
to
I'd style this a bit differently:

Clear[makeScorePanel];
makeScorePanel[names : {__String}] :=
Module[{text, k = Length@names, name = RandomChoice@names,
studentInfo, classFlag = True, buttons, class, percent, next,
correct, questions},

"internal functions";
correct[_] = 0; questions[_] = 0;
percent[name_] /; questions[name] == 0 = 0;
percent[name_] := Round[100*correct[name]/questions[name]];

"display elements";

text = "Was the answer correct?";

next := (questions[name]++; name = RandomChoice@names);

class :=
If[classFlag,
Panel[Grid[

Prepend[Transpose[{names, getQuestionCount[],
getCorrectCount[], getPercent[]}], {"Name", "Questions",
"# correct", "% correct"}], Spacings -> 3,

Dividers -> Center]], ""];
buttons :=

Row[{Button[" Yes ", correct[name]++; next],
Button[" No ", next],
Button[If[classFlag, "Hide Roster", "Show Roster"],
classFlag = ! classFlag]}];
studentInfo :=
Panel[
Grid[{{"Name", name}, {"Questions",
correct@name}, {"% correct", percent@name}},
Alignment -> Left]];

"exported functions";
Clear[getCorrectCount, getQuestionCount, getPercent,
displayPanel];
getCorrectCount[] := correct /@ names;
getQuestionCount[] := questions /@ names;
getPercent[] := percent /@ names;
displayPanel[] :=

Dynamic@Panel[Column[{studentInfo, text, buttons, class}]]
];

makeScorePanel[{"apple", "bob", "cat", "dog", "ear", "frog", "greg",
"hippo", "i9", "joe"}]
displayPanel[]

Bobby

On Sat, 23 Jan 2010 06:30:56 -0600, Leonid Shifrin <lsh...@gmail.com>
wrote:

> On Thu, Jan 21, 2010 at 12:55 PM, glenn kuhaneck

Leonid Shifrin

Jan 24, 2010, 5:41:52â€¯AM1/24/10
to
Hi Bobby,

I agree - your modification makes it more elegant. I particularly liked
your use of indexed variables and RandomChoice.

Cheers,
Leonid

DrMajorBob

Jan 24, 2010, 5:43:32â€¯AM1/24/10
to
Thanks! I've just noticed something I don't understand, however.

How can displayPanel[] be used outside the makeScorePanel function, when
it uses studentInfo, text, buttons, and class -- all of which are local to
the Module??

Bobby

On Sat, 23 Jan 2010 14:57:25 -0600, Leonid Shifrin <lsh...@gmail.com>
wrote:

Leonid Shifrin

Jan 25, 2010, 5:05:53â€¯AM1/25/10
to
Bobby,

this is one of the nice (IMO) features of local variables in Module. By
default, they have the attribute Temporary, which means that they are
destroyed once the execution exits Module. However, if some global symbols
refer to them (like the functions you've mentioned), they are not destroyed,
but kept in a symbol table. I use this trick all the time - this allows for
example to share local variables (and functions) between several functions,
which enables us to implement something similar to classes in OOP (If I
remember corerctly, this idea has been fully exploited by Roman Maeder in
his implementation of OOP in Mathematica - classes.m package).

Have a look at my post in this thread, if you will

there I implement the <pair> data type using this idea. In this thread:

I used this trick to implement traversals of nested directory structure with
possible directory skips which can be set at run-time based on a skip
function provided by the user.

One of the many other ways to use this which I find useful is to propagate
exceptions of a given type without explicitly making the exception tag
global. One particular such use I discussed in the thread:

Generally, this is a good option when you want to make an essentially
global variable available to a given set of functions but not to others -
you make it like

Module[{youvar},

f1[args]:=(body-referring-to-yourvar)

f2[args]:=(body-referring-to-yourvar)

...
];

This is a cheap way to introduce namespaces without making a full-blown
package. This allows us to use some of the nice OOP-like stuff (such as
private variables / methods) without the need to employ a general OOP
machinery (that is, when we just need nice encapsulation but not so much
OO-style polymorphism / inheritance). As long as the user never uses
variables with a dollar sign in their names (which can possibly collide with
those generated by Module), this should be safe enough.

One use of this is to make functions with "memory", similar to static local
variables in C functions - some of the function's variables remember their
values in between function calls. For example, the following function will
produce the next Fibonacci number on demand, and yet it will be as fast as
the iterative loop implementation for generation of consecutive Fibonacci
numbers (since Module is invoked only once, when the function is defined):

In[1]:= Module[{prev, prevprev, this},
reset[] := (prev = 1; prevprev = 1);
reset[];
nextFib[] := (this = prev + prevprev; prevprev = prev; prev = this)];

In[2]:=
reset[];
Table[nextFib[], {1000}]; // Timing

Out[3]= {0.01, Null}

In some cases, this can also improve performance, since some of the local
variables in a function can be made "semi-global" by this trick which may
eliminate the need of Module invocation in each function call, and the

In[4]:=
Clear[f, f1];
f[x_, y_, z_] := Module[{xl = x, yl = y, zl = z}, (xl + yl + zl)^2];

In[6]:=
Module[{xl, yl, zl},
f1[x_, y_, z_] := (xl = x; yl = y; zl = z; (xl + yl + zl)^2)];

In[8]:= test = RandomInteger[100, {10000, 3}];

In[9]:= f @@@ test; // Timing

Out[9]= {0.43, Null}

In[10]:= f1 @@@ test; // Timing

Out[10]= {0.15, Null}

I generally find this technique very useful, and also I think that it has
not been fully exploited (at least I did not fully exploit it yet). For
example, you can do nice things when you couple it with Dynamic
functionality, since Dynamic happens to work also on these Module-generated
variables.

It may however have some garbage-collection issues (I discussed this in the
first of the threads I mentioned above), since once you abandon such local
variables/functions, they will not be automatically garbage-collected by
Mathematica and can soak up memory (I have been bitten by this a few times).
Of course, this can be dealt with as well, it's just not automatic.

Regards,
Leonid

DrMajorBob

Jan 25, 2010, 5:07:21â€¯AM1/25/10
to
Huh! That's an amazing little secret trick you have, there!

I wonder, though, if it could go away in a new version.

Bobby

On Sun, 24 Jan 2010 04:11:07 -0600, Leonid Shifrin <lsh...@gmail.com>
wrote:

Leonid Shifrin

Jan 25, 2010, 5:08:16â€¯AM1/25/10
to
Bobby,

I had no doubt you'd like this one. I really hope that this feature will
stay in later versions. Besides, changes to this seem unlikely because this
is a fundamental mechanism behind scoping in Module, and it has been this
way for a very long time (I don't know since which version but I wouldn't be
surprised if it was there already in version 1).

Regards,
Leonid

Leonid Shifrin

Jan 26, 2010, 6:23:25â€¯AM1/26/10
to
Hi Andrzej,

Actually there was no Module in version 1. (e.g. see the documentation for
> Module). Module was introduced in Version 2. Version 1 had only Block.
>

somewhere (Wagner's book perhaps) - also that one of the main reasons for
its appearance was that Block is a dynamic scoping construct and lots of
people were confused by the name and using it as a lexical scoping construct
instead, which of course is very error-prone.

> Andrzej Kozlowski
>
> (Nice tricks, by the way. I have know about some of them for a long time
> but others are new to me.)

Well, thanks. Somehow all of them came out as results of my experiments with
Mathematica, at the times when I did not have Maeder's books or other
sources of information on this topic. I actually think that I would not
have come to them had I known more about Mathematica at the time.

Regards,
Leonid

E. Martin-Serrano

Jan 26, 2010, 6:26:31â€¯AM1/26/10
to
Hi Leonid,

Very useful finding!

I have been looking for these construct for years, this will simplify and
clean a lot some of my co-routines (cross invoking routines) and similar
pieces of code.

Up to now, I have overcome the gap by reserving some global variables to
simulate _local variables with persistent values between calls_. A rather
weak and error prone programming technique.

I have been tempted to post many times to ask for a canonical and
straightforward solution, and your finding seems to be the safest and
natural way. I hope WRI will document and leave the feature as it is now.

They clearly should, since it sounds as being something very very powerful;
and I do not understand why it has not been clearly documented yet.

E. Martin-Serrano

Andrzej Kozlowski

Jan 26, 2010, 6:27:38â€¯AM1/26/10
to

On 25 Jan 2010, at 11:08, Leonid Shifrin wrote:

> Besides, changes to this seem unlikely because this
> is a fundamental mechanism behind scoping in Module, and it has been this
> way for a very long time (I don't know since which version but I wouldn't be
> surprised if it was there already in version 1).

Actually there was no Module in version 1. (e.g. see the documentation for Module). Module was introduced in Version 2. Version 1 had only Block.

Andrzej Kozlowski

Richard Fateman

Jan 27, 2010, 6:23:28â€¯AM1/27/10
to
Oh dear. As I pointed out in 1992, the Module mechanism is flawed.
Module makes up names,

as you can see by g1[x_]:=Module[{r}, r]

with dollar signs. e.g. I got r\$592 one time.

The standard advice is you can avoid conflict yourself by not using
names with dollar signs in them.

Unfortunately, if you have several independent programs perhaps
written by different people and they each generate persistent names via
Module[{...,r...} ...] and you load them into the same system then
THEY can conflict.

consider
g1[x_]:=Module[{r},If[NumberQ[A[r]],Print[r],A[r]=0]]

run it a bunch of times. Save g1 and A in file f1
start up a new system
define
g2[x_]:=Module[{r},If[NumberQ[A[r]],Print[r],A[r]=1]]

run it a bunch of times. Save g2 and A in file f2.
start up a new system

the behavior of g1 and g2 will be, so far as I can tell, dependent on
the (random?) setting of some inaccessible counter in Mathematica.

Andrzej Kozlowski

Jan 27, 2010, 6:23:40â€¯AM1/27/10
to

On 27 Jan 2010, at 07:31, Richard Fateman wrote:

> Andrzej Kozlowski wrote:
>> On 25 Jan 2010, at 11:08, Leonid Shifrin wrote:
>>> Besides, changes to this seem unlikely because this

>>> is a fundamental mechanism behind scoping in Module, and it has been =
this
>>> way for a very long time (I don't know since which version but I =

wouldn't be
>>> surprised if it was there already in version 1).

>> Actually there was no Module in version 1. (e.g. see the =
documentation for Module). Module was introduced in Version 2. Version 1 =

>> Andrzej Kozlowski

>> (Nice tricks, by the way. I have know about some of them for a long =

time but others are new to me.)

-------
> Unfortunately, if you have several independent programs perhaps =
written by different people and they each generate persistent names via =
Module[{...,r...} ...] and you load them into the same system then THEY =

can conflict.
>
> consider
> g1[x_]:=Module[{r},If[NumberQ[A[r]],Print[r],A[r]=0]]
>
> run it a bunch of times. Save g1 and A in file f1
> start up a new system
> define
> g2[x_]:=Module[{r},If[NumberQ[A[r]],Print[r],A[r]=1]]
>
> run it a bunch of times. Save g2 and A in file f2.
> start up a new system
>
> load files f1 and f2.
>
> the behavior of g1 and g2 will be, so far as I can tell, dependent on
> the (random?) setting of some inaccessible counter in Mathematica.
>

This is very good, but still I think it falls short of the quality of an =
earlier example of why Richard would "shy away from a presentation that =
required the reader to own a free Mathematica player". I mean that he =
might want to write "a paper whose topic was the Mathematica Player =
itself".

I think if not exactly a world record, that post was a record of sorts =
for this forum. This one, while still pretty good, is not really keeping =
up with that standard.

I do hope some day instead of warning us of the possible problems that =
might be caused by his imaginative though somewhat contrived examples of =
"possible" user behaviour, Richard tries to produce an actual, "real =
life" example. Perhaps he could start by trying to generate this =
"possible conflict" by following the procedure he has so kindly produced =
as a warning to all of us. With luck this will keep him occupied for a =
while...

Andrzej Kozlowski=

Richard Fateman

Jan 28, 2010, 2:43:23â€¯AM1/28/10
to
Andrzej Kozlowski wrote:
.
>>
>> (RJF) the behavior of g1 and g2 will be, so far as I can tell, dependent on

>> the (random?) setting of some inaccessible counter in Mathematica.
....

In private mail, I was told that this counter is not inaccessible, but
has the name \$ModuleNumber. This still does not correct the problem,
and the rest of my comment holds.

> I do hope some day instead of warning us of the possible problems

Design errors cause possible problems. These are identified by people
who then try to correct them. Apparently AK would prefer that programs
be run (say as a control system for a laboratory experiment? a medical
treatment with x-rays? Safety for a nuclear power plant? Aircraft
control?) and then only when an actual "real-life" example causes
problems should we go back and correct the design.

After all, what are the chances that a pseudo-random 32-bit number
generator would hit a previously specified secret number? Pretty
unlikely, right? Oh one chance in 2^32 or so. How long would it take a
modern computer to run through 2^32 random numbers? Maybe less than a
minute That's one of the nice things about computers. They do
arithmetic rapidly.

that

> might be caused by his imaginative though somewhat contrived examples of

> "possible" user behaviour, Richard tries to produce an actual, "real

> life" example. Perhaps he could start by trying to generate this

> "possible conflict" by following the procedure he has so kindly produced

> as a warning to all of us. With luck this will keep him occupied for a

> while...
>
> Andrzej Kozlowski
>

Andrzej Kozlowski

Jan 28, 2010, 2:43:45â€¯AM1/28/10
to

On 27 Jan 2010, at 19:14, Richard Fateman wrote:

> Apparently AK would prefer that programs be run (say as a control
system for a laboratory experiment? a medical treatment with x-rays?
Safety for a nuclear power plant? Aircraft control?) and then only when
an actual "real-life" example causes problems should we go back and
correct the design.

Sigh... I guess I should be flattered at the influence that I obviously
threatens mankind with untold disasters unless stopped by RJF's valiant
efforts to which he devotes so much of his time (payed for by California
tax payers?) but it's all in vain - we are all doomed anyway. The worst
Mathematica can do is speed it up a little.

So while the going is good why be so depressing?
How about letting us hear some more "cheerful facts" as in:

I am the very pattern of a modern major-general :
I've information vegetable, animal, and mineral ;
I know the kings of England, and I quote the fights historical.
From Marathon to Waterloo, in order categorical ;
I'm very well acquainted, too, with matters mathematical
I understand equations, both the simple and quadratical ;
About binomial theorem I'm teeming with a lot of news
With many cheerful facts about the square of the hypotenuse ;
(Joyfully.) With many cheerful facts about the square of the
hypotenuse.

Of course you should replace major-general with "professor of computer
science" and make other suitable changes.

Andrzej Kozlowski

David Bailey

Jan 29, 2010, 7:44:36â€¯AM1/29/10
to

Clearly, Module should not be used to generate names that will be saved
and loaded into another Mathematica session. If you REALLY need several
people to independent people to generate such names without getting
clashes, it might be better to generate names based on AbsoluteTime and

To describe the Module mechanism as 'flawed' because you can't use it in
this way, seems totally unreasonable - akin to describing real
arithmetic as 'flawed' because it can't represent Sqrt[-1]!

One of the nice features of Module, is that the method of localising
names is totally explicit - so that it is easy to work out what is going
on is subtle cases.

David Bailey
http://www.dbaileyconsultancy.co.uk

Richard Fateman

Jan 29, 2010, 7:45:30â€¯AM1/29/10
to
Andrzej Kozlowski wrote:
>
>
> Sigh... I guess I should be flattered at the influence that I obviously

You could guess that, but my guess is that your sarcasm detector needs
recalibration.

... Gilbert and Sullivan ... quote from the Major-General Song..

>
> Of course you should replace major-general with "professor of computer
> science" and make other suitable changes.

Hm, I see Stephen Wolfram (perhaps played by David Ogden Stiers) and
especially Wolfram|Alpha in this. He has the accent, and the program
fits the description.

RJF

Richard Fateman

Jan 30, 2010, 7:13:48â€¯AM1/30/10
to
David Bailey wrote:
...

>
> To describe the Module mechanism as 'flawed' because you can't use it in
> this way, seems totally unreasonable - akin to describing real
> arithmetic as 'flawed' because it can't represent Sqrt[-1]!

Except that other programming languages provide the same facility
(lexical scope) while not having the same problem. That is, there is
the ordinary way of implementing lexical scope, not the Mathematica way.

>
> One of the nice features of Module, is that the method of localising
> names is totally explicit - so that it is easy to work out what is going
> on is subtle cases.

Mathematica's implementation is, I think, based on someone's reading of
an explanation of lexical scope. Perhaps, "You can think of L.S. as a
kind of renaming". But without reading the next sentence, which might
be "but of course it is not implemented that way".

I agree with your statement -- that looking at the strange renaming it
is possible to explain some phenomena that you might find puzzling. But
so many things having to do with scope in Mathematica are, at some
level, potentially puzzling, that this is a drop in the bucket. E.g.
when do names get bound during pattern matching? What is the real value
of something which is evaluated any number of times until it has no
remaining names, so that x=x+1 assigned
to x, 255+Hold[1+x], etc.

And proper lexical scope implementation is not so hard, I think. Even
given Mathematica semantics.

RJF

Noqsi

Jan 31, 2010, 5:58:19â€¯AM1/31/10
to
On Jan 30, 5:13 am, Richard Fateman <fate...@cs.berkeley.edu> wrote:
> David Bailey wrote:
>
> ...
>
>
>
> > To describe the Module mechanism as 'flawed' because you can't use it in
> > this way, seems totally unreasonable - akin to describing real
> > arithmetic as 'flawed' because it can't represent Sqrt[-1]!
>
> Except that other programming languages provide the same facility
> (lexical scope) while not having the same problem.

You like to "strain out gnats and swallow camels", don't you? Any
mechanism can be abused. In this case, inadvertent abuse seems very
unlikely, while Leonid has demonstrated productive use (very cool).
Therefore, the implementation your ideology demands would have a real
problem.

> That is, there is
> the ordinary way of implementing lexical scope, not the Mathematica way.

So? Mathematica does it better, enabling you to single out little bits
of the state and make them persistent. It's much like the
"continuations" of other languages, except it's:

1. Better targeted.

2. More transparent.

Continuations are an obfuscated, undisciplined, "shotgun" mechanism
compared to what Mathematica offers here (yes, I know, they are
beloved by CS sacerdotes).

>
>
>
> > One of the nice features of Module, is that the method of localising
> > names is totally explicit - so that it is easy to work out what is going
> > on is subtle cases.
>
> Mathematica's implementation is, I think, based on someone's reading of
> an explanation of lexical scope. Perhaps, "You can think of L.S. as a
> kind of renaming". But without reading the next sentence, which might
> be "but of course it is not implemented that way".

It doesn't match your ideology. I doubt many Mathematica users share
that ideology.

>
> I agree with your statement -- that looking at the strange renaming it
> is possible to explain some phenomena that you might find puzzling.

It's only strange because it doesn't match your ideology.

> But
> so many things having to do with scope in Mathematica are, at some
> level, potentially puzzling, that this is a drop in the bucket. E.g.
> when do names get bound during pattern matching? What is the real value
> of something which is evaluated any number of times until it has no
> remaining names, so that x=x+1 assigned
> to x, 255+Hold[1+x], etc.
>
> And proper lexical scope implementation is not so hard, I think. Even
> given Mathematica semantics.

What you call proper I call less capable. I may be using Leonid's
techniques in the future.

Martha Fateman

Feb 1, 2010, 6:09:55â€¯AM2/1/10
to
On Jan 31, 2:58 am, Noqsi <j...@noqsi.com> wrote:

>
> What you call proper I call less capable. I may be using Leonid's
> techniques in the future.

Less capable in that Mathematica is capable of errors that a correct
implementation could not commit.
I think the only "plus" is that the renaming allows one to use the
same debugging tools
some of the time.

"Continuations" which you think are too esoteric, is an underlying
implementation technique
that can, and has, been used to provide remarkable facilities, some of
which you probably would
like. Like resuming the computation of terms in a Taylor series when
you decide you need more.
Or resuming an iteration after exiting with an error indication. Or
implementing search with backtracking
in a trivial fashion. And other ideas that have clear applications in
computer algebra systems.

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