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

Popular posts from this blog

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1.