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In Less Than a Year, So Much New: Launching Version 12.1 of Wolfram Language & Mathematica

We’re pleased that despite the coronavirus pandemic and its impact on so many people and businesses we’re still able to launch today as planned… (Thanks to our dedicated team and the fact that remote working has been part of our company for decades…)

The Biggest .1 Release Ever

It’s always an interesting time. We’re getting ready to wrap up a .1 version—to release the latest fruits of our research and development efforts. “Is it going to be a big release?”, I wonder. Of course, I know we’ve done a lot of work since we released Version 12.0 last April. All those design reviews (many livestreamed). All those new things we’ve built and figured out.

But then we start actually making the list for the new version. And—OMG—it goes on and on. Different teams are delivering on this or that project that started X years ago. A new function is being added for this. There’s some new innovation about that. Etc.

We started this journey a third of a century ago when we began the development of Version 1.0. And after all these years, it’s amazing how the energy of each new release seems to be ever greater.

And as we went on making the list for Version 12.1 we wondered, “Will it actually be our biggest .1 release ever?”. We finally got the answer: “Yes! And by a lot”.

Counting functions isn’t always the best measure, but it’s an indication. And in Version 12.1 there are a total of 182 completely new functions—as well as updates and enhancements to many hundreds more.

In Less Than a Year, So Much New: Launching Version 12.1 of Wolfram Language & Mathematica

Look at It Now: HiDPI

Back in 1988 when we released Version 1.0 a typical computer display was maybe 640 pixels across (oh, and it was a CRT). And I was recently using some notebooks of mine from the 1990s (yes, they still work, which is spectacular!), and I was amazed at what a small window size they were made for. But as I write this today, I’m looking at two 3000-pixel displays. And 4k displays aren’t uncommon. So one of the things we’ve done for Version 12.1 is to add systemwide support for the new world of very-high-resolution displays.

One might think that would be easy, and would just “come with the operating system”. But actually it’s taken two years of hard work to deliver full HiDPI support. Well over a thousand carefully designed icons and other assets went from being bitmaps to being work-at-any-size algorithmic graphics. Everything about rasterization (not just for Rasterize, but for 3D graphics textures, etc. etc.) had to be redone. Sizes of things—and their interactions with the tower of kludges that operating systems have introduced over the years—had to be respecified and rethought.

But now it’s done. And we’re ready for displays of any resolution:

HiDPI icons

By the way, talking of displays, another “infrastructure” enhancement in Version 12.1 is moving to Metal and Direct3D 11 for 3D graphics rendering on macOS and Windows. Right now these just make 3D graphics modestly faster. But they also lay the groundwork for full multithreaded rendering, as well as VR, AR and more.

The Beginning of Video Computation

We’ve been working towards it for nearly 15 years… but finally it’s here: computation with video! We introduced images into the language in 2008; audio in 2016. But now in Version 12.1 we for the first time have computation with video. There’ll be lots more coming in future releases, but there’s already quite a bit in 12.1.

So… just like Image and Audio, which symbolically represent images and audio, we now have Video.

This asks for five frames from a video:

VideoFrameList
&#10005

VideoFrameList[
 Video["ExampleData/Caminandes.mp4", Appearance -> Automatic, 
  AudioOutputDevice -> Automatic, SoundVolume -> Automatic], 5]

This asks to make a time series of the mean color of every frame:

VideoTimeSeries
&#10005

VideoTimeSeries[Mean, 
 Video["ExampleData/Caminandes.mp4", Appearance -> Automatic, 
  AudioOutputDevice -> Automatic, SoundVolume -> Automatic]]

And then one can just plot the time series:

DateListPlot
&#10005

DateListPlot[%, 
 PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0], RGBColor[
   0, 0, 1]}]

Video is a complicated area, with lots of different encodings optimized for different purposes. In Version 12.1 we’re supporting more than 250 of them, for import, export and transcoding. You can refer to a video on the web as well:

Video
&#10005

Video["http://exampledata.wolfram.com/cars.avi"]

And the big thing is that video is now getting integrated into everything. So, for example, you can immediately use image processing or audio processing or machine learning functions on video. Here’s an example plotting the location of cars in the video above:

v = Video
&#10005

v = Video["http://exampledata.wolfram.com/cars.avi"]

ts = VideoTimeSeries
&#10005

ts = VideoTimeSeries[Point[ImagePosition[#, Entity["Word", "car"]]] &,
   v]

HighlightImage
&#10005

HighlightImage[
 VideoExtractFrames[v, Quantity[5, "Seconds"]], {PointSize[Medium], 
  Values[ts]}]

Let’s say you’ve got a Manipulate, or an animation (say from ListAnimate). Well, now you can just immediately make a video of it:

Video
&#10005

Video[CloudGet["https://wolfr.am/L9r00rk5"]]

You can add an audio track, then export the whole thing directly to a file, the cloud, etc.

So is this new video capability really industrial strength? I’ve been recording hundreds of hours of video in connection with a new project I’m working on. So I decided to try our new capabilities on it. It’s spectacular! I could take a 4-hour video, and immediately extract a bunch of sample frames from it, and then—yes, in a few hours of CPU time—“summarize the whole video”, using SpeechRecognize to do speech-to-text on everything that was said and then generating a word cloud:

Working session

Speaking of audio, there’s new stuff in Version 12.1 there too. We’ve redone the GUI for in-notebook Audio objects. And we’ve introduced SpeechInterpreter, which is the spoken analog of the Interpreter function, here taking an audio object and returning what airline name was said in it:

SpeechInterpreter
&#10005

SpeechInterpreter["Airline"][CloudGet["https://wolfr.am/L9r410jA"]]

In Version 12.0 we introduced the important function TextCases for extracting from text hundreds of kinds of entities and “text content types” (which as of 12.1 now have their own documentation pages). In 12.1 we’re also introducing SpeechCases, which does the same kind of thing for audio speech.

A Computer Science Story: DataStructure

One of our major long-term projects is the creation of a full compiler for the Wolfram Language, targeting native machine code. Version 12.0 was the first exposure of this project. In Version 12.1 there’s now a spinoff from the project—which is actually a very important project in its own right: the new DataStructure function.

We’ve curated many kinds of things in the past: chemicals, equations, movies, foods, import-export formats, units, APIs, etc. And in each case we’ve made the things seamlessly computable as part of the Wolfram Language. Well, now we’re adding another category: data structures.

Think about all those data structures that get mentioned in textbooks, papers, libraries, etc. Our goal is to have all of them seamlessly usable directly in the Wolfram Language, and accessible in compiled code, etc. Of course it’s huge that we already have a universal “data structure”: the symbolic expressions in the Wolfram Language. And internal to the Wolfram Language we’ve always used all sorts of data structures, optimized for different purposes, and automatically selected by our algorithms and meta-algorithms.

But now with DataStructure there’s something new. If you have a particular kind of data structure you want to use, you can just ask for it by name, and use it.

Here’s how you create a linked list data structure:

ds = CreateDataStructure
&#10005

ds = CreateDataStructure["LinkedList"]

Append a million random integers to the linked list (it takes 380 ms on my machine):

Do
&#10005

Do[ds["Append", RandomInteger[]], 10^6]

Now there’s immediate visualization of the structure:

ds
&#10005

ds["Visualization"]

Here’s the numerical mean of all the values:

Mean
&#10005

Mean[N[Normal[ds]]]

Like so much of what we do DataStructure is set up to span from small scale and pedagogical to large scale and full industrial strength. Teaching a course about data structures? Now you can immediately use the Wolfram Language, storing everything you do in notebooks, automatically visualizing your data structures, etc. Building large-scale production code? Now you can have complete control over optimizing the data structures you’re using.

How does DataStructure work? Well, it’s all written in the Wolfram Language, and compiled using the compiler (which itself is also written in the Wolfram Language).

In Version 12.1 we’ve got most of the basic data structures covered, with various kinds of lists, arrays, sets, stacks, queues, hash tables, trees, and more. And here’s an important point: each one is documented with the running time for its various operations (“O(n)”, “O(n log(n))”, etc.), and the code ensures that that’s correct.

It’s pretty neat to see classic algorithms written directly for DataStructure.

Create a binary tree data structure (and visualize it):

(ds = CreateDataStructure
&#10005

(ds = CreateDataStructure["BinaryTree", 
    3 -> {1 -> {0, Null}, Null}])["Visualization"]

Here’s a function for rebalancing the tree:

RightRotate
&#10005

RightRotate[y_] :=
 Module[{x, tmp},
  x = y["Left"]; tmp = x["Right"]; x["SetRight", y]; 
  y["SetLeft", tmp]; x]

Now do it, and visualize the result:

RightRotate
&#10005

RightRotate[ds]["Visualization"]

The Asymptotic Superfunction

You’ve got a symbolic math expression and you want to figure out its rough value. If it’s a number you just use N to get a numerical approximation. But how do you get a symbolic approximation?

Ever since Version 1.0—and, in the history of math, ever since the 1600s—there’s been the idea of power series: find an essentially polynomial-like approximation to a function, as Series does. But not every mathematical expression can be reasonably approximated that way. It’s difficult math, but it’s very useful if one can make it work. We started introducing “asymptotic approximation” functions for specific cases (like integrals) in Version 11.3, but now in 12.1 we’re introducing the asymptotic superfunction Asymptotic.

Consider this inverse Laplace transform:

InverseLaplaceTransform
&#10005

InverseLaplaceTransform[1/(s Sqrt[s^3 + 1]), s, t]

There’s no exact symbolic solution for it. But there is an asymptotic approximation when t is close to 0:

Asymptotic
&#10005

Asymptotic[InverseLaplaceTransform[1/(s Sqrt[s^3 + 1]), s, t], t -> 0]

Sometimes it’s convenient to not even try to evaluate something exactly—but just to leave it inactive until you give it to Asymptotic:

Asymptotic
&#10005

Asymptotic[
 DSolveValue[Sin[x]^2 y''[x] + x  y[x] == 0, y[x], x], {x, 0, 5}]

Asymptotic deals with functions of continuous variables. In Version 12.1 there’s also DiscreteAsymptotic. Here we’re asking for the asymptotic behavior of the Prime function:

DiscreteAsymptotic
&#10005

DiscreteAsymptotic[Prime[n], n -> Infinity]

Or the factorial:

DiscreteAsymptotic
&#10005

DiscreteAsymptotic[n!, n -> Infinity]

We can ask for more terms if we want:

DiscreteAsymptotic
&#10005

DiscreteAsymptotic[n!, n -> Infinity, SeriesTermGoal -> 5]

Sometimes even quite simple functions can lead to quite exotic asymptotic approximations:

DiscreteAsymptotic
&#10005

DiscreteAsymptotic[BellB[n], n -> Infinity]

More Math, As Always

Math is big, and math is important. And for the Wolfram Language (which also means for Mathematica) we’re always pushing the frontiers of what’s computable in math.

One long-term story has to do with special functions. Back in Version 1.0 we already had 70 special functions. We covered univariate hypergeometric functions—adding the general pFq case in Version 3.0. Over the years we’ve gradually added a few other kinds of hypergeometric functions (as well as 250 other new kinds of special functions). Typical hypergeometric functions are solutions to differential equations with three regular singular points. But in Version 12.1 we’ve generalized that. And now we have Heun functions, that solve equations with four regular singular points. That might not sound like a big deal, but actually they’re quite a mathematical jungle—for example with 192 known special cases. And they’re very much in vogue now, because they show up in the mathematics of black holes, quantum mechanics and conformal field theory. And, yes, Heun functions have a lot of arguments:

Series
&#10005

Series[HeunG[a, q, \[Alpha], \[Beta], \[Gamma], \[Delta], z], {z, 0, 
  3}]

By the way, when we “support a special function” these days, there’s a lot we do. It’s not just a question of evaluating the function to arbitrary precision anywhere in the complex plane (though that’s often hard enough). We also need to be able to compute asymptotic approximations, simplifications, singularities, etc. And we have to make sure the function can get produced in the results of functions like Integrate, DSolve and Sum.

One of our consistent goals in dealing with superfunctions like DSolve is to make them “handbook complete”. To be sure that the algorithms we have—that are built to handle arbitrary cases—successfully cover as much as possible of the cases that appear anywhere in the literature, or in any handbook. Realistically, over the years, we’ve done very well on this. But in Version 12.1 we’ve made a new, big push, particularly for DSolve.

Here’s an example (oh, and, yes, it happens to need Heun functions):

DSolveValue
&#10005

DSolveValue[(d + c x + b x^2) y[x] + a x y'[x] + (-1 + x^2) y''[x] == 
  0, y[x], x]

There’s a famous book from the 1940s that’s usually just called Kamke, and that’s a huge collection of solutions to differential equations, some extremely exotic. Well, we’ll soon be able to do 100% of the (concrete) equations in this book (we’re still testing the last few…).

In Version 12.0 we introduced functions like ComplexPlot and ComplexPlot3D for plotting complex functions of complex variables. In Version 12.1 we now also have complex contour plotting. Here we’re getting two sets of contours—from the Abs and the Arg of a complex function:

ComplexContourPlot
&#10005

ComplexContourPlot[
 AbsArg[(z^2 - I)/(z^3 + I)], {z, -3 - 3 I, 3 + 3 I}, Contours -> 30]

Also new in 12.1 is ComplexRegionPlot, which effectively solves equations and inequalities in the complex plane. Like here’s the (very much branch-cut-informed) solution to an equation whose analog would be trivial over the reals:

ComplexRegionPlot
&#10005

ComplexRegionPlot[Sqrt[z^(2 + 2 I)] == z^(1 + I), {z, 10}]

In a very different area of mathematics, another new function in Version 12.1 is CategoricalDistribution. We introduced the idea of symbolic representations of statistical distributions back in Version 6—with things like NormalDistribution and PoissonDistribution—and the idea has been extremely successful. But so far all our distributions have been distributions over numbers. In 12.1 we have our first distribution where the possible outcomes don’t need to be numbers.

Here’s a distribution where there are outcomes x, y, z with the specified probabilities:

dist = CategoricalDistribution
&#10005

dist = CategoricalDistribution[{x, y, z}, {.1, .2, .7}]

Given this distribution, one can do things like generate random variates:

RandomVariate
&#10005

RandomVariate[dist, 10]

Here’s a 3D categorical distribution:

dist = CategoricalDistribution
&#10005

dist = CategoricalDistribution[{{"A", "B", "C"}, {"D", "E"}, {"X", 
    "Y"}}, {{{2, 4}, {2, 1}}, {{2, 2}, {3, 2}}, {{4, 3}, {1, 3}}}]

Now we can work out the PDF of the distribution, asking in this case what the probability to get A, D, Y is:

PDF
&#10005

PDF[dist, {"A", "D", "Y"}]

By the way, if you want to “see the distribution” you can either click the + on the summary box, or explicitly use Information:

Information
&#10005

Information[dist, "ProbabilityTable"]

There are lots of uses of CategoricalDistribution, for example in machine learning. Here we’re creating a classifier:

cf = Classify
&#10005

cf = Classify[{1, 2, 3, 4} -> {a, a, b, b}]

If we just give it input 2.3, the classifier will give its best guess for the corresponding output:

cf
&#10005

cf[2.3]

But in 12.1 we can also ask for the distribution—and the result is a CategoricalDistribution:

cf
&#10005

cf[2.3, "Distribution"]

Information
&#10005

Information[%, "ProbabilityTable"]

The Leading Edge of Optimization

In Version 12.0 we introduced industrial-scale convex optimization. We covered most of the usual problem classes (like linear, semidefinite, quadratic and conic). But there was one straggler: geometric optimization. And now we’re adding that for 12.1:

GeometricOptimization
&#10005

GeometricOptimization[\[Pi] r (r + Sqrt[h^2 + r^2]), {1 <=  \[Pi]/
    3 h r^2 }, {h, r}]

GeometricOptimization
&#10005

GeometricOptimization[
 1/(h w d), {h <= 2 w, d <= 2 w, h*w + h*d <= 50, 2 w*d <= 20}, {h, w,
   d}]

You can solve all sorts of practical problems with GeometricOptimization—with thousands of variables if need be. As one example, consider laying out rectangles of certain sizes with a certain partial ordering in x and y. To specify the problem, you give a bunch of inequalities:

With
&#10005

With[{c1 = 0.25, c2 = 0.618}, 
  ineqs = {{c1 + w[1] + x[1] <= x[2], c1 + w[1] + x[1] <= x[3], 
     c1 + w[1] + x[1] <= x[4], c1 + w[1] + x[1] <= x[5], 
     c1 + w[1] + x[1] <= x[6], c1 + w[1] + x[1] <= x[7], 
     c1 + w[2] + x[2] <= x[3], c1 + w[4] + x[4] <= x[5], 
     c1 + w[2] + x[2] <= x[3], c1 + w[2] + x[2] <= x[5], 
     c1 + w[2] + x[2] <= x[7], c1 + w[4] + x[4] <= x[3], 
     c1 + w[4] + x[4] <= x[5], c1 + w[4] + x[4] <= x[7], 
     c1 + w[6] + x[6] <= x[5], c1 + w[8] + x[8] <= x[4], 
     c1 + w[9] + x[9] <= x[4], c1 + w[10] + x[10] <= x[4], 
     c1 + w[10] + x[10] <= x[6], c1 + w[6] + x[6] <= x[7], 
     c1 + w[8] + x[8] <= x[9], c1 + w[8] + x[8] <= x[10], x[1] >= 0, 
     x[8] >= 0, w[3] + x[3] <= \[ScriptW], w[5] + x[5] <= \[ScriptW], 
     w[7] + x[7] <= \[ScriptW]}, {c1 + h[1] + y[1] <= y[6], 
     c1 + h[1] + y[1] <= y[7], c1 + h[1] + y[1] <= y[8], 
     c1 + h[1] + y[1] <= y[9], c1 + h[1] + y[1] <= y[10], 
     c1 + h[2] + y[2] <= y[4], c1 + h[2] + y[2] <= y[9], 
     c1 + h[4] + y[4] <= y[6], c1 + h[3] + y[3] <= y[5], 
     c1 + h[5] + y[5] <= y[7], c1 + h[9] + y[9] <= y[6], 
     c1 + h[9] + y[9] <= y[10], y[1] >= 0, y[2] >= 0, y[3] >= 0, 
     h[6] + y[6] <= \[ScriptH], h[7] + y[7] <= \[ScriptH], 
     h[8] + y[8] <= \[ScriptH], 
     h[10] + y[10] <= \[ScriptH]}, {c2 <= h[1]/w[1] <= (1 + c2), 
     c2 <= h[2]/w[2] <= (1 + c2), c2 <= h[3]/w[3] <= (1 + c2), 
     c2 <= h[4]/w[4] <= (1 + c2), c2 <= h[5]/w[5] <= (1 + c2), 
     c2 <= h[6]/w[6] <= (1 + c2), c2 <= h[7]/w[7] <= (1 + c2), 
     c2 <= h[8]/w[8] <= (1 + c2), c2 <= h[9]/w[9] <= (1 + c2), 
     c2 <= h[10]/w[10] <= (1 + c2)}, {h[1] w[1] == 1, h[2] w[2] == 2, 
     h[3] w[3] == 3, h[4] w[4] == 4, h[5] w[5] == 5, h[6] w[6] == 6, 
     h[7] w[7] == 7, h[8] w[8] == 8, h[9] w[9] == 9, 
     h[10] w[10] == 10}}];

It then takes only about a second to generate an optimal solution:

GeometricOptimization

In optimization, there are usually two broad types: continuous and discrete. Our convex optimization functions in 12.0 handled the case of continuous variables. But a major new feature—and innovation—in 12.1 is the addition of support for discrete (i.e. integer) variables, and for mixed discrete and continuous variables.

Here’s a very simple example:

QuadraticOptimization
&#10005

QuadraticOptimization[
 2 x^2 + 20 y^2 + 6 x y + 5 x, -x + y >= 2, {x \[Element] Integers, 
  y \[Element] Reals}]

If x wasn’t constrained to be an integer, the result would be different:

QuadraticOptimization
&#10005

QuadraticOptimization[
 2 x^2 + 20 y^2 + 6 x y + 5 x, -x + y >= 2, {x, y}]

But—as with our other optimization capabilities—this can be scaled up, though the combinatorial optimization that’s involved is fundamentally more computationally difficult (and for example it’s often NP-complete). And actually the only reason we can do large-scale problems of this kind at all is that we’ve implemented a novel iteration-based technique that successfully unlocks mixed convex optimization.

Cracking the Vector-Plotting Problem

I’ve been trying to make good vector plots for about 40 years, but in the past it just never worked. If the vectors got too short, you couldn’t see their direction—and if you made them longer they crashed into each other. But particularly after our success in Version 12.0 in cracking the ComplexPlot problem (which had also been languishing for a long time) we decided for Version 12.1 to try to solve the vector-plotting problem once and for all. And, I’m happy to say, we seem to have been able to do that.

So now, you can just ask VectorPlot (and all sorts of related functions) to make a vector plot, and you’ll automatically get something that’s a good representation of your vector field:

VectorPlot
&#10005

VectorPlot[{2 x^2 - y^2, 3 x y}, {x, -5, 5}, {y, -5, 5}]

VectorPlot
&#10005

VectorPlot[{2 x^2 - y^2, 3 x y}, {x, -5, 5}, {y, -5, 5}, 
 VectorPoints -> 30]

What’s the trick? It’s basically about placing vectors on a hexagonal grid so they’re packed better, and are visually more uniform. (You can also make other choices if you want to.) And then it’s about using appropriately scaled color to represent vector magnitudes.

There are all sorts of other challenges too. Like being able to draw vectors in a region:

VectorPlot
&#10005

VectorPlot[{2 x^2 - y^2, 3 x y}, {x, y} \[Element] Disk[{0, 0}, 3]]

And putting together our complex-number-plotting capabilities with our new vector plotting, we also in 12.1 have ComplexVectorPlot:

ComplexVectorPlot
&#10005

ComplexVectorPlot[z^ Log[z], {z, 6}, PlotLegends -> Automatic]

Cross-Hatching & All That

Before there were gray scales, there were things like cross-hatching. Look at a book from a century ago (or less), and you’ll see all sorts of diagrams elegantly drawn with things like cross-hatching. Well, now we can do that too.

Graphics
&#10005

Graphics[Style[RegularPolygon[5], HatchFilling[]]]

Plot
&#10005

Plot[{Sin[x], Cos[x]}, {x, 0, 10}, Filling -> Axis, 
 FillingStyle -> HatchFilling[]]

Of course, everything is computable:

Graphics
&#10005

Graphics[Table[
  Style[Disk[RandomReal[10, 2]], 
   HatchFilling[RandomReal[{0, 2 \[Pi]}]]], 50]]

We also have an important generalization of cross-hatching: PatternFilling. Here are examples with named patterns:

Graphics
&#10005

Graphics[Style[Disk[], PatternFilling["DiamondBox"]]]

Graphics
&#10005

Graphics[Style[Disk[], PatternFilling[{"Checkerboard", Red, Black}]]]

You can use any image as a pattern too:

GeoGraphics
&#10005

GeoGraphics[
 Style[Polygon[Entity["Country", "UnitedStates"]], 
  PatternFilling[CloudGet["https://wolfr.am/L9r9AL5O"], 270]]]

Version 12.1 also has what one can think of as 3D generalizations of these kinds of textures:

Graphics3D
&#10005

Graphics3D[Style[Icosahedron[], HatchShading[]]]

It looks pretty good even in black and white:

Graphics3D
&#10005

Graphics3D[Style[Icosahedron[], HatchShading[]], Lighting -> "Accent"]

There’s stipple shading too:

Plot3D
&#10005

Plot3D[Exp[-(x^2 + y^2)], {x, -2, 2}, {y, -2, 2}, 
 PlotStyle -> {White, StippleShading[]}, Mesh -> None, 
 Lighting -> "Accent"]

The Beginning of Computational Topology

In the past few versions, we’ve introduced deeper and deeper coverage of computational geometry. In coming versions, we’re going to be covering more and more of computational topology too. Things like EulerCharacteristic and PolyhedronGenus were already in Version 12.0. In Version 12.1 we’re introducing several powerful functions for dealing with the topology of simplicial complexes, of the kind that are for example used in representing meshes.

This makes a connectivity graph for the dimension-0 components of a dodecahedron, i.e. its corners:

MeshConnectivityGraph
&#10005

MeshConnectivityGraph[Dodecahedron[], 0]

Here’s the corresponding result for the connectivity of lines to lines in the dodecahedron:

MeshConnectivityGraph
&#10005

MeshConnectivityGraph[Dodecahedron[], 1]

And here’s the connectivity of corners to faces:

MeshConnectivityGraph
&#10005

MeshConnectivityGraph[Dodecahedron[], {0, 2}]

It’s a very general function. Here are the graphs for different dimensional cells of a Menger sponge:

Table
&#10005

Table[MeshConnectivityGraph[MengerMesh[2, 3], d], {d, 0, 3}]

Given a mesh, it’s often useful to do what amounts to a topological search. For example, here’s a random Voronoi mesh:

vm = VoronoiMesh
&#10005

vm = VoronoiMesh[RandomReal[1, {200, 2}]]

Here are the 10 closest mesh cells to position {.5, .5} (the 2 before each index indicates that these are dimension-2 cells):

NearestMeshCells
&#10005

NearestMeshCells[vm, {.5, .5}, 10]

Now highlight these cells:

HighlightMesh
&#10005

HighlightMesh[vm, %]

Tabular Data: Computing & Editing

Dataset has been a big success in the six years since it was first introduced in Version 10.0. Version 12.1 has the beginning of a major project to upgrade and extend the functionality of Dataset.

The first thing is something you might not notice, because now it “just works”. When you see a Dataset in a notebook, you’re just seeing its displayed form. But often there is lots of additional data that you’d get to by scrolling or drilling down. In Version 12.1 Dataset automatically stores that additional data directly in the notebook (at least up to a size specified by $NotebookInlineStorageLimit) so when you open the notebook again later, the Dataset is all there, and all ready to compute with.

In Version 12.1 a lot has been done with the formatting of Dataset. Something basic is that you can now say how many rows and columns to display by default:

Dataset
&#10005

Dataset[CloudGet["https://wolfr.am/L9o1Pb7V"], MaxItems -> {4, 3}]

In Version 12.1 there are many options that allow detailed programmatic control over the appearance of a display Dataset. Here’s a simple example:

Dataset
&#10005

Dataset[CloudGet["https://wolfr.am/L9o1Pb7V"],
 MaxItems -> 5,
 HeaderBackground -> LightGreen,
 Background -> {{{LightBlue, LightOrange}}},
 ItemDisplayFunction -> {"sex" -> (If[# === 
        "female", \[Venus], \[Mars]] &)}
 ]

A major new feature is “right-click” interactivity (which works on rows, columns and individual items):

Hide column

Sort

Dataset is a powerful construct for displaying and computing with tabular data of any depth. But sometimes you just want to enter or edit simple 2D tabular data—and the user interface requirements for this are rather different from those for Dataset. So in Version 12.1 we’re introducing a new experimental function called TableView, which is a user interface for entering—and viewing—purely two-dimensional tabular data:

TableView
&#10005

TableView[{{5, 6}, {7, 3}}]

Like a typical spreadsheet, TableView has fixed-width columns that you can manually adjust. It can efficiently handle large-scale data (think millions of items). The items can (by default) be either numbers or strings.

When you’ve finished editing a TableView, you can just ask for Normal and you’ll get lists of data out. (You can also feed it directly into a Dataset.) Like in a typical spreadsheet, TableView lets you put data wherever you want; if there’s a “hole”, it’ll show up as Null in lists you generate.

TableView is actually a dynamic control. So, for example, with TableView[Dynamic[x]] you can edit a TableView, and have its payload automatically be the value of some variable x. (And, yes, all of this is done efficiently, with minimal updates being made to the expression representing the value of x.)

GANs, BERT, GPT-2, ONNX, …: The Latest in Machine Learning

Machine learning is all the rage these days. Of course, we were involved with it even a very long time ago. We introduced the first versions of our flagship highly automated machine-learning functions Classify and Predict back in 2014, and we introduced our first explicitly neural-net-based functionImageIdentify—in early 2015.

And in the years since then we’ve built a very strong system for machine learning in general, and for neural nets in particular. Several things about it stand out. First, we’ve emphasized high automation—using machine learning to automate machine learning wherever possible, so that even non-experts can immediately make use of leading-edge capabilities. The second big thing is that we’ve been curating neural nets, just like we curate so many other things. So that means that we have pretrained classifiers and predictors and feature extractors that you can immediately and seamlessly use. And the other big thing is that our whole neural net system is symbolic—in the sense that neural nets are specified as computable, symbolic constructs that can be programmatically manipulated, visualized, etc.

In Version 12.1 we’ve continued our leading-edge development in machine learning. There are 25 new types of neural nets in our Wolfram Neural Net Repository, including ones like BERT and GPT-2. And the way things are set up, it’s immediate to use any of these nets. (Also, in Version 12.1 there’s support for the new ONNX neural net specification standard, which makes it easier to import the very latest neural nets that are being published in almost any format.)

This gets the symbolic representation of GPT-2 from our repository:

gpt2 = NetModel
&#10005

gpt2 = NetModel["GPT-2 Transformer Trained on WebText Data", 
  "Task" -> "LanguageModeling"]

If you want to see what’s inside, just click—and keep clicking to drill down into more and more details:

See what's inside—click to enlarge

Now you can immediately use GPT-2, for example progressively generating a random piece of text one token at a time:

Nest
&#10005

Nest[StringJoin[#, 
   gpt2[#, "RandomSample"]] &, "Stephen Wolfram is", 20]

Hmmmm. I wonder what that was trained on….

By the way, people sometimes talk about machine learning and neural nets as being in opposition to traditional programming language code. And in a way, that’s correct. A neural net just learns from real-world examples or experience, whereas a traditional programming language is about giving a precise abstract specification of what in detail a computer should do. We’re in a wonderful position with the Wolfram Language, because what we have is something that already spans these worlds: we have a full-scale computational language that takes advantage of all those precise computation capabilities, yet can meaningfully represent and compute about things in the real world.

So it’s been very satisfying in the past few years to see how modern machine learning can be integrated into the Wolfram Language. We’ve been particularly interested in new superfunctions—like Predict, Classify, AnomalyDetection, LearnDistribution and SynthesizeMissingValues—that do “symbolically specified” operations, but do them using neural nets and modern machine learning.

In Version 12.1 we’re continuing in this direction, and moving towards superfunctions that use more elaborate neural net workflows, like GANs. In particular, Version 12.1 introduces the symbolic NetGANOperator, as well as the new option TrainingUpdateSchedule. And it turns out these are the only things we had to change to allow our general NetTrain function to work with GANs.

A typical GAN setup is quite complicated (and that’s why we’re working on devising superfunctions that conveniently deliver applications of GANs). But here’s an example of a GAN in action in Version 12.1:

NetGANModel

The Calculus of Annotations

How do you add metadata annotations to something you’re computing with? For Version 12.1 we’ve begun rolling out a general framework for making annotations—and then computing with and from them.

Let’s talk first about the example of graphs. You can have both annotations that you can immediately “see in the graph” (like vertex colors), and ones that you can’t (like edge weights).

Here’s an example where we’re explicitly constructing a graph with annotations:

Graph
&#10005

Graph[{Annotation[1 -> 2, EdgeStyle -> Red], 
  Annotation[2 -> 1, EdgeStyle -> Blue]}]

Here we’re annotating the vertices:

Graph
&#10005

Graph[{Annotation[x, VertexStyle -> Red], 
  Annotation[y, VertexStyle -> Blue]}, {x -> y, y -> x, y -> y}, 
 VertexSize -> .2]

AnnotationValue lets you query values of annotations:

AnnotationValue
&#10005

AnnotationValue[{%, 
  x}, VertexStyle]

Something important about AnnotationValue is that you can assign it. Set g to be the graph:

g = CloudGet
&#10005

g = CloudGet["https://wolfr.am/L9rgvixl"];

Now do an assignment to an annotation value:

AnnotationValue
&#10005

AnnotationValue[{g, x}, VertexStyle] = Green

Now the graph has changed:

g
&#10005

g

You can always delete the annotation if you want:

AnnotationDelete
&#10005

AnnotationDelete[{g, x}, VertexStyle]

If you don’t want to permanently modify a graph, you can just use Annotate to produce a new graph with annotations added (3 and 5 are names of vertices):

Annotate
&#10005

Annotate[{CloudGet["https://wolfr.am/L9rpqPJ0"], {3, 5}}, 
 VertexSize -> .3]

Some annotations are important for actual computations on graphs. An example is edge weighting. This puts edge-weight annotations into a graph—though by default they don’t get displayed:

Graph
&#10005

Graph[Catenate[
  Table[Annotation[i -> j, EdgeWeight -> GCD[i, j]], {i, 5}, {j, 5}]]]

This displays the edge weights:

Graph
&#10005

Graph[%, EdgeLabels -> "EdgeWeight"]

And this actually does a computation that includes the weights:

WeightedAdjacencyMatrix
&#10005

WeightedAdjacencyMatrix[
  CloudGet["https://wolfr.am/L9rtJdd9"]] // MatrixForm

You can use your own custom annotations too:

Graph
&#10005

Graph[{Annotation[x, "age" -> 10], 
  Annotation[y, "age" -> 20]}, {x -> y, y -> x, y -> y}]

This retrieves the value of the annotation:

AnnotationValue
&#10005

AnnotationValue[{CloudGet["https://wolfr.am/L9rx8Mxe"], x}, "age"]

Annotations are ultimately stored in an AnnotationRules option:

Options
&#10005

Options[CloudGet["https://wolfr.am/L9rx8Mxe"], AnnotationRules]

You can always give all annotations as a setting for this option.

A major complexity with annotations is when in a computation they should be preserved—or combined—and when they should be dropped. We always try to keep annotations whenever it makes sense:

TransitiveReductionGraph
&#10005

TransitiveReductionGraph[CloudGet["https://wolfr.am/L9rBdIZt"]]

Annotations are something quite general, that apply not only to graphs, but to an increasing number of other constructs too. But in Version 12.1 we’ve added something else that’s specific to graphs, and that handles a complicated case there. It has to do with multigraphs, i.e. graphs with multiple edges between the same vertices. Take the graph:

Graph
&#10005

Graph[{1 -> 2, 1 -> 2}]

How do you distinguish these two edges? It’s not a question of annotation; you actually want these edges to be distinct, just like the vertices in the graph are distinct. Well, in Version 12.1, you can give names (or “tags”) to edges, just like you give names to vertices:

EdgeTaggedGraph
&#10005

EdgeTaggedGraph[{1 -> 2, 1 -> 2} -> {a, b}]

In the edge list for this graph the edges are shown “tagged”:

EdgeList
&#10005

EdgeList[%]

The tags are part of the edge specification:

InputForm
&#10005

InputForm[%]

But back to annotations. Another kind of structure that can be annotated just like graphs is a mesh. This is saying to annotate dimension-0 boundary cells with a style:

Annotate
&#10005

Annotate[{MengerMesh[2], {0, "Boundary"}}, MeshCellStyle -> Red]

A completely different kind of structure that can also use annotations is audio. This annotates an Audio object with information about where there’s voice activity in the audio:

AudioAnnotate
&#10005

AudioAnnotate[ExampleData[{"Audio", "MaleVoice"}], "Voiced"]

This retrieves the value of the annotation:

AnnotationValue
&#10005

AnnotationValue[%, "Voiced"] // TimelinePlot

We’ll be rolling out annotations in lots of other things too. One that’s coming is images. But in preparation for that, in Version 12.1 we’ve added some new capabilities to HighlightImage.

Use machine learning to find what’s in the picture:

ImageBoundingBoxes
&#10005

ImageBoundingBoxes[CloudGet["https://wolfr.am/L9qz1zu4"]]

Now HighlightImage can use the annotation information:

HighlightImage
&#10005

HighlightImage[CloudGet["https://wolfr.am/L9qz1zu4"], %]

Language Innovations & Extensions

Nothing has been a big success:

{a, b, Nothing, c, Nothing}
&#10005

{a, b, Nothing, c, Nothing}

Before Nothing, you always had to poke at a list from the outside to get elements in it deleted. But Nothing is a symbolic way of specifying deletion that “works from the inside”.

Pretty much as soon as we’d invented Nothing, we realized we also wanted another piece of functionality: a symbolic object that would somehow automatically disgorge its contents into a list. People had been using idioms like Sequence@@… to do this. But Sequence is a slippery construct, and this idiom is fragile and ugly.

The functionality of our auto-inserter was easy to define. But what were we going to call it? For several years this very useful piece of functionality languished for want of a name. It came up several times in our livestreamed design reviews. Every time we would discuss it for a while—and often our viewers would offer good suggestions. But we were never happy with the name.

Finally, though, we decided we had to solve the problem. It was a painful naming process, culminating in a 90-minute livestream whose net effect was a change in one letter in the name. But in the end, we’re pretty happy with the name: Splice. Splice is a splice, like for film, or DNA—and it’s something that gets inserted. So now, as of Version 12.1 we have it:

{a, b, Splice
&#10005

{a, b, Splice[{x, y, z}], c, d}

Of course, the more common case is something like:

{a, b, x, c, d} /. x -> Splice
&#10005

{a, b, x, c, d} /. x -> Splice[{p, q, r}]

There’s a lot of strange (and potentially buggy) Flatten operations that are going to be avoided by Splice.

One of the things we’re always trying to do in developing Wolfram Language is to identify important “lumps of computation” that we can conveniently encapsulate into functions (and where we can give those functions good names!). In Version 12.1 there’s a family of new functions that handle computations around subsets of elements in lists:

SubsetCases
&#10005

SubsetCases[{a, b, a, b, a, c}, {x_, y_, x_}, Overlaps -> True]

I must have written special cases of these functions a zillion times. But now we’ve got general functions that anyone can just use. These functions come up in lots of places. And actually we first implemented general versions of them in connection with semantic-query-type computations.

But on the theory that any sufficiently well-designed function eventually gets a very wide range of uses, I can report that I’ve recently found a most unexpected but spectacular use for SubsetReplace in the context of fundamental physics. But much more on that in a little while…

Talking about physics brings me to something else in 12.1: new functions for handling time. DateInterval now provides a symbolic representation for an interval of time. And there’s an interesting algebra of ordering that needs to be defined for it. Which includes the need for the symbols InfinitePast and InfiniteFuture:

Today < InfiniteFuture
&#10005

Today < InfiniteFuture

Functional Programming Adverbs & More

We’re always working to make the Wolfram Language easier and more elegant to use, and Version 12.1 contains the latest in an idea we’ve been developing for symbolic functional programming. If you think of a built-in function as a verb, what we’re adding are adverbs: constructs that modify the operation of the verb.

A first example is OperatorApplied. Here’s the basic version of what it does:

OperatorApplied
&#10005

OperatorApplied[f][x][y]

Why is this useful? Many functions have “operator forms”. For example, instead of

Select
&#10005

Select[{1, 2, 3, 4}, PrimeQ]

you can say

Select
&#10005

Select[PrimeQ][{1, 2, 3, 4}]

and that means you can just “pick up” the modified function and do things with it:

Map
&#10005

Map[Select[PrimeQ], {{6, 7, 8}, {11, 12, 13, 14}}]

or (using the operator form of Map):

Map
&#10005

Map[Select[PrimeQ]][{{6, 7, 8}, {11, 12, 13, 14}}]

OK, so what does OperatorApplied do? Basically it lets you create an operator form of any function.

Let’s say you have a function f that—like Select—usually takes two arguments. Well, then

OperatorApplied
&#10005

OperatorApplied[f][y]

is a function that takes a single argument, and forms f[x,y] from it:

OperatorApplied
&#10005

OperatorApplied[f][y][x]

OperatorApplied allows for some elegant programming, and often lets one avoid having to insert pure functions with # and & etc.

At first, OperatorApplied may seem like a very abstract “higher-order” construct. But it quickly becomes natural, and is particularly convenient when, for example, one has to provide a function for something—like as a setting for an option, the first argument to Outer, and so on.

By default, OperatorApplied[f][y] creates an operator form to be applied to an expression which will become the first argument of f. There’s a generalized form in which one specifies exactly how arguments should be knitted together, as in:

OperatorApplied
&#10005

OperatorApplied[f, 4 -> {3, 2, 1, 4}][x][y][z][u][v]

CurryApplied is in a sense a “purer” variant of OperatorApplied, in which one specifies up front the number of arguments to expect, and then (unless specified otherwise) these arguments are always used in the order they appear. So, for example, this makes a function that expects two arguments:

CurryApplied
&#10005

CurryApplied[f, 2][x][y]

CurryApplied
&#10005

CurryApplied[f, 2][x][y][z][u][v]

Needless to say—given that it’s a purer construct—CurryApplied is itself curryable: it has an operator form in which one just gives the number of arguments to expect:

CurryApplied
&#10005

CurryApplied[2][f][x][y][z][u][v]

In Version 12.1, there’s another convenient adverb that we’ve introduced: ReverseApplied. As its name suggests, it specifies that a function should be applied in a reverse way:

ReverseApplied
&#10005

ReverseApplied[f][x, y, z]

This is particularly convenient when you’re doing things like specifying sorting functions:

Sort
&#10005

Sort[{5, 6, 1, 7, 3, 7, 3}, ReverseApplied[NumericalOrder]]

All of this symbolic functional programming emphasizes the importance of thinking about symbolic expressions structurally. And one new function to help with this is ExpressionGraph, which turns the tree structure (think TreeForm) of an expression into an actual graph that can be manipulated:

ExpressionGraph
&#10005

ExpressionGraph[{{a, b}, {c, d, e}}]

ExpressionGraph
&#10005

ExpressionGraph[{{a, b}, {c, d, e}}, VertexLabels -> Automatic]

While we’re talking about the niceties of programming, one additional feature of Version 12.1 is TimeRemaining, which, as the name suggests, tells you how much time you have left in a computation before a time constraint hits you. So, for example, here TimeConstrained said the computation should be allocated 5 seconds. But after the Pause used about 1 second, there was a little less than 4 seconds remaining:

TimeConstrained
&#10005

TimeConstrained[Pause[1]; TimeRemaining[], 5]

If you’re writing sophisticated code, it’s very useful to be able to find out how much “temporal headroom” you have, to see for example whether it’s worth trying a different strategy, etc.

Now We Can Prove That Socrates Is Mortal

In using the Wolfram Language the emphasis is usually on what the result of a computation is, not why it is that. But in Version 11.3 we introduced FindEquationalProof, which generates proofs of assertions given axioms.

AxiomaticTheory provides a collection of standard axiom systems. One of them is an axiom system for group theory:

axioms = AxiomaticTheory
&#10005

axioms = AxiomaticTheory[{"GroupAxioms", "Multiplication" -> p, 
   "Identity" -> e}]

This axiom system is sufficient to allow proofs of general results about groups. For example, we can show that—even though the axioms only asserted that e is a right identity—it is possible to prove from the axioms that it is also a left identity:

FindEquationalProof
&#10005

FindEquationalProof[p[e, x] == x, axioms]

This dataset shows the actual steps in our automatically generated proof:

Dataset
&#10005

Dataset[%["ProofDataset"], MaxItems -> {6, 1}]

But if you want to prove a result not about groups in general, but about a specific finite group, then you need to add to the axioms the particular defining relations for your group. You can get these from FiniteGroupData—which has been much extended in 12.1. Here are the axioms for the quaternion group, given in a default notation:

FiniteGroupData
&#10005

FiniteGroupData["Quaternion", "DefiningRelations"]

To use these axioms in FindEquationalProof, we need to merge their notation with the notation we use for the underlying group axioms. In Version 12.1, you can do this directly in AxiomaticTheory:

AxiomaticTheory
&#10005

AxiomaticTheory[{"GroupAxioms", "Quaternion", "Multiplication" -> p, 
  "Identity" -> e}]

But to use the most common notation for quaternions, we have to specify a little more:

AxiomaticTheory
&#10005

AxiomaticTheory[{"GroupAxioms", 
  "Quaternion", <|"Multiplication" -> p, "Inverse" -> inv, 
   "Identity" -> e, "Generators" -> {i, j}|>}]

But now we can prove theorems about the quaternions. This generates a 54-step proof that the 4th power of the generator we have called i is the identity:

FindEquationalProof
&#10005

FindEquationalProof[p[i, p[i, p[i, i]]] == e, %]

In addition to doing mathematical proofs, we can now use FindEquationalProof in Version 12.1 to do general proofs with arbitrary predicates (or, more specifically, general first-order logic). Here’s a famous example of a syllogism, based on the predicates mortal and man. FindEquationalProof gives a proof of the assertion that Socrates is mortal:

FindEquationalProof
&#10005

FindEquationalProof[
 mortal[socrates], {ForAll[x, Implies[man[x], mortal[x]]], 
  man[socrates]}]

I think it’s pretty neat that this is possible, but it must be admitted that the actual proof generated (which is 53 steps long in this case) is a bit hard to read, not least because it involves conversion to equational logic.

Still, FindEquationalProof can successfully automate lots of proofs. Here it’s solving a logic puzzle given by Lewis Carroll, that establishes (here with a 100-step proof) that babies cannot manage crocodiles:

FindEquationalProof
&#10005

FindEquationalProof[
 Not[Exists[x, And[baby[x], manageCrocodile[x]]]], {ForAll[x, 
   Implies[baby[x], Not[logical[x]]]], 
  ForAll[x, Implies[manageCrocodile[x], Not[despised[x]]]], 
  ForAll[x, Implies[Not[logical[x]], despised[x]]]}]

Geo-Everything

The Wolfram Language knows about many things. One of them is geography. And in Version 12.1 we’ve substantially updated and expanded our sources of geographic data (as well as upgrading our server-based algorithms). So, for example, the level of detail available in typical maps has increased substantially:

GeographicData map

For many years now we’ve had outstanding geodetic computation in the Wolfram Language. And we also have excellent computational geometry for doing all sorts of computations on regions in Euclidean space. But of course the Earth is not flat, and one of the achievements of Version 12.1 is to bring our region-computation capabilities to the geo domain, handling non-flat regions.

It’s an interesting exercise in geometry. We have things like the polygon of the United States defined in geo coordinates—as a lat-long region on the Earth. But to use our computational geometry capabilities we need to make it something purely Euclidean. But we can do that by using our geodesy capabilities to embed it in full 3D space.

So now we can just compute the centroid of the region that is the US:

RegionCentroid
&#10005

RegionCentroid[Polygon[Entity["Country", "UnitedStates"]]]

That third element in the geo position is a depth (in meters), and reflects the curvature of the US polygon. And, actually, we can see this directly too:

DiscretizeRegion
&#10005

DiscretizeRegion[Entity["Country", "UnitedStates"]["Polygon"]]

This is a 3D object, so we can rotate it to see the curvature more clearly:

alt

We can also work the other way around: taking geo regions and projecting them onto a flat map, then computing with them. One knows that Greenland looks very different sizes with different map projections. Here’s its “map area” in the Mercator projection (in units of degrees-squared):

Area
&#10005

Area[GeoGridPosition[Entity["Country", "Greenland"]["Polygon"], 
  "Mercator"]]

But here it is (also in degrees-squared) in an area-preserving projection:

Area
&#10005

Area[GeoGridPosition[Entity["Country", "Greenland"]["Polygon"], 
  "CylindricalEqualArea"]]

And as part of the effort to make “geo everything”, Version 12.1 also includes GeoDensityPlot and GeoContourPlot.

Advance of the Knowledgebase

Every second of every day there is new data flowing into the Wolfram Knowledgebase that powers Wolfram|Alpha and Wolfram Language. Needless to say, it takes a lot of effort to keep everything as correct and up to date as possible. But beyond this, we continue to push to cover more and more domains, with the goal of making as many things in the world as possible computable.

I mentioned earlier in this piece how we’re extending our computational knowledge by curating one particular new domain: different types of data structures. But we’ve been covering a lot of different new areas as well. I was trying to think of something as different from data structures as possible to use as an example. I think we have one in Version 12.1: goat breeds. As people who’ve watched our livestreamed design reviews have commented, I tend to use (with a thought of the Babylonian astrologers who in a sense originated what is now our scientific enterprise) “entrails of the goat” as a metaphor for details that I don’t think should be exposed to users. But this is not why we have goats in Version 12.1.

For nearly a decade we’ve had some coverage of a few million species. We’ve gradually been deepening this coverage, essentially mining the natural history literature, where the most recent “result” on the number of teeth that a particular species of snail has might be from sometime in the 1800s. But we’ve also had a project to cover at much greater depth those species—and subspecies—of particular relevance to our primary species of users (i.e. us humans). And so it is that in Version 12.1 we’ve added coverage of (among many other things) breeds of goats:

Entity
&#10005

Entity["GoatBreed", "OberhasliGoat"]["Image"]

EntityList
&#10005

EntityList[
 EntityClass["GoatBreed", "Origin" -> Entity["Country", "Spain"]]]

It may seem a long way from the origins of the Wolfram Language and Mathematica in the domain of mathematical and technical computing, but one of our great realizations over the past thirty years is just how much in the world can be put in computable form. One example of an area that we’ve been covering at great depth—and with excellent results—is food. We’ve already got coverage of hundreds of thousands of foods—packaged, regional, and as-you’d-see-it-on-menu. In Version 12.1 we’ve added for example computable data about cooking times (and temperatures, etc.):

Entity
&#10005

Entity["FoodType", "Potato"][
 EntityProperty["FoodType", "ApproximateCookingTimes"]]

ExternalIdentifier, Wikidata & More

Books have ISBNs. Chemicals have CAS numbers. Academic papers have DOIs. Movies have ISANs. The world is full of standardized identifiers. And in Version 12.1 we’ve introduced the new symbolic construct ExternalIdentifier as a way to refer to external things that have identifiers—and to link them up, both among themselves, and to the entities and entity types that we have built into the Wolfram Language.

So, for example, here’s how my magnum opus shows up in ISBN space:

ExternalIdentifier
&#10005

ExternalIdentifier["ISBN10", "1-57955-008-8"]

Right now we support 46 types of external identifiers, and our coverage will grow broader and deeper in the coming years. One particularly nice example that we’re already covering in some depth is Wikidata identifiers. This leverages both the structure of our built-in knowledgebase, and the work that we’ve done in areas like SPARQL support.

Let’s find our symbolic representation for me:

CloudGet
&#10005

\!\(\*NamespaceBox["LinguisticAssistant", 
    DynamicModuleBox[{Typeset`query$$ = "stephen wolfram", 
      Typeset`boxes$$ = 
       TemplateBox[{"\"Stephen Wolfram\"", 
         RowBox[{"Entity", "[", 
           RowBox[{"\"Person\"", ",", "\"StephenWolfram::j276d\""}], 
           "]"}], "\"Entity[\\\"Person\\\", \\\"StephenWolfram::j276d\
\\\"]\"", "\"person\""}, "Entity"], Typeset`allassumptions$$ = {}, 
      Typeset`assumptions$$ = {}, Typeset`open$$ = {1}, 
      Typeset`querystate$$ = {"Online" -> True, "Allowed" -> True, 
        "mparse.jsp" -> 0.488214`6.140155222562331, 
        "Messages" -> {}}}, 
     DynamicBox[
      ToBoxes[AlphaIntegration`LinguisticAssistantBoxes["", 4, 
        Automatic, Dynamic[Typeset`query$$], Dynamic[Typeset`boxes$$],
         Dynamic[Typeset`allassumptions$$], 
        Dynamic[Typeset`assumptions$$], Dynamic[Typeset`open$$], 
        Dynamic[Typeset`querystate$$]], StandardForm], 
      ImageSizeCache -> {117., {7., 16.}}, 
      TrackedSymbols :> {Typeset`query$$, Typeset`boxes$$, 
        Typeset`allassumptions$$, Typeset`assumptions$$, 
        Typeset`open$$, Typeset`querystate$$}], 
     DynamicModuleValues :> {}, 
     UndoTrackedVariables :> {Typeset`open$$}], 
    BaseStyle -> {"Deploy"}, DeleteWithContents -> True, 
    Editable -> False, SelectWithContents -> True]\)

Now we can use the WikidataData function to get my WikidataID:

WikidataData
&#10005

WikidataData[Entity["Person", "StephenWolfram::j276d"], "WikidataID"]

InputForm
&#10005

InputForm[%]

Let’s ask what Wikidata classes I’m a member of:

WikidataData
&#10005

WikidataData[Entity["Person", "StephenWolfram::j276d"], "Classes"]

Not that deep, but correct so far as I know.

There’s lots of data that’s been put into Wikidata over the past few years. Some of it is good; some of it is not. But with WikidataData in Version 12.1 you can systematically study what’s there.

As one example, let’s look at something that we’re unlikely to curate in the foreseeable future: famous hoaxes. First, let’s use WikidataSearch to search for hoaxes:

WikidataSearch
&#10005

WikidataSearch["hoax"]

Hover over each of these to see more detail about what it is:

WikiDataSearch["hoax"]
&#10005

WikidataSearch["hoax"]

OK, the first one seems to be the category of hoaxes. So now we can take this and for example make a dataset of information about what’s in this entity class:

WikidataData
&#10005

WikidataData[
 EntityClass[
  ExternalIdentifier["WikidataID", 
   "Q190084", <|"Label" -> "hoax", 
    "Description" -> 
     "deliberately fabricated falsehood made to masquerade as the truth"|>], All], "WikidataID"]

We could use the Wikidata ExternalIdentifier that represents geo location, then ask for the locations of these hoaxes. Not too many have locations given, and I’m pretty suspicious about that one at Null Island (maybe it’s a hoax?):

GeoListPlot
&#10005

GeoListPlot[
 Flatten[WikidataData[
   EntityClass[
    ExternalIdentifier["WikidataID", 
     "Q190084", <|"Label" -> "hoax", 
      "Description" -> 
       "deliberately fabricated falsehood made to masquerade as the \
truth"|>], All], 
   ExternalIdentifier["WikidataID", 
    "P625", <|"Label" -> "coordinate location", 
     "Description" -> 
      "geocoordinates of the subject. For Earth, please note that \
only WGS84 coordinating system is supported at the moment"|>]]]]

As another example, which gets a little more elaborate in terms of semantic querying, let’s ask for the opposites of things studied by philosophy, giving the result as an association:

WikidataData
&#10005

WikidataData[
 EntityClass[All, 
  ExternalIdentifier["WikidataID", 
    "P2579", <|"Label" -> "studied by", 
     "Description" -> 
      "subject is studied by this science or domain"|>] -> 
   ExternalIdentifier["WikidataID", 
    "Q5891", <|"Label" -> "philosophy", 
     "Description" -> 
      "intellectual and/or logical study of general and fundamental \
problems"|>]], 
 ExternalIdentifier["WikidataID", 
  "P461", <|"Label" -> "opposite of", 
   "Description" -> 
    "item that is the opposite of this item"|>], "Association"]

What Is That Molecule? Advances in Chemical Computation

You have an image of a molecular structure diagram, say from a paper. But how can you get the molecule it represents in a computable form? Well, with Version 12.1 all you need do is use MoleculeRecognize:

MoleculeRecognize
&#10005

MoleculeRecognize[CloudGet["https://wolfr.am/L9rL9B2K"]]

It’s the analog of TextRecognize, but for molecules. And what it produces is a Wolfram Language symbolic representation of the molecule. So, for example, you can then generate a 3D structure:

mol = MoleculeRecognize
&#10005

mol = MoleculeRecognize[CloudGet["https://wolfr.am/L9rL9B2K"]];

MoleculePlot3D
&#10005

MoleculePlot3D[mol]

Or you can compute the distribution of torsion angles of the structure:

Histogram
&#10005

Histogram[MoleculeValue[mol, "TorsionAngle"], 360]

You can also connect to the world of external identifiers:

MoleculeValue
&#10005

MoleculeValue[mol, "PubChemCompoundID"]

But what’s really useful about MoleculeRecognize is that it can be used programmatically. Take all the images of chemicals from a paper, “molecule OCR” them—then do things like check whether the molecules are equivalent, or make a word cloud of their 3D structures:

WordCloud
&#10005

WordCloud[
 MoleculePlot3D /@ DeleteDuplicates[MoleculeRecognize[{\!\(\*
GraphicsBox[
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"], {{0, 347.}, {373., 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->72],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSize->{57.40625, Automatic},
ImageSizeRaw->{373., 347.},
PlotRange->{{0, 373.}, {0, 347.}}]\), 
     CloudGet["https://wolfr.am/KR7FBcsM"]}], MoleculeEquivalentQ]]

Something else that’s new in 12.1—and a first sign of something big to come—is the ability to import data about molecular orbitals:

Import
&#10005

Import["ExampleData/Pyridinecarbonitrile_MO_25_29.cub", "Graphics3D"]

Making the Data Repository Easy

We launched the Wolfram Function Repository in June 2019, and there are already 1146 functions published in it. One of the innovations in the Function Repository is a very streamlined process for submitting new functions, applicable both for the public Function Repository, and for individual deployment on a single machine, or in the cloud.

In Version 12.1 we’re introducing a new, streamlined submission mechanism for the Data Repository. File > New > Repository Item > Data Repository Item gives you:

Data Resource Definition Notebook

Then if you’ve got, say, a Dataset, you just insert it in the notebook, then add examples (using the Insert ResourceObject button to insert references to the object you’re creating). When you’re done, press Deploy, and you can deploy locally, or privately or publicly to the cloud. Lots of checking just happens automatically (and if there’s something wrong you’ll usually get a suggestion about how to fix it).

My goal was to make it so that a simple Data Repository entry could be created in just a few minutes, and I think we’ve streamlined and automated things to the point where that’s now possible.

External Connectivity

We want the Wolfram Language to provide a consistent computational representation of as much as possible. And that means that in addition to things like the molecules we just discussed, we want our language to be able to represent—and seamlessly interact with—all the other kinds of computational systems that exist in the world, whether they be programs, languages, databases or whatever. The list of kinds of things we can deal with is very long—but in Version 12.1 we’ve made some significant additions.

The Wolfram Language has been able to call programs in other languages through what’s now WSTP since 1989, but in recent years we’ve been working to make it ever easier and more automatic to do this. And for example in Version 11.2 we introduced ExternalEvaluate, which provides a high-level way to directly evaluate code in external languages, and, whenever possible, to get results back in a symbolic form that can be seamlessly used in the Wolfram Language.

In Version 12.1 we’ve added Julia, Ruby and R to our collection of external languages. There are all sorts of practical issues, of course. We have to make sure that an appropriate installation exists on a user’s computer, and that the data types used in programs can be meaningfully converted to Wolfram Language.

But in the end it’s very convenient. In a notebook, just type > at the beginning of a line, select your language, and enter the code, and evaluate:

> [1,2,3+3]
&#10005

> [1,2,3+3] 

But this doesn’t only work interactively. It’s also very convenient programmatically. For example, you can create a function in the external language, that is then represented symbolically in the Wolfram Language as an ExternalFunction object, and which, when called, runs code in the external language:

> def square(x)
&#10005

> def square(x)
  x * x
end

% /@ {45, 135, 678, 34}
&#10005

% /@ {45, 135, 678, 34}

For each different language, however, one’s dealing with a whole new world of structures. But since we have built-in support for ZeroMQ (as well as having a connection to Jupyter available), we at least have the plumbing to deal with a very wide range of languages.

But particularly for languages like Python where we have full built-in connectivity, one of the significant things that becomes possible is to have functions that work just like standard Wolfram Language functions but are fully or partly implemented in a completely different language. Of course, to have this work seamlessly requires quite a bit of system support, for automated installation, sandboxing, etc. And for example, one of the things that’s coming is the ability to call functions containing Python code even in the Wolfram Cloud.

In addition to external languages, another area of expansion is external storage systems of all kinds. We’ve already got extensive support for the Bitcoin and Ethereum blockchains; in Version 12.1 we added support for the ARK blockchain. In addition, we introduced support for two external file storage systems: IPFS and Dropbox.

This is all it takes to put a Spikey into the globally accessible IPFS file system:

ExternalStoragePut
&#10005

ExternalStoragePut[CloudGet["https://wolfr.am/L9rTMCn6"], 
 ExternalStorageBase -> "IPFS"]

Here’s the content identifier:

%["CID"]
&#10005

%["CID"]

And now we can get our Spikey back:

ExternalStorageGet
&#10005

ExternalStorageGet["QmcNotbm3RZLv7caaPasU8LiHjNCCuuEemrCgbLAsN49TZ"]

You can do the same kind of thing with Dropbox, and after authentication (either through a browser, or through our new SystemCredential mechanism, discussed below) you can put expressions or upload files, and they’ll immediately show up in your Dropbox filesystem.

Given the framework that’s been introduced in 12.1, we’re now in a position to add connections to other external file storage systems, and those will be coming in future versions.

In addition to plain files, we also have a sophisticated framework for dealing with relational databases. Much of this was introduced in Version 12.0, but there are some additions in 12.1. For example, you can now also connect directly to Oracle databases. In addition, there are new functions for representing relational set operations: UnionedEntityClass, IntersectedEntityClass, ComplementedEntityClass.

And, of course, these work not only on external databases but also on our own built-in knowledgebase:

SortedEntityClass
&#10005

SortedEntityClass[UnionedEntityClass[
    \!\(\*
NamespaceBox["LinguisticAssistant",
DynamicModuleBox[{Typeset`query$$ = "EU", Typeset`boxes$$ = 
        TemplateBox[{"\"European Union\"", 
RowBox[{"EntityClass", "[", 
RowBox[{"\"Country\"", ",", "\"EuropeanUnion\""}], "]"}], 
          "\"EntityClass[\\\"Country\\\", \\\"EuropeanUnion\\\"]\"", 
          "\"countries\""}, "EntityClass"], 
        Typeset`allassumptions$$ = {{
         "type" -> "Clash", "word" -> "EU", 
          "template" -> "Assuming \"${word}\" is ${desc1}. Use as \
${desc2} instead", "count" -> "2", 
          "Values" -> {{
            "name" -> "CountryClass", 
             "desc" -> "a class of countries", 
             "input" -> "*C.EU-_*CountryClass-"}, {
            "name" -> "Unit", "desc" -> "a unit", 
             "input" -> "*C.EU-_*Unit-"}}}}, 
        Typeset`assumptions$$ = {}, Typeset`open$$ = {1, 2}, 
        Typeset`querystate$$ = {
        "Online" -> True, "Allowed" -> True, 
         "mparse.jsp" -> 0.363881`6.012504373043238, 
         "Messages" -> {}}}, 
DynamicBox[ToBoxes[
AlphaIntegration`LinguisticAssistantBoxes["", 4, Automatic, 
Dynamic[Typeset`query$$], 
Dynamic[Typeset`boxes$$], 
Dynamic[Typeset`allassumptions$$], 
Dynamic[Typeset`assumptions$$], 
Dynamic[Typeset`open$$], 
Dynamic[Typeset`querystate$$]], StandardForm],
ImageSizeCache->{232., {7., 15.}},
TrackedSymbols:>{
          Typeset`query$$, Typeset`boxes$$, Typeset`allassumptions$$, 
           Typeset`assumptions$$, Typeset`open$$, 
           Typeset`querystate$$}],
DynamicModuleValues:>{},
UndoTrackedVariables:>{Typeset`open$$}],
BaseStyle->{"Deploy"},
DeleteWithContents->True,
Editable->False,
SelectWithContents->True]\), 
    EntityClass["AdministrativeDivision", "AllUSStatesPlusDC"]], 
   "Population" -> "Descending"][{"Name", "Population"}] // Dataset

We’ve been very active over the years in supporting as many file formats as possible. In Version 12.1 we’ve added the popular new HEIF image format. We’ve also updated our DICOM importer, so you can take those CT scans and MRIs and immediately start analyzing them with Image3D and our 3D image processing.

Like, this is part of our director of R&D’s knee:

knee = Import
&#10005

knee = Import["knee_mr/DICOMDIR", {"Image3D", 1, 1, 1}];
Image3D[knee, ColorFunction -> (Blend[{{0., 
RGBColor[0.05635, 0.081, 0.07687, 0.]}, {0.0777045, 
RGBColor[0.702347, 0.222888, 0.0171385, 0.0230167]}, {0.3, 
RGBColor[1., 0.6036, 0., 0.303215]}, {0.66, 
RGBColor[1., 0.9658, 0.4926, 0.661561]}, {1., 
RGBColor[1., 0.6436, 0.03622, 1.]}}, #]& )] // ImageAdjust // 
 Blur[#, 3] &

In Version 11.3, we added MailServerConnect for connecting directly to mail servers. In 12.1, we’ve added a layer of caching, as well as a variety of new capabilities, that make the Wolfram Language uniquely powerful for programmatic mail processing. In addition, in Version 12.1 we’ve upgraded our capabilities for importing mail messages from EML and MBOX, in particular adding more controls for attachments.

Yet another new feature of 12.1 is stronger support for ZIP and TAR files, both their creation through CreateArchive, and their extraction through ExtractArchive—so you can now routinely handle tens of thousands of files, and gigabytes of data.

Whenever one is connecting to external sites or services, there are often issues of authentication. We’d had some nice symbolic ways to represent authentication for some time, like SecuredAuthenticationKey (that stores OAuth credentials). But for example in cases where you’ve got to give a username and password, there’s always been the issue of where you store those. In the end, you want to give them as part of the value for an Authentication option. But you don’t want to have them lying around in plaintext.

And in Version 12.1 there’s a nice solution to this: SystemCredential. SystemCredential ties into your system keychain—the encrypted storage that’s provided by your operating system, and secured by the login on your computer.

It’s as easy as this to store things in your system keychain:

SystemCredential
&#10005

SystemCredential["secret"] = "it's a secret"

SystemCredential
&#10005

SystemCredential["secret"]

Paclets for All

Length
&#10005

Length[PacletFind[]]

In my Wolfram Language system right now, I have 467 paclets installed. What is a paclet? It’s a modular package of code and other resources that gets installed into a Wolfram Language system to deliver pretty much any kind of functionality.

We first invented paclets in 2006, and we’ve been increasingly using them to do incremental distribution of a great many pieces of Wolfram Language functionality. Paclets are versioned, and can be set to automatically update themselves. Up until now, paclets have basically been something that (at least officially) only we create and distribute, from our central paclet server.

But as of Version 12.1, we’re opening up our paclet system so anyone can use it, and we’re making it a fully documented and supported part of the Wolfram Language. Ultimately, a paclet is a file structure that contains assets or resources of various kinds, together with a special PacletInfo.wl file that defines how the paclet should integrate itself into a Wolfram Language system.

A paclet can set up code to execute at startup time. It can set up symbols whose definitions will be autoloaded. It can install documentation. It can put items into menus. And in general it can set up assets to be used in almost any part of the fairly complex structure that is a deployed Wolfram Language system.

Typically a paclet is distributed as a single archive file, and there are many ways someone can get such a paclet file. We maintain a central paclet server that’s used by the Wolfram Language system to get automatic downloads. But in the near future, we’re also going to have a full paclet repository through which users will be able to distribute paclets. (We’re also going to make it possible for Wolfram Enterprise Private Clouds to have their own paclet repositories.)

I might mention how I see the Wolfram Paclet Repository relating to our Wolfram Function Repository. The Function Repository is built to extend the functionality of the Wolfram Language one function at a time, always maintaining the overall structure and consistency of the language. The Paclet Repository will let people distribute complete environments that may serve some particular purpose, but may not maintain the structure and consistency of the overall language. Paclets will be extremely useful, and we intend them to be as freely used as possible. So whereas the Wolfram Function Repository has a curation process to ensure a certain level of design consistency, we plan to make the Wolfram Paclet Repository basically completely open.

The goal is to allow a rich ecosystem of user-contributed paclets to develop. The Paclet Repository will serve as a smooth distribution channel for Wolfram Language material. (And by the way, I think it will be quite common for functions in the Function Repository to actually be based on code that’s distributed through the Paclet Repository—with the Function Repository serving as a streamlined and structured presentation mechanism for the functions.)

In Version 12.1 there are a variety of functions for creating and managing paclets. With the Wolfram Language, PacletObject is the symbolic representation of a paclet. Here’s a trivial example of what a paclet might be like:

PacletObject
&#10005

PacletObject[<|"Name" -> "TrivialPaclet", "Version" -> "1.0", 
  "Extensions" -> {{"Kernel", "Context" -> "TrivialPackage`"}}|>]

This can then be packaged into a .paclet file using CreatePacletArchive—and this file can then be distributed just as a file, or can be put on a paclet server. Once someone has the file, it’s just a question of using PacletInstall, and the paclet will come to life, inserting the necessary hooks into a Wolfram Language system so that its contents are appropriately used or exposed.

And Even More…

Well, that’s already a lot. But there’s even more too. FaceAlign. FindImageText. Around in Classify and ListPlot3D. SubsetCount. GenerateFileSignature. RSA digital signatures. Much faster MailSearch. Initialization tied to notebooks or cells. And so on. Here’s the whole list. Check it out!

Summary of New Features in 12.1

2 comments

  1.  

    “Nothing” has proven useful to my list processing; am looking forward to use of “Splice” as well. Congratulations on 12.1.

    MM Thomas
  2.  

    Wow. As much as open source software has progressed, Mathematica really is pushing the boundaries. Congratulations on 12.1!