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Re: A mini-language for sequences (part 1)

by BrowserUk (Pope)
on Nov 05, 2004 at 19:21 UTC ( #405594=note: print w/replies, xml ) Need Help??


in reply to A mini-language for sequences (part 1)

Tom, this isn't critism (yet:). Your's is a long article that will take several passes to fully appreciate. But.. :)

I recognised this problem. Now comparing your code with my own solution which both arrive at the same results:

my @site1 = qw( AATKKM aatkkm ); my @site2 = qw( GGGGGG gggggg ); my %counts; seq_foreach_from_spec( [ \(@site1, @site2) ], sub { seq_foreach( seq_zip( ( map seq(split//), @_ ) ), sub { $counts{"@_"}++ } ) } ); print Dumper(\%counts), "\n";

And

my @site1 = qw( AATKKM aatkkm ); my @site2 = qw( GGGGGG gggggg ); my %counts; for my $site1 ( @site1 ){ for my $site2 ( @site2 ) { $counts{ substr( $site1, $_, 1 ) . substr( $site2, $_, 1 ) + }++ for 0 .. min( length( $site1 )-1, length( $site2 )-1 ) +; } } print Dumper \%counts;

The thing that struck me was that my perl version would be reasonably clear to most people with more than a passing familiarity with perl. Whereas your version...?

Apart from the need to go off looking up the purpose, and parameters of seq_foreach_from_spec(), seq_foreach() and seq_zip(). (Oh! and seq(). Nearly missed that tucked away in there). That "one line" construction? Phew!

I'm not adverse to terse or complicated code, but anonymous subs and user subs nested inside user subs; nested inside anonymous subs, inside a user sub along with an anon. array of array references.

When I encountered that line, the only way I could begin to makes sense of it was to deconstruct the one line into 3 for 4 using temporaries and it still left me cold? Maybe if I work through the stuff again it'll make it easier.

But given that it almost exactly the same number of lines/bytes of code; I doubt it is any quicker; certainly isn't clearer; what, besides novelty value and a cool implementation of FP, is the point?

Please don't misread that. It is a question, not a statement. I keep reading about FP, and have now installed 3 different FP langauges and downloaded 4 FP/langauge books trying to get what people are raving about.

So far, FP is interesting, and in some ways quite elegant, but it is also way over sold in some aspects. And I have to say, beyond encouraging me to continue to make full use of map, grep and List::Util where they fit, which I've taken some critisism for in the past, I don't quite get the idea of going further with FP in Perl?


Examine what is said, not who speaks.
"Efficiency is intelligent laziness." -David Dunham
"Think for yourself!" - Abigail
"Memory, processor, disk in that order on the hardware side. Algorithm, algorithm, algorithm on the code side." - tachyon

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Re^2: A mini-language for sequences (part 1)
by tmoertel (Chaplain) on Nov 05, 2004 at 21:46 UTC
    First, thanks for your comments and your questions.

    Regarding the general thrust of your questioning, which I believe can be summarized, Why do all this crazy FP stuff when the equivalent "normal" Perl code is perfectly reasonable and probably easier for most Perl programmers to understand? I have a couple of answers.

    First, don't take my examples as the poster children for functional programming. This is a meditation, and a tutorial-style one at that. The examples are slanted towards meditation – mind stretching calisthenics and all that monastic stuff – not practicality. The point of the examples is not to say, This is the way to do it, but rather, Meditate on this code. The journey is the reward. (Further, my code isn't truly functional. It's a Perl-functional hybrid that nobody ought to mistake for real functional programming. Consider it "functionally inspired.")

    Second, after studying the code and understanding the thinking behind it, that's when you can see the benefits. The benefits aren't so much visible in my meditation's examples as they are upon reflection on the spectrum of problems that can be solved using the exact same techniques. Even though the shape of the problems may change, the solutions stay the same. The traditional approach, in contrast, is effectively hard-coded for one shape.

    Consider the code snippets that you compared in your post. Both operate on two input arrays @site1 and @site2. That's the shape of this particular problem. But what if the problem isn't static? What if we're given a list of arrays to process? The shape changes. The code you wrote embodies the static structure of the 2-array problem. Add an array, and it breaks. The outer pair of for loops is a hard-coded solution for the 2-array case. The fp-inspired approach that I show is independent of the shape of the problem. Instead of passing in \(@site1,@site2) you can pass in any number of input arrays. And the same code works, as is.

    Now, in real life, we usually know the shapes of our problems ahead of time. So what does the shape-flexing capability buy us? It buys the ability to code at a higher level and, more importantly, to think at a higher level, which gets to the heart of your question below:

    The thing that struck me was that my perl version would be reasonably clear to most people with more than a passing familiarity with perl. Whereas your version...?

    The reason you perceive the fp-inspired code as hard to follow is because your brain hasn't "internalized" the meaning of products and zips and the rest of our new vocabulary. To you, these words probably seem like gratuitous functional jargon.

    But the words have meaning. When I saw the example problem as first posted by the Seeker of Perl Wisdom, I immediately thought, "Ah, he wants the count of each value from the zipped products." And the code immediately followed.

    That's the reward. To have the right words to describe the problem. To have more ways of stringing together ideas. To reduce the gulfs between problem and understanding, understanding and code.

    If this sounds like worthless babble, that's OK. I didn't appreciate functional programming until I had used it to solve real-world sized problems for over a year. Only then could I "feel" fp in my bones and appreciate its potential.

    So far, FP is interesting, and in some ways quite elegant, but it is also way over sold in some aspects.
    Have you read Why Functional Programming Matters? If not, read it.

    I love Perl Monks, but don't look at the fp examples on Perl Monks as your window into the fp world. Take a look at the stuff going on in the Haskell Communities Report. Read some of the papers from the ICFP proceedings. That's where you'll find the good stuff.

    ... I don't quite get the idea of going further with FP in Perl?
    Perl leaves much to be desired as a functional programming language. Not having currying, pattern matching, proper tail recursion, and real garbage collection are limiting, painful even. But Perl is much better for fp than most other mainstream languages. For that, I'm thankful.

    Is it worth the effort to use functional-programming ideas with Perl? Part of my motivation for this meditation was to explore that question. I wanted to take some common fp ideas and run with them in Perl. How far can we go? What's painful? What's beautiful?

    My belief is that fp is a useful and even powerful way to program in Perl. But Perl isn't Haskell or O'Caml, and some things are better off left to the old ways. I think the most practical approach, then, is a functionally inspired style for Perl.

    Thanks again for your thoughtful response to my meditation.

    Cheers,
    Tom

      Okay, the title was meant to be inflammatory. Don't read too much into it.

      Perl leaves much to be desired as a functional programming language. Not having currying, pattern matching, proper tail recursion, and real garbage collection are limiting, painful even.

      What frustrates me most about Perl (from a FP perspective, at least) is the distinction between builtins and user functions. If anyone can tell me how to get around that without writing sub {$_[0] + $_[1]} everywhere, I'd be most grateful.

      Update: I realize that this complaint is a bit vague. What I really want to do is take coderefs to builtins and operators and pass them to higher-order functions. To the best of my knowledge, this is not an easy thing to do.

      --
      Yours in pedantry,
      F o x t r o t U n i f o r m

      "Anything you put in comments is not tested and easily goes out of date." -- tye

        When Perl 6 is ready, you will be able to use:

        my $plus_op_subref = \&infix:+; my $not_op_subref = \&prefix:!; say $plus_op_subref.(3, 2); # 5

        not tested ;)

        Have you tried prototypes?

      The following are some (lightly edited :) initial reactions to your follow post.

      Note: Initial.

      There are some positive reactions lower down, but I thought that I would get these out of the way first in the hope that the later reactions would erase any bitter taste these might leave.

      1. I never said "this crazy FP stuff".
      2. I get the FP -v- FP-inspired difference.
      3. Phrases like "spectrum of problems", "shape of the problems", "shape-flexing capability", "code at a higher level" & "think at a higher level" all leave me...um...suspicious.

        What is he trying to conceal with flowery language?

        Put that down to having suffered the hyperbole of too many "new programming paradyms". (I realise FP isn't new.)

      4. "The reason you perceive the fp-inspired code as hard to follow is because your brain hasn't "internalized" the meaning of products and zips and the rest of our new vocabulary. To you, these words probably seem like gratuitous functional jargon."

        My question is: "Do I need this new vocabulary?".

      5. "But the words have meaning. When I saw the example problem as first posted by the Seeker of Perl Wisdom, I immediately thought, "Ah, he wants the count of each value from the zipped products." And the code immediately followed."

        Fair enough, but then you had to construct a fairly complicated library of intrastructure in order to be able to code the solution in terms of your vocabulary.

        When I saw the example problem, I thought: "He want's to iterate over the two arrays of strings and count the unique pairs of aligned characters".

        1. "...iterate over the two arrays..."

          Nested for loops.

        2. "...count the unique pairs...".

          Increment a hash.

        3. "...pairs of aligned characters."

          Two substrs and another for loop.

        The code I posted fell straight out of that. No need for any auxillary vocabulary. No need to code a bunch of extra infrastructure. Perl already has all the vocabulary and in-built infrastructure required.

      6. "That's the reward. To have the right words to describe the problem. To have more ways of stringing together ideas. To reduce the gulfs between problem and understanding, understanding and code." & "If this sounds like worthless babble, that's OK."

        {nods} Yes. It does a bit :)

      7. I didn't appreciate functional programming until I had used it to solve real-world sized problems for over a year. Only then could I "feel" fp in my bones and appreciate its potential.

        I've been trying to get into FP since I first starting reading articles on Lisp in Byte magazine circa. 1979/80. I currently have Common Lisp, Scheme, and most recently Haskell.

      8. Have you read Why Functional Programming Matters? If not, read it. Now."

        I pulled the PDF and tried to read it. After a little while it began to seem familiar. Maybe it appeared on Byte, or CoACM or something else I've read a long time ago? Maybe it was referenced from such an article and I tracked it down and tried to read it? Maybe it's just the code examples that seem familiar from some other FP tutorial I've read down the years? Whatever, the PDF sits open on my desktop and I will endevour to try and read it to the end, but I doubt that I made it that far last time either.

        The problem is that in common with many other FP (and, notoriously OO) tutorials, it selects very specifically limited and condusive (to their aims) examples. It then spends an inordinate amount of time concentrating on how to construct the solution--in it's own, very esoteric, terms. Your real world examples are infinitely better (IMO).

        But the single line that sums up my feelings of what I've seen of FP tutorials and advocacy comes from the Abstract on the page you linked to above.

        Since modularity is the key to successful programming, functional languages are vitally important to the real world.

        The problem with that statement is that it implies that the only way to achieve modularity, is through FP--which is clearly not the case.

      9. I love Perl Monks, but don't look at the fp examples on Perl Monks as your window into the fp world. Take a look at the stuff going on in the Haskell Communities Report. Read some of the papers from the ICFP proceedings. That's where you'll find the good stuff.

        As I mentioned, my FP exploration goes back a long way, and my research is definitly not confined to what I see on PM. My exploration of Haskell is very new (and did come about because of articles mentioning it here at PM), so thankyou for those links--they will give me many hours of good reading.

      I hope that doesn't all seem negative? It isn't meant to be. I'm continuing to enjoy reading your article and pursue various angles that lead from it. Thankyou for writing it and I look forward to Part 2.

      For me, the most significant thing that came out of your response to my initial post is:

      Instead of passing in \(@site1,@site2) you can pass in any number of input arrays. And the same code works, as is.

      Now that is a benefit that I can understand and internalise :).

      I did try to download your code and try this out, but there seem to be some bits missing to allow the examples to work? Perhaps in part 2. This is what I extracted from your OP: {**Code moved to bottom of post**} but I couldn't work out how to fix it up so that I could try out the example with 3 or more lists?

      Anyway, the concept of writing a generic solution to the nested loops problem is one that I've been trying to get to grips with for some time. The upshot of your article (for me) is that I've finally gotten around to putting some concerted effort into it, and I came up with a solution. It doesn't look much like FP in it's construction, but it does rely heavily of constructing anonymous subroutine iterators, and constructing lists of these. And combining these iterator functions to contruct a higher order iterator. So, having (probably) butchered the FP vocabulary to death :), this is what my solution looks like:

      #! perl -slw use strict; sub loops { my @iters = map{ my @list = ( @$_, undef ); sub { $list[ @list ] = shift @list || () }; } @_; my @rv = map{ $iters[ $_ ]() } 0 .. $#iters; return sub { my $rv = [ @rv ]; for my $i ( reverse 0 .. $#iters ) { $rv[ $i ] = $iters[ $i ]() and return $rv; $rv[ $i ] = $iters[ $i ](); } return; }; } my $iter = loops [ 'a' .. 'd' ], [ 1 .. 4 ], [ 'me', 'you' ]; print "@$_" while $_ = $iter->(); __END__ [13:51:21.14] P:\test>loops a 1 me a 1 you a 2 me a 2 you a 3 me a 3 you b 1 me b 1 you b 2 me b 2 you b 3 me b 3 you c 1 me c 1 you c 2 me c 2 you c 3 me

      I'm pretty pleased with that. Your article has been invaluable to me because it has prompted me to look at this problem in a completely different way to that in which I have been looking at it in the past. And it has triggered a lot of thoughts about using similar techniques to generalise several other pieces of code I have kicking around.

      So thankyou again for posting your very thought provoking article. I really look forward to part 2.

      I think the conclusion I'm drawing is that it's possible to learn and adopt techniques from other languages and styles of coding, without having to adopt the vocabularies and working practices wholesale. Once you understand (if I have?), the underlying techniques, it becomes possible to utilise them whilst retaining the flavour of the language in which your writing.


      The code I extracted from your OP. Moved to the bottom to avoid interupting the main post. (Wouldn;t it be nice if readmore tags only expanded when asked? Even if viewing the containing post directly).


      Examine what is said, not who speaks.
      "Efficiency is intelligent laziness." -David Dunham
      "Think for yourself!" - Abigail
      "Memory, processor, disk in that order on the hardware side. Algorithm, algorithm, algorithm on the code side." - tachyon
        First, thank you for your most-recent response. There are many interesting ideas within it, and I want to get to them. But before we do that, let's put our discussion within proper context.

        I must make clear that we have wandered onto dangerous ground. We are using my meditation as the testing ground for your conclusions about the merits functional programming, and it was never indented to be anything other than a meditation.

        To be blunt: If you're looking at my mediation as anything other than a meditation, you're making a mistake. My meditation is not meant to be practical. It is not meant to be advocacy for functional programming. It is a meditation: a discourse intended to express my focusing on one idea – sequences – and going with it as deep as I can.

        The point of the meditation is not to say that "this way is better." The point is to ignore earthly concerns that normally hold us back (such as practicality) and to keep moving forward with an idea, just to see where it takes us. Maybe we'll end up at a really cool place. Maybe not. Regardless, we hope to learn something from the journey, if not from the destination.

        At certain points along our journey you have asked, "Why did we walk this strange path to get to this spot? I know a road that most people would think is better."

        But we're not trying to find the best path just now. We're trying to walk a different path to see if we can learn anything. Instead of comparing our path to the road and dismissing it as inferior after the first few miles, can we not keep on walking, just to walk, and to see if we encounter anything new or valuable?

        In the end, maybe we will still prefer the road, but then again, maybe knowing the path will come in handy. Maybe some destinations will best be reached by traveling both road and path. Maybe the path will be a valuable addition to our options for future travel.

        To come at it practically, don't judge my examples. Judge the ideas the examples represent. But only later, after due consideration.

        Now, on to specific items.

        My question is: "Do I need this new vocabulary?"
        Whether you need any new vocabulary depends on the vocabulary and on the problems you are trying to solve. If the cost of learning the vocabulary and then using the vocabulary to solve your problems is less than the cost of solving your problems otherwise, then you probably ought to use the vocabulary.

        Do you need my meditation's sequence vocabulary? Most likely, no. It exists only for this meditation. If I were to create a CPAN module from it, as some have suggested, the resulting library would look and be organized much differently than presented here. The goals of meditation and of practical programming are far apart.

        I would, however, argue that much of the vocabulary of ideas and techniques within the meditation are useful and could earn its place in most personal programming toolboxes.

        But the single line that sums up my feelings of what I've seen of FP tutorials and advocacy comes from the Abstract on the page you linked to above.
        Since modularity is the key to successful programming, functional languages are vitally important to the real world.
        The problem with that statement is that it implies that the only way to achieve modularity, is through FP--which is clearly not the case.
        I don't think that Hughes implies that FP is the only way to achieve modularity. Rather, he argues that FP (especially modern FP) reduces the cost of modularity to the point where it can be used, and its benefits reaped, almost ubiquitously – not just at object or function boundaries but at much finer granularities.

        In most languages, the smallest unit of modularity is the object or function. But in Haskell, for example, I can go much finer. I can slice function calls in half, turn operators into functions and functions into operators, flip arguments around, and do a whole lot more to reshape code until it fits my needs. The potential for code reuse goes up dramatically because I can modularize on a much finer granularity than before; the artificial distinctions that other languages impose between language and library cease to exist. Libraries aren't merely collections of functions but real vocabularies that extend the language.

        I've been trying to get into FP since I first starting reading articles on Lisp in Byte magazine circa. 1979/80. I currently have Common Lisp, Scheme, and most recently Haskell.
        To help me understand and quantify where you're coming from, how many hours would you conservatively estimate that you have spent writing functional code in the last year?
        The problem is that in common with many other FP (and, notoriously OO) tutorials, ("Why Functional Programming Matters") selects very specifically limited and condusive (to their aims) examples. It then spends an inordinate amount of time concentrating on how to construct the solution--in it's own, very esoteric, terms. Your real world examples are infinitely better (IMO).
        The problem for people who write about FP to non-FP audiences is that the introductory slant of their writing prevents the use of complex example problems, where FP's merits are readily visible. Anybody can solve simple problems in any language, and so readers understandably wonder why FP is so great when their "normal" solutions to the same example problems would arguably be better.

        For this reason, some FP authors will pick simple example problems that just happen to be hard to solve in non-FP languages. While the intent is to keep the problems simple enough that readers can follow them and yet "hard" enough to show the merits of FP, many readers conclude that the examples are contrived or even rigged to put FP in an artificially glowing light.

        Sadly, this is a case when the limitations of the medium undo the message.

        FP languages let you manipulate pieces of programming logic as easily as most languages let you manipulate data. As a result, it is easy to create FP programs that morph their shape to match the structure of the problems they solve. Thus the straightforward FP solution is often general enough to scale from the simplest to the most complex problems within a wide spectrum of related problems.

        This is deeply cool magic, but you won't appreciate it until you experience it. And you won't experience it in introduction-to-FP papers. You can't put examples like that into a paper. The best way, then, to read the papers is to study the examples carefully, understand the thinking behind them, and then meditate on how far that thinking will take you. Often, it will take you surprisingly far.

        I hope that doesn't all seem negative? It isn't meant to be. I'm continuing to enjoy reading your article and pursue various angles that lead from it.
        No, I don't take it as negative. But I do see a slight danger of your drawing conclusions about FP without due consideration. Be deliberate about forcing your mind to remain wide open while you invest serious time in studying and coding functional programs.

        The best advice I can give you is to force yourself to write functional code, the more the better. If you're already looking at Haskell, that's your best bet. (There is no finer modern FP language, IMHO.) Get a copy of Paul Hudak's The Haskell School of Expression: Learning Functional Programming through Multimedia and work through it. Solve the problems. Don't skip the proofs.

        For me, the most significant thing that came out of your response to my initial post is:
        Instead of passing in \(@site1,@site2) you can pass in any number of input arrays. And the same code works, as is.
        Now that is a benefit that I can understand and internalise :). I did try to download your code and try this out, but there seem to be some bits missing to allow the examples to work?
        I think the problem is that you commented out the OO portion of my code but forgot to recode seqsub accordingly. Add the following to your code (or uncomment the OO portion that you commented out), and you should be good to go:
        sub seqsub(&) { $_[0] }
        Here's our example solution put into a subroutine so that we can call it repeatedly with differently "shaped" inputs:
        sub count_zipped_combo_vals { my %counts; seq_foreach_from_spec( \@_, sub { seq_foreach( seq_zip( ( map seq(split//), @_ ) ), sub { $counts{"@_"}++ } ) }); \%counts; }
        Now we can see how will it handles problems of different shapes. We'll try 1-, 2-, and 3-array cases, but the same code should work for any number of arrays greater than zero.
        # a few problem cases to try my @site1 = qw( AATKKM aatkkm ); my @site2 = qw( GGGGGG gggggg ); my @site3 = qw( XXXXXX ++++++ yyyyyy ); # ... and so on ... my @sites = \( @site1, @site2, @site3, # ... and so on ... ); # try 1-, 2-, and 3- array cases print Dumper( count_zipped_combo_vals( @sites[0..$_] ) ) for 0 .. $#sites; # $VAR1 = { # 'A' => 2, # 'k' => 2, # 'a' => 2, # 'M' => 1, # 'T' => 1, # 'K' => 2, # 'm' => 1, # 't' => 1 # }; # $VAR1 = { # 'K G' => 2, # 'A G' => 2, # 'm g' => 1, # 'a g' => 2, # 'A g' => 2, # 'M G' => 1, # 'k g' => 2, # 'k G' => 2, # 'T G' => 1, # 'a G' => 2, # 'm G' => 1, # 't G' => 1, # 'K g' => 2, # 'M g' => 1, # 't g' => 1, # 'T g' => 1 # }; # $VAR1 = { # 't G y' => 1, # 'T G +' => 1, # 'K G X' => 2, # 'a G X' => 2, # 'm G +' => 1, # 'a G +' => 2, # 'T g y' => 1, # 'm g +' => 1, # 'a g +' => 2, # 'k G X' => 2, # 'a g X' => 2, # 'm g X' => 1, # 'T g X' => 1, # 't G +' => 1, # 'M G y' => 1, # 'k g y' => 2, # 't g y' => 1, # 'm G X' => 1, # 't G X' => 1, # 'K G +' => 2, # 't g +' => 1, # 'a g y' => 2, # 'T G X' => 1, # 'm g y' => 1, # 'm G y' => 1, # 'A G X' => 2, # 'M g +' => 1, # 'k g +' => 2, # 'k g X' => 2, # 'A g X' => 2, # 'A g +' => 2, # 'M g y' => 1, # 'k G +' => 2, # 'a G y' => 2, # 'M G +' => 1, # 'A g y' => 2, # 'K g X' => 2, # 'K G y' => 2, # 'K g y' => 2, # 'K g +' => 2, # 'k G y' => 2, # 'A G y' => 2, # 'T g +' => 1, # 't g X' => 1, # 'M G X' => 1, # 'M g X' => 1, # 'T G y' => 1, # 'A G +' => 2 # };
        Thanks again for taking the time to consider our discussion. I hope that in the end your will find the journey to be the reward.

        Cheers,
        Tom

Re^2: A mini-language for sequences (part 1)
by dragonchild (Archbishop) on Nov 05, 2004 at 21:27 UTC
    I think the gain comes from two things:
    1. You can take an intermediate form of seq_foreach_from_spec() and pass it around, similar to lexical filehandles. (Remember how cool that was when 5.6 added it?)
    2. You can extend it a lot more easily than for-loops. In the trivial case, which is what you're describing, it's very easy to see how the for-for version is easier to handle than the FP version. What if you're working 5-10 nested loops? What if you're working a loop with a bunch of double-nested pairs, all of which are similar?

    Let's say that you can create an intermediate representation that allows you to pass in N lists and do XYZ to it. What XYZ does is irrelevant, except to say that you need it done to more than one group of lists. Think about it as templates for algorithms and I think you'll see the gain.

    Being right, does not endow the right to be rude; politeness costs nothing.
    Being unknowing, is not the same as being stupid.
    Expressing a contrary opinion, whether to the individual or the group, is more often a sign of deeper thought than of cantankerous belligerence.
    Do not mistake your goals as the only goals; your opinion as the only opinion; your confidence as correctness. Saying you know better is not the same as explaining you know better.

Re^2: A mini-language for sequences (part 1)
by tmoertel (Chaplain) on Nov 06, 2004 at 16:03 UTC
    One thing I forgot to point out in my previous response is that the example you cite, in particular, is crippled by the small size of our vocabulary (to date). Had we the space to discuss concatmap, the following implementation would have been available:
    seq_from_spec( \(@site1, @site2) ) ->seq_concatmap( sub { seq_zip(map seq(split//), @_) } ) ->seq_foreach( sub { $counts{"@_"}++ } );
    This implementation is much clearer (assuming you know concatmap).

    Concatmap combines concat and map; it maps a sequence to a sequence of sequences, whose values are then concatenated to yield the output sequence. For example:

    sub upto { 1 .. $_[0] } sub supto { seq(upto(@_)); } seq(1..3) -> seq_map(\&upto) -> enumerate; # 0 => 1 # 1 => 1 2 # 2 => 1 2 3 seq(1..3) -> seq_map(\&supto) -> seq_concat -> enumerate; # 0 => 1 # 1 => 1 # 2 => 2 # 3 => 1 # 4 => 2 # 5 => 3 seq(1..3) -> seq_concatmap(\&supto) -> enumerate; # 0 => 1 # 1 => 1 # 2 => 2 # 3 => 1 # 4 => 2 # 5 => 3

    Cheers,
    Tom

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