4 ms·

That's altering the syntax for function application by allowing f x to mean f(x), which is fine but changing the rules. If we do go with that, then "a b c" sho

by kmill 9y ago

That's altering the syntax for function application by allowing f x to mean f(x), which is fine but changing the rules. If we do go with that, then "a b c" should be "Apply a (Apply b c)" because of the associativity of application --- traditional math tends not to have functions which return functions. There is an ambiguity which comes up when you allow both f x and f(x) syntax, which is you cannot tell the difference between f(x,y) and f((x,y)) if your language has tuples. (One solution: make tuples be like Mathematica's Sequence, thus establishing the associativity of Cartesian products once and for all.) A gotcha is that you can't write f(x)(y) if your function does return a function, since this will parse as f(x y). You would need to write (f(x))(y) instead. (If you insist the associativity rule should be the other way, then 3f(x) would be (3f)x, which is possibly ok, and sin cos x would be (sin(cos))(x).)

yorwba 9y ago
I don't think you can have all of traditional mathematical notation (which is often highly overloaded) in a programming language while only using syntax for disambiguation. Mathematicians mostly depend on the semantics of an expression to disambiguate it. You could get something close by making use of a type system to filter all possible parses. For example, sin cos x would parse as sin(cos(x)), (sin(cos))(x), (sin * cos)(x), ... and most of these would be discarded by the type checker. If you do it naively, the complexity will be exponential, but I think you could use dynamic programming to handle that. In my example, it would be more problematic that (sin * cos) might be interpreted as multiplication of functions, but this could be solved by having separate operators for numbers and other types, and only allowing some to be represented by juxtaposition. Overall, I'm not convinced that such a highly ambiguous system would be beneficial in a general purpose programming language, but when you just want to type in a specific mathematical formula, disambiguate it once, and then work with the unambiguous representation (maybe using a pretty-printer), I think it would be useful.
- alphaalpha101 9y ago
  >If you do it naively, the complexity will be exponential Hindley-Milner type inference is doubly exponential already but is still widely used. If people never actually use extremely nested expressions in real code that require this sort of disambiguation then it will probably be completely fine.
- kmill 9y ago
  > I don't think you can have all of traditional mathematical notation (which is often highly overloaded) in a programming language while only using syntax for disambiguation. I think it is reasonable to be able to handle certain subsets of mathematical notation, but yeah, in total it seems to be context dependent. (I've tried designing languages to deal with this before, and my previous comments were about actual issues I had encountered making it all consistent.) > For example, sin cos x would parse as sin(cos(x)), (sin(cos))(x), (sin * cos)(x), ... and most of these would be discarded by the type checker Unfortunately, all of those make some sense! The first is what anyone who took precalculus would read, but the second could be substituting cos into the power series of sin, and the third could be the pointwise product (sin * cos)(x) := sin(x) * cos(x), which is particularly important for the ring of functions on the real line. > Overall, I'm not convinced that such a highly ambiguous system would be beneficial in a general purpose programming language, but when you just want to type in a specific mathematical formula, disambiguate it once, and then work with the unambiguous representation (maybe using a pretty-printer), I think it would be useful. One thing I've been enjoying about Mathematica is the ability to have the source code be written in 2D graphical notation. I'm not convinced that it would be useful to have a special way to input and disambiguate mathematical formulae for a general purpose programming language because 1) actual formulae are infrequent and 2) it tends not to be too challenging to translate a formula into the syntax of the programming language. I think the main use case for trying to interpret traditional math notation is for exploratory calculating/graphing/programming where you would only ever use the equation at most a few times.
alphaalpha101 9y ago
>That's altering the syntax for function application by allowing f x to mean f(x), which is fine but changing the rules. If we do go with that, then "a b c" should be "Apply a (Apply b c)" because of the associativity of application --- traditional math tends not to have functions which return functions. That's the standard way of writing function application in functional programming languages. And in those languages, f a b c == (((f a) b) c) because they're curried: a function of 2 arguments (f: a -> b -> c) is a function f from type a to type (b -> c). And mathematics definitely has functions that return functions. For example, any function where the codomain is a space of sequences is technically a function that returns functions, as a sequence in X is a function from the natural numbers to X. >There is an ambiguity which comes up when you allow both f x and f(x) syntax, which is you cannot tell the difference between f(x,y) and f((x,y)) if your language has tuples. (One solution: make tuples be like Mathematica's Sequence, thus establishing the associativity of Cartesian products once and for all.) You don't need to tell the difference. If you have a curried function, then you write 'f x y' which isn't the same as 'f (x y)' (which is like the C-style 'f(x(y))'. If you have a non-curried function, then 'multiple arguments' are just a tuple, at least that's how it works in ML. A function that takes a tuple and a function with multiple arguments are equivalent. That's related to the fact that ((A AND B) IMPLIES C) is equivalent to (A IMPLIES (B IMPLIES C)). >A gotcha is that you can't write f(x)(y) if your function does return a function, since this will parse as f(x y). You would need to write (f(x))(y) instead. (If you insist the associativity rule should be the other way, then 3f(x) would be (3f)x, which is possibly ok, and sin cos x would be (sin(cos))(x).) sin cos x is okay notation in mathematics because sin can't take cos as an argument, so it obviously has to mean 'sin(cos x)' and not 'sin(cos)(x)'. But when dealing with computers, I would far rather just write sin (cos x) which is how people write it in Haskell/ML/Lisp. Not sure if you mean existential quantification or just the number 3 in '3f(x)' but as I said, I don't think that it's a good idea to support 'x f' as meaning '(lambda (a) (* 3 (f a)))' or '(a => 3 * f(a))' or '\a -> 3 * (f a)' or whatever your preferred function notation is for that scaling operation.
- kmill 9y ago
  Just to keep things in perspective, all I'm saying is that "the ability to write multiplication naturally" comes at a cost. I'm aware of Haskell/ML-like syntax for curried functions, but it is at odds with juxtaposition for multiplication. I'm speaking from the position of having tried designing languages with these features and not being able to find a way to make everything consistent. > And mathematics definitely has functions that return functions Recall, I said "tends not to". My point was that the notation of mathematics is set up to prefer the case of functions returning values. > Not sure if you mean existential quantification or just the number 3 in '3f(x)' I meant 3f(x) to mean 3*f(x). I believe you'd want to be able to write 3f(x) since you should be able to have the "ability to write multiplication naturally," in your words. > sin can't take cos as an argument But sin can be represented as a power series and cos can be substituted in with cos^n meaning iterating cos n times. This may or may not be reasonable; I don't think it is "obvious."

by kmill 9y ago

yorwba 9y ago

I don't think you can have all of traditional mathematical notation (which is often highly overloaded) in a programming language while only using syntax for disambiguation. Mathematicians mostly depend on the semantics of an expression to disambiguate it. You could get something close by making use of a type system to filter all possible parses. For example, sin cos x would parse as sin(cos(x)), (sin(cos))(x), (sin * cos)(x), ... and most of these would be discarded by the type checker. If you do it naively, the complexity will be exponential, but I think you could use dynamic programming to handle that. In my example, it would be more problematic that (sin * cos) might be interpreted as multiplication of functions, but this could be solved by having separate operators for numbers and other types, and only allowing some to be represented by juxtaposition. Overall, I'm not convinced that such a highly ambiguous system would be beneficial in a general purpose programming language, but when you just want to type in a specific mathematical formula, disambiguate it once, and then work with the unambiguous representation (maybe using a pretty-printer), I think it would be useful.

alphaalpha101 9y ago
>If you do it naively, the complexity will be exponential Hindley-Milner type inference is doubly exponential already but is still widely used. If people never actually use extremely nested expressions in real code that require this sort of disambiguation then it will probably be completely fine.
kmill 9y ago
> I don't think you can have all of traditional mathematical notation (which is often highly overloaded) in a programming language while only using syntax for disambiguation. I think it is reasonable to be able to handle certain subsets of mathematical notation, but yeah, in total it seems to be context dependent. (I've tried designing languages to deal with this before, and my previous comments were about actual issues I had encountered making it all consistent.) > For example, sin cos x would parse as sin(cos(x)), (sin(cos))(x), (sin * cos)(x), ... and most of these would be discarded by the type checker Unfortunately, all of those make some sense! The first is what anyone who took precalculus would read, but the second could be substituting cos into the power series of sin, and the third could be the pointwise product (sin * cos)(x) := sin(x) * cos(x), which is particularly important for the ring of functions on the real line. > Overall, I'm not convinced that such a highly ambiguous system would be beneficial in a general purpose programming language, but when you just want to type in a specific mathematical formula, disambiguate it once, and then work with the unambiguous representation (maybe using a pretty-printer), I think it would be useful. One thing I've been enjoying about Mathematica is the ability to have the source code be written in 2D graphical notation. I'm not convinced that it would be useful to have a special way to input and disambiguate mathematical formulae for a general purpose programming language because 1) actual formulae are infrequent and 2) it tends not to be too challenging to translate a formula into the syntax of the programming language. I think the main use case for trying to interpret traditional math notation is for exploratory calculating/graphing/programming where you would only ever use the equation at most a few times.

alphaalpha101 9y ago

>That's altering the syntax for function application by allowing f x to mean f(x), which is fine but changing the rules. If we do go with that, then "a b c" should be "Apply a (Apply b c)" because of the associativity of application --- traditional math tends not to have functions which return functions. That's the standard way of writing function application in functional programming languages. And in those languages, f a b c == (((f a) b) c) because they're curried: a function of 2 arguments (f: a -> b -> c) is a function f from type a to type (b -> c). And mathematics definitely has functions that return functions. For example, any function where the codomain is a space of sequences is technically a function that returns functions, as a sequence in X is a function from the natural numbers to X. >There is an ambiguity which comes up when you allow both f x and f(x) syntax, which is you cannot tell the difference between f(x,y) and f((x,y)) if your language has tuples. (One solution: make tuples be like Mathematica's Sequence, thus establishing the associativity of Cartesian products once and for all.) You don't need to tell the difference. If you have a curried function, then you write 'f x y' which isn't the same as 'f (x y)' (which is like the C-style 'f(x(y))'. If you have a non-curried function, then 'multiple arguments' are just a tuple, at least that's how it works in ML. A function that takes a tuple and a function with multiple arguments are equivalent. That's related to the fact that ((A AND B) IMPLIES C) is equivalent to (A IMPLIES (B IMPLIES C)). >A gotcha is that you can't write f(x)(y) if your function does return a function, since this will parse as f(x y). You would need to write (f(x))(y) instead. (If you insist the associativity rule should be the other way, then 3f(x) would be (3f)x, which is possibly ok, and sin cos x would be (sin(cos))(x).) sin cos x is okay notation in mathematics because sin can't take cos as an argument, so it obviously has to mean 'sin(cos x)' and not 'sin(cos)(x)'. But when dealing with computers, I would far rather just write sin (cos x) which is how people write it in Haskell/ML/Lisp. Not sure if you mean existential quantification or just the number 3 in '3f(x)' but as I said, I don't think that it's a good idea to support 'x f' as meaning '(lambda (a) (* 3 (f a)))' or '(a => 3 * f(a))' or '\a -> 3 * (f a)' or whatever your preferred function notation is for that scaling operation.

kmill 9y ago
Just to keep things in perspective, all I'm saying is that "the ability to write multiplication naturally" comes at a cost. I'm aware of Haskell/ML-like syntax for curried functions, but it is at odds with juxtaposition for multiplication. I'm speaking from the position of having tried designing languages with these features and not being able to find a way to make everything consistent. > And mathematics definitely has functions that return functions Recall, I said "tends not to". My point was that the notation of mathematics is set up to prefer the case of functions returning values. > Not sure if you mean existential quantification or just the number 3 in '3f(x)' I meant 3f(x) to mean 3*f(x). I believe you'd want to be able to write 3f(x) since you should be able to have the "ability to write multiplication naturally," in your words. > sin can't take cos as an argument But sin can be represented as a power series and cos can be substituted in with cos^n meaning iterating cos n times. This may or may not be reasonable; I don't think it is "obvious."