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Symbolics.jl: A Modern Computer Algebra System for a Modern Language (julialang.org)
379 points by natemcintosh 38 days ago | hide | past | favorite | 94 comments



Rackauckas has done a lot of work to create the whole scientific machine learning (SciML) ecosystem (with collaborators).

Symbolics.jl is the new step to a universal CAS tool in Julia to bridge the gap between symbolic manipulation and numerical computation

It is especially useful as a foundation for equation-based simulation package ModelingToolkit.jl. In the foreseeable future, I expect ModelingToolkit.jl can be comparable with the Modelica ecosystem to provide fast, accurate modeling and simulation capability and easy integration with machine learning methods for Julia ecosystem, which is crucial for engineering application and scientific research.


Thanks! The tutorials on the Modelica-like features of ModelingToolkit.jl are just starting to roll out. For example: https://mtk.sciml.ai/dev/tutorials/acausal_components/ . Indeed, there's a lot to do in this space, but we already have some pretty big improvements that we'll start writing down and share (hopefully) at JuliaCon 2021!


Hi Chris, I am a very heavy Modelica user and having an equivalent system in Julia is very welcome. Using Sciml's solvers in our models would be a killer feature. Obviously, I am tempted to contrast ModelingToolkit.jl with Modelica and I have some questions, if you don't mind:

Could you elaborate on the design choice to model components as functions as opposed to objects ? functions seem compose well from your example but what about initial conditions and type safety ? In Modelica each component is an object where you can specify initial conditions inside each of them with constraints and can eventually make a big system, and if it compiles you can be reasonably confident you aren't mismatching units, missing outputs etc.

Do you have any plans for graphical representation ? Some of the systems I work on in Motorsports are absolutely massive with 150k equations, and having a diagram to see the connections are really helpful. An auto generated one from code would be more than good enough.

How do you handle FFI and interaction with other Julia objects inside ModelingToolkit.jl since it requires symbolic reduction ?

The FMI standard is a very popular export standard for these models. Any plans to support it here ?

I understand these are early days and I am very excited to know there's more on the pipeline. Thanks for your contribution.


Most of the answers to this will be in a big JuliaCon talk this summer. But I'll give a few hints.

>Could you elaborate on the design choice to model components as functions as opposed to objects ? functions seem compose well from your example but what about initial conditions and type safety ? In Modelica each component is an object where you can specify initial conditions inside each of them with constraints and can eventually make a big system, and if it compiles you can be reasonably confident you aren't mismatching units, missing outputs etc.

There's things for type verification and all of that. The decision comes from how it interacts with the compiler. You can easily redefine functions and create them on the fly, less so for structs. That turns out to make it easier to do features like inheritance easily in a functional workflow. Symbolic computing seems to work very well in this setup.

>Do you have any plans for graphical representation ? Some of the systems I work on in Motorsports are absolutely massive with 150k equations, and having a diagram to see the connections are really helpful. An auto generated one from code would be more than good enough.

Somewhat. Auto-generated ones from code already exist in the Catalyst.jl extension library (https://catalyst.sciml.ai/dev/). That kind of graphing will get added to MTK: we just added the dependency graph tooling to allow it to happen, so it's just waiting for someone to care.

A full GUI? There's stuff we're thinking about. More at JuliaCon.

>How do you handle FFI and interaction with other Julia objects inside ModelingToolkit.jl since it requires symbolic reduction ?

All of the functions are Julia functions, and you can easily extend the system by registering new Julia functions to be "nodes" that are not traced. See https://symbolics.juliasymbolics.org/dev/manual/functions/#R... . So if you do `f(x,y) = 2x^2 + y`, then it will eagerly expand f in your equations. If you do `@register f` (and optionally add derivative rules), then it'll keep it as a node in the graph and put its function calls into the generated code. This is how we're doing FFI for media libraries in a new HVAC model we're building.

>The FMI standard is a very popular export standard for these models. Any plans to support it here ?

There is a way to read in FMI that will be explained much more at JuliaCon, with some details probably shared earlier. It's somewhat complex so I'll just wait till the demos are together, but yes we have examples with FMU inputs already working.


Thank you


Thank you for your hard work on SciML ecosystem and community, Rackauckas!

The new Modelica-like usage of ModelingToolkit.jl is really a game changer for acasual DAE system simulation in Julia. It makes composable hierarchical component-based simulation of large and complex system possible with pure Julia.

Based on full featured dynamic simulation capability and machine learning ecosystem (Flux.jl etc), Julia will be a perfect choice for reinforcenment learning research, where simulation and training process can all be implemented in pure Julia with promising performance. It delivers the promise of "solve two language problem" of Julia language.


This is very cool. Computer Algebra Systems are like a super power. I’ve been able to use the TI-89’s arbitrary precision to do pretty sophisticated cryptography work. Symbolics.jl has the ability to modify some of the rules of the algebra, which should allow even easier implementation of number theory related cryptographic concepts (as well as compression and error correcting codes... or quaternions or bra-ket quantum mechanics) without needing specific libraries. I love this as I’m trying to teach myself the fundamental concepts in mathematical terms and not just something in a specialized black box library. (And without paying the insanely high price of Mathematica if I ever want to use it professionally.)

I’ve looked briefly into Julia in the past, but if stuff like this becomes pretty standard (in the way numpy is for Python), I think I could become pretty comfortable in Julia.


We already have a very good open source CAS called Maxima, which is implemented in Lisp. It is a descendent of Macsyma, the oldest CAS, created at the MIT.


Maxima is quite awkward to use IME. Because of its age it has very nonstandard syntax for some things, like ":" for assignment, and "(..., ..., ...)" for blocks of commands (equivalent to e.g. "{...; ...; ...}" in C, JavaScript, Java, etc.).

I prefer Sympy.


You are quite right to draw attention to this fact, but let the "geniuses" reinvent the wheel worse ;)


CASes are one of the most difficult things to write and they’re never complete. There are always bugs, performance concerns, anemic mathematical domains, etc. Every major free or commercial CAS that’s still in use is under active development to battle these inadequacies. Dozens of CASes have come and gone in the past 30 years, the majority of which have bitrotted or stopped being maintained. And not a single CAS has reigned supreme as the CAS that beats all CASes.

It’s very exciting to see more work in CASes being done, but I worry that “starting a CAS from scratch” isn’t the right approach. The Axiom [0, 1] project rightly identified that building a general-purpose CAS for working practitioners of computational mathematics is an effort requiring nearly generational timespans [2], and that you must have the right language to describe mathematical objects and their relationships. They had a literate programming policy, where all math code must be accompanied by publication-quality documentation, precisely because it’s so hard to build and maintain these systems. Some of the greatest computational discoveries and expositions came out of the development of Axiom, like the richest and most complete implementation of the renowned Risch algorithm for doing symbolic integrals.

Axiom fell into disuse for a variety of reasons, but from my perspective, found new life in a fork called FriCAS [3, 4], which is actively developed and allows a more “software engineer friendly” approach to the development of the system. The code they have is enormously complex and has mountains of knowledge from foremost experts in computer algebra.

I really wish new computer algebra initiatives attempted in earnest to make use of and extend Axiom/FriCAS so that we could continue to build up our knowledge of this exceedingly delicate and tricky subject without constantly starting from zero. Axiom has a manual that is over 1,000 pages of dense mathematics and that’s really hard to rebuild correctly.

(The only project I know who honestly tried to build upon and subsequently extend CAS functionality is Sage [5], which builds upon a plethora of existing open source general-purpose and specialized computational math systems.)

[0] https://en.m.wikipedia.org/wiki/Axiom_(computer_algebra_syst...

[1] http://www.axiom-developer.org/

[2] Quote from Axiom manual (http://fricas.sourceforge.net/doc/book.pdf):

> With that in mind I’ve introduced the theme of the “30 year horizon”. We must invent the tools that support the Computational Mathematician working 30 years from now. How will research be done when every bit of mathematical knowledge is online and instantly available? What happens when we scale Axiom by a factor of 100, giving us 1.1 million domains? How can we integrate theory with code? How will we integrate theorems and proofs of the mathematics with space-time complexity proofs and running code? What visualization tools are needed? How do we support the conceptual structures and semantics of mathematics in effective ways? How do we support results from the sciences? How do we teach the next generation to be effective Computational Mathematicians? The “30 year horizon” is much nearer than it appears.

[3] https://en.m.wikipedia.org/wiki/FriCAS

[4] http://fricas.sourceforge.net/

[5] https://www.sagemath.org/


From looking at the FriCAS github:

"Additionally, FriCAS algebra library is written in a high level strongly typed language (Spad), which allows natural expression of mathematical algorithms."

One could argue that Julia also allows natural expression of mathematical algorithms. Coupled with Julia features like multiple dispatch, high performance (due to Julia's LLVM backend) and growing ecosystem of AD and ML libraries, it seems like Julia is probably the more “software engineer friendly” approach at this point. It doesn't seem odd that the Julia folk would want to implement their CAS in Julia. That's not to say that maybe bridges from Julia to FriCAS couldn't be built as has been done with both R and Python.


There are lots of things which may go wrong trying to make a mathematics fit into a type system. In particular you need a particularly general typesystem. I think Julia has a better chance than haskell/rust/ml (let alone C++) because types are less rigid so functions may return values of types that are functions of the arguments. The difficulty is subtyping.

For a trivial example, the integers under addition form a group. But they also form a ring, which is that group plus a different monoid structure (and extra rules), and the real numbers form a field (two connected groups with extra structure) which meets with the ring structure of the integers. If I write an algorithm about groups, I want a way to be able to apply it to the additive or multiplicative groups of the reals. You need the ability to easily talk about forgetting extra structure but then taking results back from the simpler structure.

Another example is dimensions. It is, as far as I know, basically impossible to encode something like “force (ie mass times distance per time squared) divided by time is commensurate with current times voltage” in a type system like that in ocaml/haskell/rust (you can maybe do it with c++ but the compiler can’t check that your template is correct)

The next problem with general algorithms is that there are isomorphisms and then there are isomorphisms so things that work in mathematics (this group is cyclic and g is a generator so for my element h, there is some integer a such that g^a=h) are sometimes infeasible on computers (the process described above is trivial in integers under addition, hard in integers mod p under multiplication, and very hard in elliptic curve groups (of rank 1), even though they are all cyclic).

I don’t think these are reasons not to try and I think Julia feels like a good language for a combination of performance and interface ergonomics, as well as the library being useful in the ecosystem of other scientific or mathematical libraries.


Regarding your point about dimensions, it really isn't impossible; it requires a little bit of type-level mechanisms, but not very much and quite a few languages already have it.

Here are some examples in some languages.

The author of Libra (which I /think/ was the first to use this technique) gave a good talk [4] on the mechanism.

[1] https://to-ithaca.github.io/libra/ (Scala)

[2] https://hackage.haskell.org/package/dimensional (Haskell)

[3] https://docs.rs/dimensioned/0.7.0/dimensioned/ (Rust)

[4] https://www.youtube.com/watch?v=CrwVD0Pco5Q


Ok I guess I concede that it is technically possible in haskell and rust. But note that the way it works in both cases is quite hacks, relying on typeclasses with associated types (ie effectively type-level functions) to fake numbers (up to some maximum), and then representing the dimension of something as a fixed length vector of these numbers. This is fine for dimensional analysis of real world physical things I guess but as soon as you’re trying to look at dimensions of something to see if a calculation makes sense it will fail because there won’t be an appropriate slot in your vector, or will need you to reimplement or modify the hacks.

I don’t think it is sustainable to need to jam that kind of trickery into every step.


I don't understand why you call it "trickery or "fake". Church encoding of natural numbers is the same technique used in Agda, Coq and Idris to represent the Peano numbers. It's a completely valid encoding and isomorphic to any other representation.

You don't need to use a fixed-length array either - you can used a recursive linked list at the type-level for an unbounded encoding [1]. The Scala library is an example of that; the Github page even has an example of encoding arbitrary units like sheep and wheat.

[1] https://github.com/lloydmeta/frunk


The trickery here is basically pushing the type system to the extreme for relatively simple things. I wonder how the type system might cope for more complicated things. Another consequence is type errors. You won’t get something about trying to add a length to a time but rather something like “can’t unify type Pos1 with Zero,” or “error Units<f64,(z,(z,(z,(z,(s<z>,(z,()))))))> does not implement trait Add<Units<f64,(z,(z,(z,(s<z>,(z,(z,()))))))>>.” This is confusing enough in a trivial example but I wonder how this would go if the example were buried deep in the code.

The encodings used in the haskell library were not a church encoding (they could only express integers between about minus 9 and 9) though to be fair they claimed to be only there until better type level literals. I don’t know about the rust implementation.

Coq and Idris are quite weird languages. The reason they use Peano arithmetic at the type level is that it is easy for them to reason about. But in dependently typed languages where the type and value levels aren’t really separate, if you want to prove things about your programs, they also need to use these numbers which means you either need special language support for making them fast or you have slow programs, which is not suitable for the topic of efficient systems for computer mathematics. The other problem is that the computer can’t help much with the types and they require a lot of manually specifying things which many practitioners would consider trivial enough to be tacit. This is not good for having an expressive system for computer mathematics.

Another example of a system which tried to offer better mathematical foundations is fortress. They started out as something like “fortran but better at mathematics and without so much history” and ended up with something a lot more like haskell with a typeclass-like mechanism and mostly function language. They had a lot of issues with making the type system work with mathematics (one case is that it is hard to specify that something can be a (eg) a monoid. In mathematics, it’s often obvious or said very simply, but computers aren’t so good at knowing what is obvious. It can be complicated to say eg “this is a monoid with identity 1 and operation x,” or “this is a monoid with identity 0 and operation +” because your type system needs a way to disambiguate between the two monoids (or the language will need some bad ergonomics where it is very manually expressed). If your type system ends up needing to know about the names of things and having eg type classes dispatch on the operation or something, it could become tricky.


Rust is getting real type-level numbers in a couple of weeks. It’s just the bare minimum though. More to come in the future.


Do you know whether there's going to be any near term support for ordering constraints (i.e. N < M) with the type-level numbers?


I don't.


We definitely can, should, and will interface with it. There's no reason to throw away great work. Just like how DifferentialEquations.jl always had a focus on pure Julia solvers, it made sure to wrap every solver it could. Building such wrappers is required for good research anyways, it's required for both timing and calculating efficiency in other ways (like amount of simplification or percentage of integrals solved).


Axiom, in its original commercial incarnation, took 20 years to build. My comment meant to suggest not spending another 20 for the dubious promise of a bit of additional “software engineer friendliness”, which is why FriCAS forked Axiom instead of building one from scratch.


I'll make one (probably unpopular) comment.

On HN every time the subject of language wars or platform wars or browser wars comes up, the idea also comes up that "everything is about the number of users". The self-reinforcing cycle is that the language/platform/browser with the most users also has the most development time put into it, which then attracts more users, etc.

Fine, I don't deny that the phenomenon exists. But I think that it's often overlooked that it's not just about the number of users. It's also about the quality of the users. If we could quantify "an excellent developer", I wouldn't claim that Julia has more of those than Python. But I'm convinced that Julia has more "excellent developers who are also excellent at numerics" than Python. I think the idea that the productivity as function of "developer excellence percentile" is a power-law applies even more strongly in multi-domain expertise situations, like numerical computing. So forget about 100x coders. The contributions of some people like Chris et al are closer to 10_000x as significant as that of an ok contributor.

It's not just about the quantity, it's also about quality.


I won’t comment on the relative numbers because there are top-notch developers in many language communities.

I think the more important point is that Julia has attracted enough first-rate people to self-sustainably build out an ecosystem — and even more keep joining. Several aspects of Julia’s design and core tooling interact to provide compounding leverage to this group. I think it’s a similar situation to the development of the NumPy ecosystem where standardizing on a common array data structure and API led to an explosion of interoperable libraries. Julia arguably takes that a step further by allowing any code to interoperate with high performance and fluent APIs. Julia’s performance characteristics also reduce the barrier to entry because people can make deep contributions throughout the ecosystem without needing to develop low-level programming expertise on top of their domain-specific knowledge.


Computer Algebra is criminally underused - it has the potential to make math-heavy critical-path code a lot more clear, testable, observable and bug-free by design.

I feel I need to mention "Structure and Interpretation of Classical Mechanics" by Wisdom and Sussman, and the accompanying "scmutils" library, which first implemented in Scheme many of the same features, although this Julia library seems to be more complete.

There's also a great one-to-one port to Clojure by Colin Smith [2] in case you want to use it on a more production-friendly environment. The Strange Loop talk [3] is a good showcase of the power and simplicity of these kind of systems.

[1] https://groups.csail.mit.edu/mac/users/gjs/6946/

[2] https://github.com/sicmutils/sicmutils

[3] https://www.youtube.com/watch?v=7PoajCqNKpg&ab_channel=Cloju...


SICMUtils co-author here, if anyone has any questions on the Clojure port.

One beautiful thing about a Clojure computer algebra system is that it can run completely in the browser. This includes automatic differentiation, numerical integration, all of the hardcore Lagrangian and Hamiltonian mechanics work, differential geometry... it is startling stuff.

For example, here's a(n interactive!) derivation of Kepler's Third Law in the browser (thanks to Nextjournal's lovely integration), if anyone wants to play: https://nextjournal.com/try/sicm/ex-1-11

Many more exercises live here: https://nextjournal.com/sicm/


Yes, we found these and this (along with Mathematica) was the impetus for building automated Latexification into Symbolics.jl. Here for example is a teaching notebook used in Alan Edelman's MIT Computational Thinking course where Symbolics.jl is used to visualize the numerical methods as they iterate, and all of the outputs convert to LaTeX:

https://computationalthinking.mit.edu/Spring21/newton_method...

Thanks for the ideas!


Those are gorgeous. How were they done? Using Julia packages?


This is a Pluto notebook over Symbolics.jl and ForwardDiff. All of the packages used are at the top of the page. You can click the edit button on the top right to open it up.


Forgot to mention: Shashi cites SICMUtils as a big influence for the rewrite system SymbolicUtils.jl, which is the rewrite system underneath Symbolics.jl.

https://discourse.julialang.org/t/ann-symbolicutils-jl-groun...

He took Sussman's course and became interested in the field from that! Great work.


Whenever SICM comes up here on HN it receives a lot of admiration. One thought that a lot of people have is whether there would be advantages in trying to do something like that in a statically typed language. Have you thought about types and SICMUtils?


I made a scratch like visual programming system for sicmutils, clj-tiles[0]. Types are inspectable by right-clicking->Inspect on a specific block. Clj-tiles uses Clojure spec under the hood. One note: I am not sure whether a static type system can improve on this for someone who wants to learn Physics. After all, the types of function parameters change during the running program, as the same functions are called several times. But see for yourself:

[0] https://kloimhardt.github.io/cljtiles.html?page=pendulumfina...


Hey, I have... I'm a co-author of Algebird[0], which has many ideas that I'd pull over.

I'm hoping to introduce Clojure's "spec" or "schema" libraries so that the types at play can at least be inspectable inside the system. In a fully typed language, I'd implement the extensible generics as typeclasses.

I suspect it would make it quite a bit tougher (at least in the approach I'm imagining) for folks to write new generic functions, due to many type constructors...

On the other hand, the complexity is there, even if you don't write it down!

It would be a big project, and a worthy effort, to write down types for everything in SICM.

[0] https://github.com/twitter/algebird


I was amazed the first time I used Mathematica. I have used later professionally Maxima to compute some Taylor series and also simpy for some hobby projects. I found simpy less powerful than Maxima (not to mention Mathematica) but the ability to integrate it with the rest of the program is wonderful.


Using Maxima in my day-to-day work has been a complete game changer. I use it via org-mode and use the `tex()` command to have it output TeXified results. These automatically get formatted into beautiful readable equations.


Cool! I managed to do something similar with org-mode + sicmutils, works quite well.

Render (pt_BR): https://github.com/hcarvalhoalves/math-fin-training/blob/mas... Source: https://raw.githubusercontent.com/hcarvalhoalves/math-fin-tr...

I would love to see how you're using org-mode for that if possible :)


> "Structure and Interpretation of Classical Mechanics" by Wisdom and Sussman

I come across this book every so often and find it really tough to read due to the complete lack of "types" in any of the code. Math, especially physics, relies heavily on units and function signatures to be understood.


I won't pretend to understand all or this, but from what I can understand the Julia ecosystem is about to be light years ahead of anything else out there. It's an amazing community at the intersection of programming language theory and numerics/math. Exciting to see what's going to emerge.


I've used Julia a bit and I fail to see that "light years ahead" point. Julia is sure a cool language to work with, with a very nice type system and a quite smart compiler, but well, the stuff I've used is still much of meta-programming (like Python does, except here it's JITed right). But maybe my epxerience is so limited I don't see the interesting bits...


Were you doing anything with:

1. custom units 2. custom GPU kernels 3. Custom array types 4. custom bayesian priors. 5. AD through custom types 6. Task based parallelism 7. symbolic gradients with modeling toolkit 8. Agent based modeling 9. physics informed neural networks 10. abstract tables types ...

or various combinations of the above?


I think the biggest thing that is "light years ahead" is that numpy is built right in and insanely extensible. Libraries can be very lightweight / maintainable because they don't need to roll their own numpy like tensorflow pytorch etc. do (which makes them depend on bigcorps to improve and maintain). In julia somehow everything is compatible and composable. On the other hand counting from 1 is is quite a big issue for me and leads to constant errors when I switch between python and julia. Also the startup time it takes is a huge bummer. They should have an interpreted mode for debugging (as long as the debugger is unusable for e.g. breakpoints)


I think the key is less that it has good multi-dimensional arrays built in, and more that Multiple dispatch makes Julia more composable than python. For an example of this, consider that BandedMatrices is a package that can be used in conjunction with any library that is expecting a Matrix like object, despite the fact that most of them weren't designed to do so.


I haven't tried it yet as it's still a release candidate, but startup time is said to be much improved in v1.6.0.


"Please don't think that Julia is only useful for 1. custom units 2. custom GPU kernels 3. Custom array types 4. custom bayesian priors. 5. AD through custom types 6. Task based parallelism 7. symbolic gradients with modeling toolkit 8. Agent based modeling 9. physics informed neural networks 10. abstract tables types ..."


Is Julia better for these use-cases? I'd interested and would love some links/examples.


> 2. custom GPU kernels

You can write them in Julia, whereas in Python you have to write them in C/C++ and then use the FFI to call them.


You can write them in python using numba


Unfortunately not. The Python ecosystem has much more already in both symbolics and numerics, and a much bigger community.


Could someone explain to me what the difference between a computer algebra system like Symbolics.jl and a theorem prover like Coq is?

Is that more in the nuances or is there a fundamental difference between these two (referring to the terms and not their specific implementations in Symbolics.jl and Coq respectively)?

Or is this question unreasonable to ask in the first place?


Computer algebra systems are usually large, heuristic systems for doing algebraic manipulation of symbolic expressions by computer. Roughly, they’re there to aid a human in doing mechanical algebra by working with symbols and not just numbers. Generally, the results coming out of a CAS are not considered to be “proved correct”, and ought to be verified by the programmer/user.

Proof assistants aim to allow one to write down a mathematics assertion in a precise manner, and to help the user write a formally verifiable proof for that theorem.

Extremely crudely, a CAS is like a super-powered calculator, while a proof assistant is like a super-powered unit test framework.


The intersection is also interesting when you look at equational proofs in Mathematica.

https://reference.wolfram.com/language/ref/FindEquationalPro...


Not unreasonable at all.

A rough way to think about a tool like Coq or Isabelle is that it provides a very simple core foundation built out of some mathematical logic and mechanisms for manipulating statements in that logic ("if P implies Q, then X"). Proofs end up being sequences of applications of rules that manipulate the statements until you reach some conclusion. People build up theories on top of this core (and other theories), which introduce new mathematical constructs and theorems/lemmas that represent their properties.

A computer algebra system (CAS) tends to have a more complex core because instead of only having some logic as its basis, it might know about higher level constructs like polynomials and such. This allows it to more easily operate on those constructs directly via specialized algorithms and heuristics without having to build up a huge foundation of theories that they live on top of.

That said, you could implement lots of things a CAS does in a theorem prover - it just would probably be pretty awkward to work with, and possibly quite slow. Similarly, lots of CAS tools (like Mathematica and Maple) provide features for doing proofs that are similar to theorem provers. One place I would expect those to differ though is that the small, simple core of a theorem prover allows people to careful verify it such that the theories atop it inherit the verification evidence of the core. I do not know if any such verification evidence exists when it comes to the large kernels that make up tools like Mathematica/Maple.


After playing around with them for a while types seem like the wrong paradigm for symbolic work.

Term rewriting systems based on rules capture most mathematical manipulation far better than ones based on functions and classes.

Differentiation was a breeze to solve when I sat down and figured out how to formulate it so everything had an understandable normal form. Then I could feed the output of that into another algorithm that brought simple algebraic equations into a normal form.

It's not a way I've seen much programming done outside of niches in academia, but it was extremely powerful.


I disagree with this point of view. Fundamentally, types appear all over the place in math.

If I give you a term

    x * y - y * x
and ask you to simplify it, you may tell me this term is just 0, but if I had intended for x and y to represent quaternions or square matrices that'd be a very foolish thing to do!

The rulesets that apply to a term depend on the types of the elements of that term. You could bundle this information into an `Assumptions` object that carries around some rules like

    commutes_under_multiplication(x) => false
    commutes_under_multiplication(y) => false
but imo, leveraging types are a very natural way to carry around this particular class of metadata.


Types only appear in type theory in mathematics. The rest of it is based around objects built on top arbitrarily tall towers of sets which types are simply unable to capture.

The smarts in mathematics is based in the rewrite rules, not the objects they apply to. Since you do not specify what rules apply that is just a term that is already in its normal form since we have no other rules we can apply.

That you assume we are dealing with real numbers merely stems from the fact that books are too lazy to state the rules explicitly in every calculation (and with good reason) and expressions with real numbers are the most common ones that people have exposure to in the average undergraduate course.


>The smarts in mathematics is based in the rewrite rules

Can't it be argued that types are simply a way to associate a considerable amount of rewrite rules and constraints with expressions and therefore spare you the trouble of repeatedly stating them again and again ?

If the type system is rich enough that custom types encode rules just as rich, powerful, concise, etc... as native types while being interoperable with them, I would say this is much, much better than a blind rewriter who either does nothing until you provide it with a rulebook or does everything it can unless you explicitly forbid it. You can just say that that every object/expression has a rulebook attached to it instead and add meta-rules to specify how rulebooks are defined, used and extended.


In the same way that objects are just tables in disguise. Sure it's true in some way, but the impedance mismatch is huge and horrible.

There is a reason why the internals of all major cas systems use rewrite rules instead of objects with types.

As a toy test try and implement a system that simplifies an expressions of a given group to normal form for that group. Using rewrite rules you end up with a program that is dozen of lines long and even better is provably correct by hand. Using types and functions on types you end up with a monstrosity that is 10 times as long and just feels clunky. Worse even something as simple as adding a super group of which your example is a sub group requires a rewrite of the program. Using rewrite rules on the other hand merely adds a few more if/else clauses in the main program.

Types are a fetish of the programming community and are used in places where they are actively detrimental. This stems from the fact the majority of programmers do not understand the difference between types and constraint rules.


This is why we've ended up at a mixed approach. Types in Julia are used to "distinguish between things". Multiple dispatch then is a great system for defining rules that are specific to groups: `AbstractMatrix` types have non-commutative rulesets, and we then let them have different rules from `Real` and let subtyping control at a finer grain what rules apply to which pieces, and what rules apply when you combine them.


Is there a tutorial somewhere for absolute beginners with some examples of use? I used octave in the past and I used pandas/numpy before, but reading documentation linked to from the github site I have no idea how this can be used.


Nice to see the interest! https://symbolics.juliasymbolics.org/dev/tutorials/symbolic_... is a tutorial that incorporates some nice features like building parallel sparse matrix code from symbolic arithmetic.


I've always found computer algebra systems a particularly fascinating niche in computing, implementations in accessible languages like Julia are really doing a service to enthusiasts and learners everywhere!


One problem with CAS systems in general is that they either make restrictive assumptions about the types of algebraic objects, or they require you to provide extremely detailed type information yourself, beyond that which most people are capable of explicitly expressing. There are so many ambiguities and domain-specific assumptions people make when they perform algebraic manipulation that only a very small fraction of people are actually equipped to express. This is especially problematic if you mix multiple kinds of algebraic extensions - for example, I’ve had a hard time getting CAS systems to correctly deal with functions over tensors over Clifford (sub)algebras.

I think the only way you could get something like that to work would be a type system so aggressive that it would turn off most mathematicians, who tend to have a narrow understanding of type theory.


I think Axiom/FriCAS thought about this problem hard and mostly solved it.

The dominating CAS paradigm for most popular CASes is to assume everything is a real number and see how far you get. Later many major CASes bolted on features to say “well, except, this might be a complex number” and the like. This of course flies right in the face of the kind of thing you want to do. Axiom didn’t take that approach and instead created a robust system for specifying and implementing mathematical objects.

I tried doing quarternion algebra in Maxima many moons ago and it was painful. They’ve since added packages for doing Clifford algebra but it’s not exactly well integrated in the rest of the machinery.


Symbolics.jl is keeping things generic from the getgo.


Julia solves this by using interfaces (functions defined over one or more custom types that abstract over concrete information).

Just overload those on your type, and voila it works with the CAS.


What if one type admits several different interfaces? Like, an Integer type admits several equally valid group structures which may be written as an interface.


I'm skeptical that duck typing can really be considered a full "solution", but at the very least it seems like a good bet.


There is also FriCAS: https://fricas.github.io/


Really interesting work! I'm really happy that people at MIT are pushing forward Julia. In my opinion, Julia is incredibly powerful once you understand the language. It often feels like writing pseudocode that performs like C++. The ecosystem is still pretty barebones compared to Python, but, while I'm using both tools extensively, I begin to prefer Julia over Python more and more.


> The ecosystem is still pretty barebones compared to Python

There's help: https://github.com/JuliaPy/PyCall.jl https://github.com/JuliaInterop/RCall.jl

Works like a charm.


That seems to work really well. I didn't have use cases where I would prefer to call Python from Julia instead of using Python directly.

Additionally, I think you also have to think about things like Documentation and Tooling.


While this is an decent interim solution, I wouldn't want to call out to Python for anything running in production from Julia. Good for prototyping at home perhaps.

A lot of the "scientific" Python packages (NumPy, SciPy, etc.) actually just call out to C libraries for the majority of their computation. I imagine Julia can do that already, or it can call out to something similar in the stdlib, so it doesn't really need integration with these Python packages other than just API familiarity. But is that worth the cost in performance?


People definitely have used PyCall in production.

Julia at this point covers everything in NumPy, SciPy and much more. For optimization, bayesian stuff, scientific, and the convergence of the above with ML, it's far ahead- https://sciml.ai/

Even has relatively mature web frameworks (https://github.com/GenieFramework/Genie.jl)


I don't care about production or performance much; I care about data analysis and machine/deep learning for NLP. Whatever lets me use the best language and packages is best and so far it's Julia with the ecosystems and tools from Python and R.

I'm very certain that you can just import the underlying packages directly but this way is easiest, especially since I'm not familiar with R.


How do you deal with multiple dispatch? It really doesn't match up to any mental metaphors for me, I've always preferred my bag-of-functions to do things. I've tried and tried with Julia, is there any good resource for how dispatch programs are supposed to be built and thought about?


I actually find it far more linguistic, that is, more akin to natural languages.

In my view, it's not multiple dispatch per se that is the bigger departure from traditional OOP, it's the fact that methods are no longer contained in the classes. Julia draws a separation between data (structs: the nouns) and behaviors (functions: the verbs). Traditional OOP never really made sense to me; why should each class define and own its own methods? It feels far more sensible to me to just have global behaviors that are well defined. Those verbs can sometimes apply to _anything_ (leaning on more rudimentary operations that you need to define; duck-typing), and sometimes they require you to explicitly define your behavior, and sometimes you just want to add an additional optimization that's available in your particular situation.

Once you have that mental model down, multiple dispatch is just how Julia chooses which method to call... and it's really not much different from single-dispatch.


> why should each class define and own its own methods?

State mutations. That's it. By ensuring that your data can only be mutated by a your API, it can never get "corrupted".


Sure, there is a subset of behaviors for which this style makes sense, but it's just as well supported by simply defining your own functions alongside the struct.


Multiple dispatch is not too different to method overloading, so you could start there for comparable examples, maybe in programming languages you are more familiar with: https://en.wikipedia.org/wiki/Function_overloading#Rules_in_...

Bag of functions isn't a bad way to think about it, even for Julia. In some languages, only the name of the function determines which function is called. In others, the number of arguments is used too, so eg foo/1 and foo/2 may be different functions. In Julia, the types matter too, so foo(::Int) and foo(::String) are different functions ("methods" in Julia terminology), and which is used is based on the type of the argument, rather than the number.

That's where the magic in Julia happens, as if you define a function foo(x), without specifying any types, then the specific functions that foo call will only be determined once the type of the arguments to foo are known. But once they are known, that type information can ripple all the way down, picking up the specific implementation for each function depending on the actual types used.


Objects aren't bag-of-functions though (they have state, inheritance, initializers/destructors, interface/abstract classes, classes vs objects and tons of other concepts and patterns) and any complex program can become a large hierarchic tree of classes and graph of objects that goes way beyond a simple bag-of-function. Even modules that are almost literally bag-of-functions will scale quickly to something more complex.

The point is that simple concepts are nice to explain for a beginner, but what actually built your intuition in how to use objects is the years and years learning and experiencing it's benefits and pitfalls. With multiple dispatch it's the same, but since few languages use it (and even fewer, if any, pushes it everywhere like Julia does) most people didn't experience this process.

For me when I'm using a function I just consider them as self-contained abstractions over the arguments. For example there are hundreds of implementations of sum (+), which in practice I ignore and only think about the concept of addition no matter what arguments I give and I trust the compiler/library to find the optimal implementation of the concept or fail (meaning I have to write one myself). If I'm writing a method (or function) I consider arguments as whatever acts the way I need so that I can implement the the concept on them (for example if I'm writing a tensor sum I just consider arguments as n-dimensional iterable arrays and implement assuming that - and declare for the compiler when my method is applicable, without having to care about all other implementations of sum - if anyone needs a scalar sum them that person can implement it and through collaboration we all expand the concept of sum).

And the fact that whoever uses a function can abstract away the implementation, and whoever writes a function can abstract away the whole extension of the arguments (through both duck typing and the fact that the compiler will deal with choosing the correct implementation of the concept) means everything plays along fine without having to deal with details of each side.


First and foremost you should have a general understanding of type inference. You also have to understand the difference between a function and a method (in Julia's terms), see: Functions: https://docs.julialang.org/en/v1/manual/functions/ Methods: https://docs.julialang.org/en/v1/manual/methods/

Once that's understood, multiple dispatch is simply using all of a function's arguments inferred types to choose which method should be invoked.


Maybe you’re using the term loosely, but one definitely shouldn’t have to understand type inference to write working Julia programs. Unlike static type systems like Haskel or ML where inference is part of the spec, inference in Julia is just an optimization and doesn’t affect behavior at all.


Relating to other programming languages, it was the first term that came to mind. You have to know the type before you can specialize a function, don't you?

I think it's a good mental model to constantly keep the types in mind, but I have made the experience that people working exclusively in dynamically typed languages, i.e. the majority of data scientists, don't share that mental model.


I found this article by Chris Rackaukas to be pretty helpful: https://www.stochasticlifestyle.com/type-dispatch-design-pos...


Please add ways to donate for your cause. Looking forward for a CAS in a modern language rather than what we have now.


Can I use this now? Is there a getting-started guide or gallery of examples?



Please don’t call this “Symbolics,” that name is already taken by the Lisp Machine vendor that had its own excellent CAS, Maxima, derived from the original MACSYMA at MIT.


First of all, we call it "Symbolics.jl", not "Symbolics" (same with all julia packages) so that lifts the ambiguity already ambiguity, but even still almost every word imaginable has been used in multiple, conflicting contexts. It's fine.


Macsyma was the name of the CAS from the defunct Symbolics, Inc..

Maxima is an open source variant, based on an earlier version derived from the original Macsyma code base at MIT.


Computer algebra systems are incredibly useful. It's a shame another project is started that's not in C. The world is really waiting for a library that can be used in effectively all programming languages.


Julia will be able to generate c-compatible .so/.dlls in a not so distant future.


This sounds very interesting. Do you have any more information on this?


Currently the best you can do is to build a somewhat unwieldy relocatable “bundle” with PackageCompiler.jl [1], but there are apparently plans for actual static compilation using an approach more similar to that used in GPUCompiler.jl [2], and the latter approach should as far as I understand allow for the creation of proper .so/.dlls.

[1] https://github.com/JuliaLang/PackageCompiler.jl

[2] https://github.com/JuliaGPU/GPUCompiler.jl




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