The difference between imperative and functional programming

Have you ever had this frustrating experience?  You need to drive somewhere that you’ve not been before, so you look that place up on the web.  You find its site, and start looking through the pages for the address and postcode.  Instead, you find this kind of thing:

To get to us from the south/east (for example, if driving from London), the easiest way is to get onto the A4136 heading southwest, then turn right onto Morse Road which takes you straight to Ruardean.  Our house is on the right as you approach from this direction, almost immediately after the 30-mile-per-hour speed-limit sign that marks the boundary of the village.

You didn’t want instructions on how to find the place.  You just wanted to know what the address is: you know how to reach a place with a known address, because you have Google Maps and a satellite navigation system and a helpful wife who will shout “Left!  Left!  No, LEFT!  Why didn’t you turn LEFT?” when required.

The postcode GL17 9XF tells you what the destination is; the series of instructions on the unhelpful web-site tell you how to reach the destination.

And this seems to be the core of the difference between imperative programming (which is what most of us do most of the time) and functional programming (which is what exotic academic types like to do in their ivory towers, never actually getting anything done but writing a lot of papers).

Except, wait, didn’t we just agree that functional addresses are better than imperative ones?  They give you more freedom in figuring out how to implement them, and they are not sensitive to global state such as whether you happen to be coming from London in the first place.  What if you’re coming from Cardiff?  If you follow the imperative address (and you are of a literal turn of mind) you’ll drive to London, then follow the directions quoted above(*), which is hardly optimal.

So could it be that functional programming is better than imperative programming after all?

What does a functional program look like?  At the risk of perpetuating a cliché, consider the toy problem of factorials.  (I nearly said “the problem of calculating factorials, but even saying it that way presupposes an imperative approach to solving the problem.)  A factorial is the product of a specified integer and all smaller positive integers, so that factorial(5) is 1x2x3x4x5 = 120.  And the obvious way to write a function for that is of course:

def factorial(x)
res = 1
while (x > 1)
res = res*x
x = x-1

This is nice, simple code, and hard to get wrong.  (Although as a matter of full disclosure, I guess I should admit that I did get it wrong when I typed it in just now for this article: I forgot the x = x – 1 and got into an infinite loop.  Oops.)

That’s an imperative program, which tells you how to calculate a factorial.  But a functional program just says what a factorial is — the number you first thought of times the factorial of the number one less than that.  (I’m using Ruby as a sort of executable pseudocode for both versions of the factorial function so that language-wars  don’t obscure the conceptual point.)

def factorial(x)
if x < 2

And this version also works just fine.  Of course I cheated a bit, by not saying up front how this version avoids getting into an infinite loop of its own.  The answer is that the factorial of 1 is just 1 — there’s no need to go figure out the factorial of one-less-than-one.

Which version is better?  That’s a matter of judgement and, yes, opinion.  Leaving aside all performance considerations (which we probably don’t understand as well as we think we do anyway), and thinking only about comprehensibility, which is clearest?  I really couldn’t say.

But of course this function is so short and simple anyway that both versions are bound to be easy to read and understand.  The real question is: which approach scales better to realistic-sized problems? And the answer to that seems to depend as much on the temperament of the individual as it does on the intrinsic merits of one approach of the other.

Like a lot of programmers, I was raised imperative but have been repeatedly drawn to functional — often seeing the appeal, but never yet able to make the leap into actually doing anything in a functional style beyond artificial exercises.  It’s time that changed: I am going to make a serious attempt at Lisp, the canonical functional language, and I’m going to blog it as I go.

Let me be clear: I am not setting myself up to teach anyone anything about this subject, since I know close to nothing about it myself.  I’m just hoping my notes might be useful to fellow travellers, if only so that they can avoid making the same dumb mistakes as I do.  And those of you who are further along the road than I am, please do shout when I wander off the path!

We start next week!

(*) A favourite joke of mine (not original to me, but I don’t remember where it’s from).  The university’s Health and Safety officer is checking that the faculty know what to do in the event of a fire.  He asks an engineer, “What would you do if you were locked in the laboratory and a fire broke out?”  The engineer thinks for a moment and says “The centrifuge in the lab is pretty heavy: I’d use that to smash the door down and make my escape”.  “Very good”, says the H&S guy, and turns to a mathematician.  “And what would you do”, he asks, “if you were up on the roof when a fire broke out?”  Quick as a flash, she replies “I’d lock myself in the engineering lab, thus reducing the problem to one that has already been solved.”

Thank you, thank you, I’m here all week.  Don’t forget to tip your waitress.

*** Special bonus EoPS extract ***

As I’ve continued my re-reading of The Elements of Programming Style, I ran across this old favourite from page 70, which I thought you might enjoy:

“Finally, the two inner loops (on K2) are handled differently even though they perform similar functions.  This tips us off that one of them is incorrect.  (As a matter of fact both are, but the details are not worth pursuing.)”

This, by the way, in the section illustrating the rule “Don’t patch bad code — rewrite it.


29 responses to “The difference between imperative and functional programming

  1. The school where I received my undergraduate CS degree a decade ago has since started teaching their introductory CS course in Scheme. I think the CS department takes the opinion that if you can’t handle thinking in a functional language, you shouldn’t be taking CS, and you’d better find that out as soon as possible.

    Great blog by the way; picked it up with a link to the “whatever happened to programming” post and I’m definitely enjoying it. Look forward to your Lisp experiences.

  2. Thanks, KR. I guess I sort of agree with your school’s approach — it reminds me of Joel Spolsky’s two-point-test for potential hires, which is: can they cope with recursion and pointers?

    I nearly didn’t post this article because I am kind of ashamed of never having got properly to grips with functional programming after all this time. But I figured that if I made a public commitment to remedy that deficiency, I’d be more likely to stick to it :-) And hopefully, some of this blog’s readers will be inspired to do the same.

    At the very least, coming to terms with Lisp should make me a better Ruby programmer.

  3. I have been fighting myself over imperative and functional programming for a long time. But still looking at the two code snippets you have shown; I still find the first one more natural. Recursion in particular is like a paranormal entity to me :)

  4. Lisp is terribly unpopular. Perhaps you should try Phosphorous – a Popular Lisp:

    Click to access phosphorous.pdf

  5. I applaud your project. It’s something I’ve wanted to do for a long time, so I’ll follow your efforts with great interest. I’ve done many toy projects in Scheme but have never done “realistic” projects. I’ve also played around with Ruby and Clojure.

    One reason I am particularly interested in learning functional languages is because I think they provide the best way forward for handling concurrent applications with thread safety and performance. As processors add more cores, concurrency gets more important. It’s nearly impossible to write code that is both safe and scalable with OO languages when the number of threads gets large.

    Good luck!

  6. Xiong Chiamiov

    Most of the functional weenies I know swear by Haskell.

  7. Yeah, I know that Haskell is the new kid on the block; but Lisp is canonical, and I want to be in a position where no-one can tell me “You say you’ve learned functional but you don’t know Lisp!” OCaml is another trendy alternative which I’d like to learn one day, but Lisp is just The Functional Language, isn’t it?

    (Also, I am interested in the ability to get Lisp programs to read and write other Lisp programs due to their lack of syntax.)

  8. Go with Lisp. It’s plasticity actually makes it easier to write algorithms that take advanage of multiple processors. The Thinking Machine, a SIMD machine with 8,192 processors, was programmed in an extension to Lisp that borrowed ideas from APL and the hardware interprocessor architecture.

  9. I applaud your project too!
    I have also learn Ocaml for the same reason.
    I have always programmed in an imperative style and I was just curious about another way to “think”.
    So, I have chosen two application target(s):
    * Brute force, validation of Fermat’s theorem (bignum…)
    * network application(multithread), port scanning, which looks for kademlia nodes.

    I have wrote these application twice.. the first time while learning, the second time when I was used to functional style.
    It has been a funny experience, after a few weeks thinking: “This guys are simply crazy, spending their time in useless jokes” … I have started writing program in different ways.
    I have also noted I was able to produce code at impressive (high) speed.

    My last note:
    I have acquired a “trasversal” way to look at algorithm’s implementation and this is useful when coming back to imperative programming but I have also understood that functional programming is (for me) not the natural way to solve problems.
    Time ago I have had few months without programming but after coming back to the keyboard I was able to program in few seconds(imperative).
    Now, thinking about functional style programming, I’m sure I need a full day in order to get back to functional programming.
    Yes this coud be related I was raised in imperative paradigm, but for me functional still sounds a strange way to program.(quick nice and elegant)

  10. If it boils down to teaching a course, I would pick Haskell as the language for the course.

    However if you want to stay with lisp, that’s allright to.

    However the comment “You say you’ve learned functional but you don’t know Lisp!” is something most people that did do their homework on FP will plainly laugh at you and tell you you’re not serious in your attempt at FP.

    By the way Erlang is a cool functional language too if you want to work with distributed systems.

  11. Xiong Chiamiov

    We used Scheme for my programming languages course, and it worked really well for making a lispy language, because it has native parsers.

    Making a non-lispy language is actually much more difficult.

  12. Andrew Raybould

    Funny you should raise the credibility issue; there was a time I used Common Lisp extensively, and now I feel I should dip back in to find out what modern functional languages are like. I’m thinking of Haskell, OCaml or Erlang, with a bias towards the last because of its concurrency support and extensive ‘real-world’ use.

    On the other hand, I think Lisp will give you a sound basis in functional style. Just be aware that most, if not all, Lisp dialects support an imperative programming style as well as functional. I’m no fundamentalist in this regard, but as you want to experience the latter, watch out that you don’t slip into a Basic style of programming!

  13. Hmmph, well, er, hmmm. Having spent the last two summers wrestling with XSL and overflowing stacks and for-next loops that take 12 lines of code I’m still unconvinced of the practical applications of functional languages over imperative ones.

    On the other hand, I’m no genius, but at least I’m open-minded enough to keep a lookout for applications that could benefit from a functional approach. No luck there yet, however.

    Having an average brain is, I think, what makes me an OK programmer – my code is simple because my cave-man mind needs to be able to understand it.

    My problem is not with functional languages per-se, but the difficulty I constantly run into when it takes me several hours to implement what seems to be to be a simple enhancement to an existing functional program.

    Of course it could just be that my brain doesn’t, uh, function properly.

    At any rate, I have one specific thought to share; how many large applications (with a user interface, please) can we name that are written in any functional language? Isn’t this an indication of something? I’m not talking about a web-app here but a real end-user application such as iTunes or Word. How about any games written in functional languages? If there are no (commercial) examples then there must be a reason for this, right? Right?

    Probably I’m just missing something really obvious here…

  14. Chris,

    These questions bother me, too.

    But, at least, I don’t think we should use the extreme horribleness of XSLT as a reason to write off the entire functional paradigm. It is truly difficult to invent a more ugly, inconvenient and error-prone syntax for anything, and that surely conceals whatever intrinsic beauty the underlying XSLT model might have. (Why the world ever switched from the elegant Lisp-like syntax of DSSSL to something as foul as XSLT has always astounded me: it seems to be simply a manifestation of the Angle Brackets Good fever that took over the world a decade ago.)

  15. Mark Arvieux

    Now it’s XML:

    <title>Empire Burlesque</title>
    <artist>Bob Dylan</artist>

    Now it’s Lisp:

    '(title '(Empire Burlesque))
    '(artist '(Bob Dylan))
    '(country USA)
    '(company Columbia)
    '(price 10.90)
    '(year 1985)

    This is easy!

  16. Mark, you don’t need all those quote marks.

    ‘(catalog (cd (title (Empire Burlesque)) (artist (Bob Dylan)) (country (USA)) … ))

    If you want to execute code in there, use ` instead: this was the sort of thing backquote was *made* for.

    (I wrote a Lisp->XML translator *long* ago, in Lisp macros of course. It makes writing XML not quite so much like stabbing your eyes out with a toasting fork.)

    Oh, btw, DSSSL wasn’t just ‘Lisp-like syntax’: it *was* Lisp, or rather a dialect of Scheme. I wish the world never switched, but, hey, it could be worse. We could be using XExpr to write our shell scripts. (The standard for it — a dead draft — is notable for being the only standard I’ve ever seen to include syntax errors in virtually every one of its examples. If a language is too ugly for even its inventors to get right, it’s a bad language.)

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  18. Hey, great blog, just wanted to make a few points:

    – Lisp itself doesn’t necessarily imply functional programming. Common Lisp users especially dislike the label. The dialect Scheme tends to be very functional, so if you decide to go Lispy for your functional fix I would suggest you take a look at that one.

    An especially helpful set of books for this are The Little Schemer and The Seasoned Schemer. They start with simple exercises to get you used to thinking recursively, but by the end you will have had a lovely brainmelt ^_^

    A good, easy-to-setup Scheme implementation is PLT Scheme, soon renamed PLT Racket.

    – Regarding Chris’ post with the questions that bothered you, two points. First, I’m curious which language left him with stack overflows (all functional languages implement tail call elimination, and it’s one of Scheme’s only features).

    And second, I’m wary of the “how many large applications have been built in X?” argument as a basis for discrediting X. There was a time when the best medical treatment available was a good bleeding; to look at something new and note that it hasn’t immediately overtaken the establishment, and inferring that it’s lacking is shortsighted. This field is very new, especially the point at which we citizens can have computers of our own.

    Further, it depends on what you mean by large software. Darcs, XMonad, and the Scheme used in Naughty Dog’s video games are contenders, I would argue ^_^

  19. Thanks, Paul, good points. I did intend to get The Little Schemer, but that approach was overtaken by events, which I’ll write about in a forthcoming post. It looks like I will be using PLT Scheme/Racket, though.

    I think that, for a paradigm that’s been around since 1958, “How many large applications have been built using it?” is a fair question to ask. For all I know, the answer is “lots”; the point is that I don’t know, and if functional advocates out there can point me at some examples, I’ll be the wiser for it.

  20. A late comment – it strikes me that a lot of the problem with functional programming is similar to your comment in the article about closures – at times it seems that essentially simple concepts are made harder to understand by choice of terminology.

    (I can understand why – it’s an area that is still closer to the mathematical roots of computer science, so the terminology tends to be more academic and less colloquial).

  21. Because all those advantages are also disadvantages.

    Stateless programs; No side effects

    Real-world programs are all about side effects and mutation. When the user presses a button it’s because they want something to happen. When they type in something, they want that state to replace whatever state used to be there. When Jane Smith in accounting gets married and changes her name to Jane Jones, the database backing the business process that prints her paycheque had better be all about handling that sort of mutation. When you fire the machine gun at the alien, most people do not mentally model that as the construction of a new alien with fewer hit points; they model that as a mutation of an existing alien’s properties.

    When the programming language concepts fundamentally work against the domain being modelled, it’s hard to justify using that language.

    Concurrency; Plays extremely nice with the rising multi-core technology

    The problem is just pushed around. With immutable data structures you have cheap thread safety at the cost of possibly working with stale data. With mutable data structures you have the benefit of always working on fresh data at the cost of having to write complicated logic to keep the data consistent. It’s not like one of those is obviously better than the other.

    Programs are usually shorter and in some cases easier to read

    Except in the cases where they are longer and harder to read. Learning how to read programs written in a functional style is a difficult skill; people seem to be much better at conceiving of programs as a series of steps to be followed, like a recipe, rather than as a series of calculations to be carried out.

    Productivity goes up (example: Erlang)

    Productivity has to go up a lot in order to justify the massive expense of hiring programmers who know how to program in a functional style.

    And remember, you don’t want to throw away a working system; most programmers are not building new systems from scratch, but rather maintaining existing systems, most of which were built in non-functional languages. Imagine trying to justify that to shareholders. Why did you scrap your existing working payroll system to build a new one at the cost of millions of dollars? “Because functional programming is awesome” is unlikely to delight the shareholders.

    Imperative programming is a very old paradigm (as far as I know) and possibly not suitable for the 21th century

    Functional programming is very old too. I don’t see how the age of the concept is relevant.

    Don’t get me wrong. I love functional programming, I joined this team because I wanted to help bring concepts from functional programming into C#, and I think that programming in an immutable style is the way of the future. But there are enormous costs to programming in a functional style that can’t simply be wished away. The shift towards a more functional style is going to happen slowly and gradually over a period of decades. And that’s what it will be: a shift towards a more functional style, not a wholesale embracing of the purity and beauty of Haskell and the abandoning of C++.

    I build compilers for a living and we are definitely embracing a functional style for the next generation of compiler tools. That’s because functional programming is fundamentally a good match for the sorts of problems we face. Our problems are all about taking in raw information — strings and metadata — and transforming them into different strings and metadata. In situations where mutations occur, like someone is typing in the IDE, the problem space inherently lends itself to functional techniques such as incrementally rebuilding only the portions of the tree that changed. Many domains do not have these nice properties that make them obviously amenable to a functional style.

  22. Thanks, babor, you make some excellent points here, not least that most programming is maintenance of existing (imperative) systems rather than building new (functional or imperative) ones.

    I have a post lined up on what I call side-effect schizophrenia: that it’s easier to reason about code that doesn’t have side-effects, but ultimately the only reason we run programs at all is for the side-effects. (Dammit, now I just summarised the entire post in 24 words — is there any point in writing the post?)

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  26. > That’s an imperative program, which tells you *how to* calculate a factorial. But a functional program just says what a factorial *is*

    I disagree with this statement. I know a lot of people like to present functional programming that way, but in all functional languages I’m aware of (except maybe Haskell, not sure) there’s a reasonably strict definition saying which function calls and side effects happen in which order. The program is very clearly describing a process, a *how*. It may be describing it in a slightly more declarative way, but it’s still a *how*.

    I’m currently writing 70% functional, 10% OO, 20% procedural code in Scala, which has prompted me to think about what the essence of functional code is. Here’s my current best attempt at articulating it: your code is written in a functional style if and to the degree each snippet of code (e.g. each function) doesn’t refer to any part of the rest of the program unless it’s passed in as an argument, and it doesn’t impact the future execution of the program except via its return value. Note that this permits mutation within a function. I guess your code is more functional-style if it’s functional-style at more levels of zoom (The imperative factorial is functional at the function level but not at the statement level), but the real gain lies in being able to read a function and understand it fully simply by understanding the range of arguments to it and knowing what it’s supposed to do.

    Being able to work concisely with simple immutable data (preferably with good support for sum types and pattern matching) facilitates writing in a functional style: then you can’t write back into the arguments, so you’re prevented from being non-functional in this particular way.

    Having an expressive type system, in particular with good support for sum types, lets you represent failures as return values rather than thrown exceptions. If you can further shape the input types so as to make invalid states impossible to represent, you prevent a source of failures.

    If you construct your functions such that arguments and return values tell the whole story, a cognitive will be lifted from you: you don’t need to keep track of what state the program is in outside your function, or how it got there; you only need to think of your one particular function. I think that’s real benefit of functional programming.

    This implies that you can write e.g. C and Java in both a more functional style and a less functional style.

    I might want to relax the definition of functional style a little: a snippet can also be considered to be in a functional style if it only impacts the future execution of the program in ways that are explicitly declared (and preferably checked by your compiler, at least if you share my preferences). With this relaxation, passing a mutable reference around in Rust is still functional [I’m being generous here], with the caveat that at the point where control returns to the owner of the mutable-reference object, that owning function/method doesn’t know what state the object is in. That is, *some* knowledge has been destroyed but not all; much ease of reading and understanding is still maintained. [Again, more functional means functional at more levels of zoom, so it’s more functional to pass immutable references. But performance is also a thing; clarity and low-overhead readability are not the only concerns.]

    TL;DR: functional means the impacts of a function on the rest of the program and vice versa are explicit and (preferably, per your best approximation) compiler-checked.

  27. Well, Jonas, I don’t much disagree with any part of what you write here, except that you seem in the end to more or less define “functional programming” as “good programming”. I think you’d struggle to find anyone who doesn’t agree that elements of a program should, so far as possible, be free of side-effects. But it’s also obviously the case that, except when simply performing a calculation, side-effects are what we run a program for — whether that’s updating a value in a database, moving Mario around the screen, or ordering groceries. The trick is to constrain where and how such side-effects happen; and maybe to define them away by saying that when they done in this predictable way, they’re not side-effects, they’re effects.

    (Anyway, the point of my definition was to sum up the imperative/functional distinction in 20 words. Of course you can build a more comprehensive description if you’re prepared to stretch to 450 words, but that’s rather a different goal.)

  28. Excellent reply.

    Scrolling up, I see I wrote “a cognitive will be lifted”. I meant to say “burden” before “will”. Oops.

    > [We run programs for their effects]
    I agree, as does Simon Peyton-Jonas, Mr. Haskell himself (

    > [Not side-effects, just effects]
    I’m trying to adopt the label “external effects”, meaning external to the program that sets them in motion. Writing to disk, reading from the keyboard and mouse, etc.; this is distinct from “internal effects” which basically means mutating memory you will read in the future, and whatever else an OS process can do on its own.

    > [constrain where, define them away]
    I think we can reasonably divide program activities into four kinds: reading data from input devices, moving data around, computing new data from preexisting data, and using data to drive output devices.

    There’s a design structure/pattern called “functional core, imperative shell”. One way of describing it is (I think) “separate the computation from all the other parts”. The input and output parts are inherently unable to be functional, but the computation can be. The motivation for this structure is to maximize the benefits you can get out of writing in a functional style.

    Another way of saying the same thing, with a different framing, is “separate the I/O from all the other parts”. Often the I/O layer can be made very thin and simple, especially if you can introduce a simple data model that describes how to drive it, plus a small layer which executes this model. (Which I guess can be as simple as the functional part conveying an integer specifying how many bytes/lines/whatever of data to grab from the TCP session next, or what state your application protocol is in).

    I think this at the very least rhymes with what you’re saying; maybe it’s the same.

    > [My goal was a 20-word summary]
    That’s a worthwhile goal. I still think what you’re describing is not functional programming but rather declarative programming. My hot take: declarative programming is all about taking some predefined generic process, often a search process, and specializing it to your particular application. Makefiles, constraint solver libraries and logic programming in miniKanren are all declarative programs (as far as I’m concerned), and they’re all about describing relationships between things which constrain the solution space, not a way of computing a solution. They focus on the what, not the how.

    > [everyone advocates minimizing side effects]
    Not my Java teachers in school (write your getters and setters, kids). Not the Java library designers (mutable collections, mutable java.util.Date). Arguably not the Java language designers, by not providing constructs that make writing in a functional style easy.

    Arguably also not me. There are benefits to writing in a functional style, but also costs, see the next section.

    > [you’re defining “functional programming” as “good programming”]
    At work I implemented a small thing in a functional style, complete with higher-order procedures and immutable data and it was very concise, readable and slow. Then I wrote a C-style imperative solution with explicit loops and mutable arrays which was 100x faster. Based on my performance measurements of the system, I think the second is better.

    I’ve also written constraint solvers for sudoku-like problems, mutating state as they narrow down the search space. I don’t think those programs would become better if I inserted a bunch of copying of intermediate results followed by a deallocation of the old copy (i.e. if I took the necessary steps to make them more functional). On the contrary: all the deeply iterated data dependencies (essential complexity) would still be there, but the code would obscure what’s going on by adding noise. And it would be slower.

    Writing in a functional style makes your program better along some but not all dimensions of goodness. The other dimensions also matter.

    I should question myself, though, so I will: there’s a big difference between defining X in terms of objective and value-free criteria versus defining X to inherently be good. One should never conflate those, or worse motte-and-bailey them. Did I do any of this?

    Maybe, kind of? The closest I think I got is defining the essence of functional style to be that aspect which occurs across code in multiple functional languages (and functional code in multi-paradigm languages) which I think provides the most value. Maybe the *essence* of functional style isn’t exactly the same as the *best parts* of functional style.

    Let’s think about the other aspects of functional programming and compare.

    I think using a higher-order List[A].replace_elements(A => A) => void is less functional than a List[A].map(A => B) => List[B], even if we require A = B. This still holds if A is a sum type, and also if replace_elements is implemented in a (tail-)recusive fashion but map isn’t. At least that’s how I see it.

    So immutability across function boundaries seems like the most essential feature, compared to the other ones.

    There’s one more aspect common across many functional languages (and some multiparadigmatic ones): good support for higher-order functions and inline anonymous functions. I think that lets you combine separate pieces of code easily. Combining things is the essence of programming (if the things are code and/or data). Using higher-order and anonymous inline functions doesn’t inherently make your code more functional, but having them available facilitates it, simply because it facilitates all kinds of programming.

    For those interested, here’s a talk I find excellent about the pragmatic, un-academic benefits of functional programming: TL;DR: 1. map/filter/reduce; 2. parametric types; 3. sum types (and pattern matching); 4. monad-friendly syntactical constructs. I think one big benefit of map/filter/reduce is that they let you operate on your data in collections by computing a new collection rather than mutating the old one. (They’re also more concise than writing your own loop, and they feel more declarative.)

    With hopes that this discussion will (continue to?) be informative and enjoyable for everyone — cheers :-)

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