6 ms·

If you think APL is less expressive than Matlab, you probably haven't really grasped it, IMO. Having said that, Matlab is optimized for manipulating matrices an

by patrec 6y ago

If you think APL is less expressive than Matlab, you probably haven't really grasped it, IMO. Having said that, Matlab is optimized for manipulating matrices and replicating the notation of normal linear algebra and has excellent implementations of basically any numerical algorithm that frequently comes up in this . So writing something like chol(X'*X+diag(eig(X))) in an APL will look uglier and quite possibly slower and less accurate and depending on the numerical routines you need, require extra implementation work on your part. But that overhead is constant factor, more or less anything you can express well in matlab can be expressed straightforwardly in APL, too, if you have the right numerical routines. That's not true in the other direction though: there's a lot of stuff in APL you cannot express adequately in matlab at all. For example J (and these days Dyalog as well IIRC) have an operation called under which basically does this: u(f,g) = x => f^-1(g(f(x)). So you can write geometric_mean = u(mean, log). It is completely impossible to implement something like "under" in matlab. Admittedly the J implementation at least of deriving a generalized inverse for an arbitrary function f is a somewhat ill-defined hack, but this is still something that is both conceptually and practically quite powerful. Also, whilst Matlab is really clunky for anything that is not a 2D array and hardcodes matrix multiplication as the one inner-product, APL has more powerful abstractions for manipulating arbitrary rank arrays and a more general concept of inner products. Also, APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function.

henrikeh 6y ago
> It is completely impossible to implement something like "under" in matlab. I’m a little curious about this. Does J have a notion of the relationship between certain functions and their inverse? What is it that enables “under” in J which makes it impossible in Matlab?
- klibertp 6y ago
  > Does J have a notion of the relationship between certain functions and their inverse? Yes. Many built-in words have inverses assigned, and you can assign inverse functions to your own words with :. https://code.jsoftware.com/wiki/Vocabulary/codot https://code.jsoftware.com/wiki/Vocabulary/codot EDIT: and here's a table with predefined inverses: https://code.jsoftware.com/wiki/Vocabulary/Inverses https://code.jsoftware.com/wiki/Vocabulary/Inverses
  - henrikeh 6y ago
    That can definitely be implemented in Matlab with the symbolic math toolbox. But interesting nonetheless.
    - patrec 6y ago
      I have used matlab quite a bit in the past, but not the symbolic math toolbox: can you make a "normal" function definition symbolic retroactively? I thought you need to define the function explicitly as symbolic to start with, or am I wrong?
      - henrikeh 6y ago
        Yes, the symbolic functions can be made "retroactively". The various conventional functions are overloaded with the symbolic input. >> syms x >> f = @(x) log(sqrt(x)).^2 f = function_handle with value: @(x)log(sqrt(x)).^2 >> f(x) ans = log(x^(1/2))^2 >> finverse(f(x)) ans = exp(2*x^(1/2)) And to implement under: function u = under(f, g) syms x g_inv = matlabFunction(finverse(g(x))); u = @(x) g_inv(f(g(x))); end
        patrec 6y ago
        So if you have odddouble.m with function y=odddouble(a,b) y=2*x+1 endfunction you can do >>> h = under(@(x) 1/x, odddouble) >>> h(3) ? If so, yeah, I agree you can implement under in matlab (as long as you have the symbolic toolbox as well); in which case it's probably one of very few non-CAS systems where you can define it.
        henrikeh 6y ago
        Just tested it and it worked fine. Note that your function takes too many parameters in its definition and that functions can't be passed as parameters by name. It is definitely not as elegant as the built-in facility in J, but definitely doable and usable in Matlab. In fact, I think any language with flexible enough function overloading should be able to implement such a feature.
- patrec 6y ago
  Here is toy example: I define a function f(x):=1+x2. J will automatically work out the inverse for me, as you can see below: f=:(1+*&2) f 1 2 3 4 3 5 7 9 (f^:_1)f 1 2 3 4 1 2 3 4 (f^:_1) 1 2 3 4 0 0.5 1 1.5 Now obviously not every function is bijective or even if bijective, trivial to invert -- and J doesn't (or at least didn't) have a super well specified way of computing those generalized inverses. But still: "under" is actually pretty cool, even just conceptually I find it quite valuable.
moonchild 6y ago
> Admittedly the J implementation at least of deriving a generalized inverse for an arbitrary function f is a somewhat ill-defined hack Have you seen the version used by dzaima/apl[1]? The equivalent of '(-&.:{:) i.5' works and results in 0 1 2 3 _4. > APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function Klong[2] is a partial attempt to resolve this. I won't repeat the arguments in favour of ambivalent functions, as I guess you've heard them a dozen times before > u(f,g) = x => f^-1(g(f(x)). Other way round; it's g^-1(f(g(x))) 1. https://github.com/dzaima/apl https://github.com/dzaima/apl 2. https://t3x.org/klong/ https://t3x.org/klong/
- patrec 6y ago
  Is there an argument for ambivalent function definitions other than "keyword" recycling (possibly in a mnemonic fashion)? I hadn't seen Dzaima's APL, thanks! I like that he made a processing binding; APL always seemed like it would be such an obvious choice for doing dweet style graphics code golfing that I wondered why no one seemed to be doing it. A web-based APL would be a better choice though.
  - moonchild 6y ago
    > web-based In that case you'll be wanting ngn/apl[1], which runs in a browser and compiles to js. > ambivalent The arguments are mostly linguistic. Natural language is also context-sensitive, so we are well-equipped to parse such formations; and they allow us to reuse information. The monadic and dyadic forms of '-' are related, so it's less cognitive overhead to recognize its meaning. 1. https://gitlab.com/n9n/apl https://gitlab.com/n9n/apl
  - dzaima 6y ago
    Keyboard space is another somewhat important factor. My layout for dzaima/APL already uses all altgr keys, so I could definitely not afford multiplying the number of needed characters by 2. Not having ambivalently callable operators would also mean needing 2 versions of most of them. dzaima/APL being written in Java means getting it to run in a browser would be a bit hard, and ngn has given up on ngn/apl, but BQN[0] could definitely get a web canvas based graphics interface. Somewhat interesting to add to the conversation about Under is that, in my impl, calling a function, calling its inverse, or doing something under it (i.e. structural under) are all equally valid ways to "use" a function, it's just a "coincidence" that there's direct syntax for invoking only one. (Dyalog does not yet have under, but it definitely is planned.) 0. https://mlochbaum.github.io/BQN/ https://mlochbaum.github.io/BQN/

by patrec 6y ago

henrikeh 6y ago

> It is completely impossible to implement something like "under" in matlab. I’m a little curious about this. Does J have a notion of the relationship between certain functions and their inverse? What is it that enables “under” in J which makes it impossible in Matlab?

klibertp 6y ago
> Does J have a notion of the relationship between certain functions and their inverse? Yes. Many built-in words have inverses assigned, and you can assign inverse functions to your own words with :. https://code.jsoftware.com/wiki/Vocabulary/codot https://code.jsoftware.com/wiki/Vocabulary/codot EDIT: and here's a table with predefined inverses: https://code.jsoftware.com/wiki/Vocabulary/Inverses https://code.jsoftware.com/wiki/Vocabulary/Inverses
- henrikeh 6y ago
  That can definitely be implemented in Matlab with the symbolic math toolbox. But interesting nonetheless.
  - patrec 6y ago
    I have used matlab quite a bit in the past, but not the symbolic math toolbox: can you make a "normal" function definition symbolic retroactively? I thought you need to define the function explicitly as symbolic to start with, or am I wrong?
    - henrikeh 6y ago
      Yes, the symbolic functions can be made "retroactively". The various conventional functions are overloaded with the symbolic input. >> syms x >> f = @(x) log(sqrt(x)).^2 f = function_handle with value: @(x)log(sqrt(x)).^2 >> f(x) ans = log(x^(1/2))^2 >> finverse(f(x)) ans = exp(2*x^(1/2)) And to implement under: function u = under(f, g) syms x g_inv = matlabFunction(finverse(g(x))); u = @(x) g_inv(f(g(x))); end
      - patrec 6y ago
        So if you have odddouble.m with function y=odddouble(a,b) y=2*x+1 endfunction you can do >>> h = under(@(x) 1/x, odddouble) >>> h(3) ? If so, yeah, I agree you can implement under in matlab (as long as you have the symbolic toolbox as well); in which case it's probably one of very few non-CAS systems where you can define it.
        henrikeh 6y ago
        Just tested it and it worked fine. Note that your function takes too many parameters in its definition and that functions can't be passed as parameters by name. It is definitely not as elegant as the built-in facility in J, but definitely doable and usable in Matlab. In fact, I think any language with flexible enough function overloading should be able to implement such a feature.
patrec 6y ago
Here is toy example: I define a function f(x):=1+x2. J will automatically work out the inverse for me, as you can see below: f=:(1+*&2) f 1 2 3 4 3 5 7 9 (f^:_1)f 1 2 3 4 1 2 3 4 (f^:_1) 1 2 3 4 0 0.5 1 1.5 Now obviously not every function is bijective or even if bijective, trivial to invert -- and J doesn't (or at least didn't) have a super well specified way of computing those generalized inverses. But still: "under" is actually pretty cool, even just conceptually I find it quite valuable.

moonchild 6y ago

> Admittedly the J implementation at least of deriving a generalized inverse for an arbitrary function f is a somewhat ill-defined hack Have you seen the version used by dzaima/apl[1]? The equivalent of '(-&.:{:) i.5' works and results in 0 1 2 3 _4. > APL has some really dumb but cherished-by-the-community ideas that make the language less expressive and much more awkward to learn, e.g. the idea of replicating the terrible defect of normal mathematical notation where - is overloaded for negation and subtraction to every other function Klong[2] is a partial attempt to resolve this. I won't repeat the arguments in favour of ambivalent functions, as I guess you've heard them a dozen times before > u(f,g) = x => f^-1(g(f(x)). Other way round; it's g^-1(f(g(x))) 1. https://github.com/dzaima/apl https://github.com/dzaima/apl 2. https://t3x.org/klong/ https://t3x.org/klong/

patrec 6y ago
Is there an argument for ambivalent function definitions other than "keyword" recycling (possibly in a mnemonic fashion)? I hadn't seen Dzaima's APL, thanks! I like that he made a processing binding; APL always seemed like it would be such an obvious choice for doing dweet style graphics code golfing that I wondered why no one seemed to be doing it. A web-based APL would be a better choice though.
- moonchild 6y ago
  > web-based In that case you'll be wanting ngn/apl[1], which runs in a browser and compiles to js. > ambivalent The arguments are mostly linguistic. Natural language is also context-sensitive, so we are well-equipped to parse such formations; and they allow us to reuse information. The monadic and dyadic forms of '-' are related, so it's less cognitive overhead to recognize its meaning. 1. https://gitlab.com/n9n/apl https://gitlab.com/n9n/apl
- dzaima 6y ago
  Keyboard space is another somewhat important factor. My layout for dzaima/APL already uses all altgr keys, so I could definitely not afford multiplying the number of needed characters by 2. Not having ambivalently callable operators would also mean needing 2 versions of most of them. dzaima/APL being written in Java means getting it to run in a browser would be a bit hard, and ngn has given up on ngn/apl, but BQN[0] could definitely get a web canvas based graphics interface. Somewhat interesting to add to the conversation about Under is that, in my impl, calling a function, calling its inverse, or doing something under it (i.e. structural under) are all equally valid ways to "use" a function, it's just a "coincidence" that there's direct syntax for invoking only one. (Dyalog does not yet have under, but it definitely is planned.) 0. https://mlochbaum.github.io/BQN/ https://mlochbaum.github.io/BQN/