Skip to main content

custom notation - Going full functional (Haskell style)


I'm trying to define some notation so that Mathematica code would be more functional, similar to Haskell (just for fun): currying, lambdas, infix operator to function conversion, etc.. And I have some questions about it:



  • Is it possible to make all Mathematica h_[x1_,x2_,...] functions to work as h[x1][x2][..]?

  • Can I distinguish inside Notation box between <+1> and <1+>, how do I check for + there?

  • How to define right-associate apply operator with highest precedence ($)?


This is what I have so far:


<< Notation`;


lapply[x_, y_] := x[y]
rapply[x_, y_] := x[y]
InfixNotation[ParsedBoxWrapper["\\"], lapply]
InfixNotation[ParsedBoxWrapper["$"], rapply]
x_ \ y_ \ z_ := x[y][z]
x_ $ y_ $ z_ := x[y[z]]

Notation[ParsedBoxWrapper[
RowBox[{
RowBox[{"λ", " ", "x__"}], "->", 

     "y_"}]] ⟺ ParsedBoxWrapper[
RowBox[{"Function", "[", 
RowBox[{
RowBox[{"{", "x__", "}"}], ",", "y_"}], "]"}]]]

Notation[ParsedBoxWrapper[
RowBox[{"〈", 
RowBox[{"op_", " ", "x_"}], 
     "〉"}]] ⟺ 
  ParsedBoxWrapper[

RowBox[{
RowBox[{"#", "op_", " ", "x_"}], "&"}]]]
AddInputAlias["f" -> ParsedBoxWrapper[
RowBox[{"〈", 
RowBox[{"\[Placeholder]", "\[Placeholder]"}], 
     "〉"}]]]

Notation[ParsedBoxWrapper[
RowBox[{"{", 
RowBox[{

RowBox[{"x_", " ", ".."}], " ", "y_"}], 
     "}"}]] ⟺ ParsedBoxWrapper[
RowBox[{"Range", "[", 
RowBox[{"x_", ",", "y_"}], "]"}]]]

filter[f_][x_List] := Select[x, f]
map[f_][x__List] := Map[f, x]

filter\PrimeQ $ map\〈-1〉 $ map\ \
(λ x -> 2^x)\ {1 .. 100}


EDIT: Also did some kinda lazy lists, soon it will be haskell inside Mathematica :)


SetAttributes[list, HoldAll]
list[h_, l_][x_] := list[h, l[x]]
list[x_] := list[x, list]

map[f_][list] := list
map[f_][list[x_, xs_]] := list[f[x], map[f][xs]]

take[0][_] := list

take[_][list] := list
take[n_Integer][list[x_, xs_]] := list[x, take[n - 1][xs]]

range[n_Integer] := range[1, n]
range[m_, n_] := list[m, range[m + 1, n]]
range[n_, n_] := list[n, list]

show[list] := "[]"
show[list[x_, l_]] := ToString[x] <> "," <> show[l]


show $ (take[10] $  map\ (λ x -> x^2) $ range[10000])

Answer



That's how I finally defined haskell operators:


rapply[x_] := x
rapply[x_, y__] := x[rapply[y]]
InfixNotation[ParsedBoxWrapper["|"], rapply]

lapply[x_] := x
lapply[x__, y_] := lapply[x][y]
InfixNotation[ParsedBoxWrapper["∘"], lapply]


InfixNotation[ParsedBoxWrapper["·"], Composition]

Now $\circ$, $\dot{}{}$ and | act exactly like haskell's space, . and $ respectfully. Also if we have only single left application then @ is still helpful and it can be hidden with escape characters :@:.


And beautiful code like $\bf{show\cdot take\ 10\cdot map\ (\lambda\ x\to x{}^{\wedge}2)\cdot range | \infty }$ is possible. There are invisible @'s between take and 10, map and ($\lambda\ x\to x{}^{\wedge}2)$. In haskell the same would look like $\bf{show . take\ 10.map(\backslash x->x{}^{\wedge}2)$[1..]}$.


With double left application, $\circ$ is necessary: $\bf{map\circ(\lambda\ x\to x+1)\circ \{1,2,3\}}$


UPDATE: I made this TextCell hack to make partial infix operators:


infix[f_String] := Block[{x,y},Head[ToExpression["x" <> f <> "y"]]]

〈TextCell[s_][x_]〉 := infix[s][#, x] &

〈x_[TextCell[s_]]〉 := infix[s][x, #] &

We again can use invisible @ for application. Aliases can be made with TextCell on the left and on the right within AngleBrackets to enter them conveniently. Now stuff like <~Mod~10>, <2^>, <^3> also works.


Comments

Post a Comment

Popular posts from this blog

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...
Post a Comment
Read more