Reduce.jl

Symbolic parser generator for Julia language expressions using REDUCE algebra term rewriter

• Reduce.jl
• Introduction
• Setup
• Background
• Usage
• Troubleshooting
• Reduce.jl Library
• Index
• Reduce Interface
• Tools & Options

Introduction

The premise behind Reduce.jl is based on the idea that Symbol and Expr types can be translated into computer algebra rewrite commands and then automatically parsed back into Julia ASTs, essentially extending the Julia language into a fully programable symbolic AST rewrite environment.

REDUCE is a system for general algebraic computations of interest to mathematicians, scientists and engineers:

• exact arithmetic using integers and fractions; arbitrary precision numerical approximation;
• polynomial and rational function algebra; factorization and expansion of polynomials and rational functions;
• differentiation and integration of multi-variable functions; exponential, logarithmic, trigonometric and hyperbolic;
• output of results in a variety of formats; automatic and user controlled simplification of expressions;
• substitutions and pattern matching of expressions; quantifier elimination and decision for interpreted first-order logic;
• solution of ordinary differential equations; calculations with a wide variety of special (higher transcendental) functions;
• calculations involving matrices with numerical and symbolic elements; general matrix and non-commutative algebra;
• powerful intuitive user-level programming language; generating optimized numerical programs from symbolic input;
• Dirac matrix calculations of interest to high energy physicists; solution of single and simultaneous equations.

Interface for applying symbolic manipulation on Julia expressions using REDUCE's term rewrite system:

• reduce expressions are RExpr objects that can parse into julia Expr objects and vice versa;
• interface link communicates and interprets via various reduce output modes using rcall method;
• high-level reduce-julia syntax parser-generator walks arbitrary expression to rewrite mathematical code;
• import operators from REDUCE using code generation to apply to arbitrary computational expressions;
• interactive reduce> REPL within the Julia terminal window activated by } key;
• extended arithmetic operators +,-,*,^,/,// compute on Symbol and Expr types;
• provides hundreds of internal and external methods each supporting many argument types.

Additional packages that depend on Reduce.jl are maintained at JuliaReducePkg.

The upstream REDUCE software created by Anthony C. Hearn is maintained by collaborators on SourceForge.

This package is a heavily modifed version of Nathan Smith's Maxima.jl with many additional features.

Setup

The Reduce package provides the base functionality to work with Julia and Reduce expressions, provided that you have redcsl in your path. On GNU/Linux/OSX/Windows, Pkg.build("Reduce") will automatically download a precompiled binary for you. If you are running a different Unix operating system, the build script will download the source and attempt to compile redcsl for you, success depends on the build tools installed. Automated testing for Travis CI and appveyor using Linux, OSX, and Windows are fully operational using Reduce.

julia> Pkg.add("Reduce"); Pkg.build("Reduce")
julia> using Reduce
Reduce (Free CSL version, revision 4521),  11-March-2018 ...

For linux users who wish to speed up frequent precompilation, it is possible to disable extra precompilation scripts by setting the environment variable ENV["REDPRE"] = "0" in julia (only effective when Reduce is being compiled).

View the documentation stable / latest for more features and examples.

This Reduce package for the Julia language was created by github.com/chakravala for mathematics and computer algebra research with the upstream developed REDUCE software. Please consider donating to show your thanks and appreciation to this Julia project for interfacing the upstream REDUCE language at liberapay, GitHub Sponsors, Patreon, or contribute (documentation, tests, examples) in the repository.

Background

The Reduce package currently provides a robust interface to directly use the CSL version of REDUCE within the Julia language and the REPL. This is achieved by interfacing the abstract syntax tree of Expr objects with the parser generator for RExpr objects and then using an IOBuffer to communicate with redpsl.

REDUCE is a system for doing scalar, vector and matrix algebra by computer, which also supports arbitrary precision numerical approximation and interfaces to gnuplot to provide graphics. It can be used interactively for simple calculations but also provides a full programming language, with a syntax similar to other modern programming languages. REDUCE has a long and distinguished place in the history of computer algebra systems. Other systems that address some of the same issues but sometimes with rather different emphasis are Axiom, Macsyma (Maxima), Maple and Mathematica. REDUCE is implemented in Lisp (as are Axiom and Macsyma), but this is completely hidden from the casual user. REDUCE primarily runs on either Portable Standard Lisp (PSL) or Codemist Standard Lisp (CSL), both of which are included in the SourceForge distribution. PSL is long-established and compiles to machine code, whereas CSL is newer and compiles to byte code. Hence, PSL may be faster but CSL may be available on a wider range of platforms.

Releases of Reduce.jl enable the general application of various REDUCE functionality and packages to manipulate the Julia language to simplify and compute new program expressions at run-time. Intended for uses where a symbolic pre-computation is required for numerical algorithm code generation.

Julia is a high-level, high-performance dynamic programming language for numerical computing. It provides a sophisticated compiler, distributed parallel execution, numerical accuracy, and an extensive mathematical function library. Julia’s Base library, largely written in Julia itself, also integrates mature, best-of-breed open source C and Fortran libraries for linear algebra, random number generation, signal processing, and string processing. The strongest legacy of Lisp in the Julia language is its metaprogramming support. Like Lisp, Julia represents its own code as a data structure of the language itself. Since code is represented by objects that can be created and manipulated from within the language, it is possible for a program to transform and generate its own code. This allows sophisticated code generation without extra build steps, and also allows true Lisp-style macros operating at the level of abstract syntax trees.

Usage

The extended algebraic symbolic expression mode of Reduce.jl is activated with ForceImport.jl by

@force using Reduce.Algebra

This locally extends native Julia functions to Symbol and Expr types in the current module without extending global methods. Alternatively, the methods it provides can be accesed by prefixing Algebra. in front of the method.

Reduce expressions encapsulated into RExpr objects can be manipulated within julia using the standard syntax. Create an expression object either using the RExpr("expression") string constructor or R"expression". Additionally, arbitrary julia expressions can also be parsed directly using the RExpr(expr) constructor. Internally RExpr objects are represented as an array that can be accessed by calling *.str[n] on the object.

When Reduce is used in Julia, standard arithmetic operations are now extended to also work on Symbol and Expr types.

julia> 1-1/:n
:((n - 1) // n)

julia> ans^-:n
:(1 // ((n - 1) // n) ^ n)

julia> limit(ans,:n,Inf)
ℯ = 2.7182818284590...

Julia abstract syntax trees are automatically converted into sequences of reduce statements (using RExpr constructor) that are in return parsed into julia quote blocks usig parse. The rcall method is used to evaluate any type of expression.

julia> :(int(sin(im*x+pi)^2-1,x)) |> rcall
:((1 - (ℯ ^ (4x) + 4 * ℯ ^ (2x) * x)) // (8 * ℯ ^ (2x)))

However, there are often multiple equivalent ways of achieving the same result:

julia> int(sin(im*:x+π)^2-1,:x)
:((1 - (ℯ ^ (4x) + 4 * ℯ ^ (2x) * x)) // (8 * ℯ ^ (2x)))

The output of rcall will be the same as its input type.

julia> "int(sin(y)^2, y)" |> rcall
"( - cos(y)*sin(y) + y)/2"

Use rcall(expr,switches...) to evaluate expr using REDUCE mode switches like :expand, :factor, and :latex.

julia> :((x+im+π)^2; int(1/(1+x^3),x)) |> RExpr
^(+(x,i,pi),2);
int(/(1,+(1,^(x,3))),x);

julia> rcall(ans,:horner) |> parse
quote
    ((π + 2x) * π + 2 * (π + x) * im + x ^ 2) - 1
    ((2 * sqrt(3) * atan((2x - 1) // sqrt(3)) - log((x ^ 2 - x) + 1)) + 2 * log(x + 1)) // 6
end

Mathematical operators and REDUCE modes can be applied directly to Expr and RExpr objects.

julia> Expr(:function,:(fun(a,b)),:(return 4x^4-44x^3+61x^2+270x-525)) |> horner
:(function fun(a, b)
        return ((4 * (x - 11) * x + 61) * x + 270) * x - 525
    end)

Additionally, REDUCE switch statements can be used as macros to control evaluation of expressions.

julia> @rounded @factor x^3-2x+1
:((x + 1.61803398875) * (x - 1) * (x - 0.61803398875))

Most core features have a corresponding Julia method, but language features that have not been implemented yet can also be directly evaluated with rcall using a synergy of julia syntax.

julia> Expr(:for,:(i=2:34),:(product(i))) |> rcall
:(@big_str "295232799039604140847618609643520000000")

The squash function provides a way to reduce full program blocks into simplified functions, e.g.

julia> Expr(:function,:(example(a,b)),quote
           z = 3
           target = z * :a * :b
           z -= 1
           target += z*(1-:a)*(1-:b)
       end) |> squash |> factor
:(function example(a, b)
        (5b - 2) * a - 2 * (b - 1)
    end)

where z is a program variable and :a and :b are symbolic variables.

Loading packages

Packages which come shipped with REDUCE can be loaded with the load_package method. For example, the optimize method is available with

julia> load_package(:scope)

julia> Algebra.optimize(:(z = a^2*b^2+10*a^2*m^6+a^2*m^2+2*a*b*m^4+2*b^2*m^6+b^2*m^2))
quote
    g40 = b * a
    g44 = m * m
    g41 = g44 * b * b
    g42 = g44 * a * a
    g43 = g44 * g44
    z = g41 + g42 + g40 * (2g43 + g40) + g43 * (2g41 + 10g42)
end

Other packages can be loaded, but not all of them come with pre-defined Julia dispatch methods.

Matrices

Some special support for symbolic matrices has also been added to Reduce.Algebra methods,

julia> [:x 1; :y 2]^-1
2×2 Array{Any,2}:
 :(2 / (2x - y))   :(-1 / (2x - y))
 :(-y / (2x - y))  :(x / (2x - y))

The jacobian method has been added to the ReduceLinAlg package, which is dedicated to the LINALG extra package included with Reduce binaries.

julia> using ReduceLinAlg

julia> eqns = [:x1-:x2, :x1+:x2-:x3+:x6t, :x1+:x3t-:x4, 2*:x1tt+:x2tt+:x3tt+:x4t+:x6ttt, 3*:x1tt+2*:x2tt+:x5+0.1*:x8, 2*:x6+:x7, 3*:x6+4*:x7, :x8-sin(:x8)]
8-element Array{Expr,1}:
 :(x1 - x2)
 :(x1 - ((x3 - x6t) - x2))
 :((x3t - x4) + x1)
 :(x4t + x6ttt + x3tt + x2tt + 2x1tt)
 :((10x5 + x8 + 20x2tt + 30x1tt) // 10)
 :(2x6 + x7)
 :(3x6 + 4x7)
 :(x8 - sin(x8))

julia> vars = [:x1, :x2, :x3, :x4, :x6, :x7, :x1t, :x2t, :x3t, :x6t, :x7t, :x6tt, :x7tt];

julia> jacobian(eqns, vars) |> Reduce.mat
8×13 Array{Any,2}:
 1  -1   0   0  0  0  0  0  0  0  0  0  0
 1   1  -1   0  0  0  0  0  0  1  0  0  0
 1   0   0  -1  0  0  0  0  1  0  0  0  0
 0   0   0   0  0  0  0  0  0  0  0  0  0
 0   0   0   0  0  0  0  0  0  0  0  0  0
 0   0   0   0  2  1  0  0  0  0  0  0  0
 0   0   0   0  3  4  0  0  0  0  0  0  0
 0   0   0   0  0  0  0  0  0  0  0  0  0

The package also provides a demonstration of how additional Reduce methods can be imported into Julia.

Output mode

Various output modes are supported. While in the REPL, the default nat output mode will be displayed for RExpr objects.

julia> :(sin(x*im) + cos(y*MathConstants.φ)) |> RExpr

     (sqrt(5) + 1)*y
cos(-----------------) + sinh(x)*i
            2

This same output can also be printed to the screen by calling print(nat(r)) method.

It is possible to direclty convert a julia expression object to LaTeX code using the latex method.

julia> print(@latex sin(x) + cos(y*MathConstants.φ))
\begin{displaymath}
\cos \left(\left(\left(\sqrt {5}+1\right) y\right)/2\right)+\sin \,x
\end{displaymath}

Internally, this command essentially expands to rcall(:(sin(x) + cos(y*MathConstants.φ)),:latex) |> print, which is equivalent.

In IJulia the display output of RExpr objects will be rendered LaTeX with the rlfi REDUCE package in latex mode.

REPL interface

Similar to <kbd>?</kbd> help and <kbd>;</kbd> shell modes in Julia, Reduce provides a reduce> REPL mode by pressing <kbd>shift</kbd>+<kbd>]</kbd> as the first character in the julia terminal prompt. The output is in nat mode.

reduce> df(atan(golden_ratio*x),x);

          2              2
 sqrt(5)*x  + sqrt(5) - x  + 1
-------------------------------
           4      2
       2*(x  + 3*x  + 1)

Troubleshooting

If the reduce> REPL is not appearing when } is pressed or the Reduce pipe is broken, the session can be restored by simply calling Reduce.Reset(), without requiring a restart of julia or reloading the package. This kills the currently running Reduce session and then re-initializes it for new use.

Otherwise, questions can be asked on gitter/discourse or submit your issue or pull-request if you require additional features or noticed some unusual edge-case behavior.

AbstractTensors interoperability

By importing the AbstractTensors.jl module, the Reduce is able to correctly bypass operations on TensorAlgebra elements to the correct methods within the scope of the Reduce.Algebra module. This requires no additional overhead for the Grassmann.jl or Reduce packages, because the AbstractTensors interoperability interface enables separate precompilation of both.