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The Egison Programming Language
Egison is the pattern-matching-oriented purely functional programming language. We can directly represent pattern-matching against lists, multisets, sets, trees, graphs and any kind of data types. This is the repository of the interpreter of Egison.
For more information, visit our website.
Non-Linear Pattern-Matching against Non-Free Data Types
We can do non-linear pattern-matching against non-free data types in Egison. An non-free data type is a data type whose data have no canonical form, a standard way to represent that object. It enables us to write elegant programs.
Twin Primes
We can use pattern-matching for enumeration. The following code enumerates all twin primes from the infinite list of prime numbers with pattern-matching!
Poker Hands
The following code is the program that determines poker-hands written in Egison. All hands are expressed in a single pattern.
Mahjong
We can write a pattern even against mahjong tiles. We modularize patterns to represent complex mahjong hands.
Graphs
We can pattern-match against graphs. We can write program to solve the travelling salesman problem in a single pattern-matching expression.
Aren't these exciting? The pattern-matching of Egison is very powerful. We can use it for pattern-matching also against graphs and tree-structures such as XML.
Egison as a Computer Algebra System
As an application of Egison, we implemented a computer algebra system on Egison. The most part of this computer algebra system is written in Egison and extensible in Egison.
Symbolic Algebra
Unbound variables are treated as symbols.
> x
x
> (** (+ x y) 2)
(+ x^2 (* 2 x y) y^2)
> (** (+ x y) 10)
(+ x^10 (* 10 x^9 y) (* 45 x^8 y^2) (* 120 x^7 y^3) (* 210 x^6 y^4) (* 252 x^5 y^5) (* 210 x^4 y^6) (* 120 x^3 y^7) (* 45 x^2 y^8) (* 10 x y^9) y^10)
We can handle algebraic numbers, too.
> (sqrt x)
(sqrt x)
> (sqrt 2)
(sqrt 2)
> (sqrt 4)
2
> (+ x (sqrt y))
(+ x (sqrt y))
Complex Numbers
The symbol i is defined to rewrite i^2 to -1 in Egison library.
> (* i i)
-1
> (* (+ 1 (* 1 i)) (+ 1 (* 1 i)))
(* 2 i)
> (** (+ 1 (* 1 i)) 10)
(* 32 i)
> (* (+ x (* y i)) (+ x (* y i)))
(+ x^2 (* 2 i x y) (* -1 y^2))
Square Root
The rewriting rule for sqrt is also defined in Egison library.
> (* (sqrt 2) (sqrt 2))
2
> (* (sqrt 6) (sqrt 10))
(* 2 (sqrt 15))
> (sqrt x)
(sqrt x)
> (* (sqrt (* x y)) (sqrt (* 2 x)))
(* x (sqrt 2) (sqrt y))
The 5th Roots of Unity
The following is a sample to calculate the 5th roots of unity.
> (q-f' 1 1 -1)
[(/ (+ -1 (sqrt 5)) 2) (/ (+ -1 (* -1 (sqrt 5))) 2)]
> (define $t (fst (q-f' 1 1 -1)))
> (q-f' 1 (* -1 t) 1)
[(/ (+ -1 (sqrt 5) (sqrt (+ -10 (* -2 (sqrt 5))))) 4) (/ (+ -1 (sqrt 5) (* -1 (sqrt (+ -10 (* -2 (sqrt 5)))))) 4)]
> (define $z (fst (q-f' 1 (* -1 t) 1)))
> z
(/ (+ -1 (sqrt 5) (sqrt (+ -10 (* -2 (sqrt 5))))) 4)
> (** z 5)
1
Differentiation
We can implement differentiation easily in Egison.
> (d/d (** x 3) x)
(* 3 x^2)
> (d/d (** e (* i x)) x)
(* i (** e (* i x)))
> (d/d (d/d (log x) x) x)
(/ -1 x^2)
> (d/d (* (cos x) (sin x)) x)
(+ (* -1 (sin x)^2) (cos x)^2)
Taylor Expansion
The following sample executes Taylor expansion on Egison. We verify Euler's formula in the following sample.
> (take 8 (taylor-expansion (** e (* i x)) x 0))
{1 (* i x) (/ (* -1 x^2) 2) (/ (* -1 i x^3) 6) (/ x^4 24) (/ (* i x^5) 120) (/ (* -1 x^6) 720) (/ (* -1 i x^7) 5040)}
> (take 8 (taylor-expansion (cos x) x 0))
{1 0 (/ (* -1 x^2) 2) 0 (/ x^4 24) 0 (/ (* -1 x^6) 720) 0}
> (take 8 (taylor-expansion (* i (sin x)) x 0))
{0 (* i x) 0 (/ (* -1 i x^3) 6) 0 (/ (* i x^5) 120) 0 (/ (* -1 i x^7) 5040)}
> (take 8 (map2 + (taylor-expansion (cos x) x 0) (taylor-expansion (* i (sin x)) x 0)))
{1 (* i x) (/ (* -1 x^2) 2) (/ (* -1 i x^3) 6) (/ x^4 24) (/ (* i x^5) 120) (/ (* -1 x^6) 720) (/ (* -1 i x^7) 5040)}
Tensor Index Notation
Egison supports tesnsor index notation. We can use Einstein notation to express arithmetic operations between tensors.
The method for importing tensor index notation into programming is discussed in Egison tensor paper.
The following sample is from Riemann Curvature Tensor of S2 - Egison Mathematics Notebook.
;; Parameters
(define $x [|θ φ|])
(define $X [|(* r (sin θ) (cos φ)) ; = x
(* r (sin θ) (sin φ)) ; = y
(* r (cos θ)) ; = z
|])
;; Local basis
(define $e_i_j (∂/∂ X_j x~i))
e_i_j
;[|[|(* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
; [|(* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]
; |]_#_#
;; Metric tensor
(define $g__ (generate-tensor 2#(V.* e_%1_# e_%2_#) {2 2}))
(define $g~~ (M.inverse g_#_#))
g_#_#;[| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |]_#_#
g~#~#;[| [| (/ 1 r^2) 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) |] |]~#~#
;; Christoffel symbols
(define $Γ_j_k_l
(* (/ 1 2)
(+ (∂/∂ g_j_l x~k)
(∂/∂ g_j_k x~l)
(* -1 (∂/∂ g_k_l x~j)))))
(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
Γ~1_#_#;[| [| 0 0 |] [| 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
Γ~2_#_#;[| [| 0 (/ (cos θ) (sin θ)) |] [| (/ (cos θ) (sin θ)) 0 |] |]_#_#
;; Riemann curvature tensor
(define $R~i_j_k_l
(with-symbols {m}
(+ (- (∂/∂ Γ~i_j_l x~k) (∂/∂ Γ~i_j_k x~l))
(- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
R~#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
R~#_#_1_2;[| [| 0 (sin θ)^2 |] [| -1 0 |] |]~#_#
R~#_#_2_1;[| [| 0 (* -1 (sin θ)^2) |] [| 1 0 |] |]~#_#
R~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#
Differential Forms
By designing the index completion rules for omitted indices, we can use the above notation to express a calculation handling the differential forms.
The following sample is from Curvature Form - Egison Mathematics Notebook.
;; Parameters and metric tensor
(define $x [| θ φ |])
(define $g__ [| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |])
(define $g~~ [| [| (/ 1 r^2) 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) |] |])
;; Christoffel symbols
(define $Γ_i_j_k
(* (/ 1 2)
(+ (∂/∂ g_i_k x~j)
(∂/∂ g_i_j x~k)
(* -1 (∂/∂ g_j_k x~i)))))
(define $Γ~i_j_k (with-symbols {m} (. g~i~m Γ_m_j_k)))
;; Connection form
(define $ω~i_j (with-symbols {k} Γ~i_j_k))
;; Curvature form
(define $d
(lambda [%A]
!((flip ∂/∂) x A)))
(define $wedge
(lambda [%X %Y]
!(. X Y)))
(define $Ω~i_j (with-symbols {k}
(df-normalize (+ (d ω~i_j)
(wedge ω~i_k ω~k_j)))))
Egison Mathematics Notebook
Here are more samples.
Comparison with Related Work
There are a lot of existing work for pattern-matching.
The advantage of Egison is that it achieves all of the following features at the same time.
- Modularization of the way of pattern-matching for each data type
- Pattern-matching with multiple results (backtracking)
- Non-linear pattern-matching with lexical scoping
- Parametric polymorphism of pattern-constructors
The Pattern-Matching Mechanism section in Egison developer's manual explains how we achieve that.
Please read our paper on arXiv.org for details.
Installation
If you are using Linux, please install libncurses-dev at first.
% sudo apt-get install libncurses-dev # on Debian
To compile Egison, you also need to install Haskell Platform.
After you installed Haskell Platform, run the following commands on the terminal.
% cabal update
% cabal install egison
Now, you can try Egison.
% egison
Egison Version X.X.X(C) 2011-2014 Satoshi Egi
https://www.egison.org
Welcome to Egison Interpreter!
> ^D
Leaving Egison Interpreter.
If you are a beginner of Egison, it would be better to install egison-tutorial.
% cabal update
% cabal install egison-tutorial
% egison-tutorial
Egison Tutorial Version 3.7.4 (C) 2013-2017 Satoshi Egi
Welcome to Egison Tutorial!
** Information **
We can use a 'Tab' key to complete keywords on the interpreter.
If we type a 'Tab' key after a closed parenthesis, the next closed parenthesis will be completed.
*****************
==============================
List of sections in the tutorial.
1: Calculate numbers
2: Basics of functional programming
3: Basics of pattern-matching
4: Pattern-matching against various data types
5: Symbolic computation
6: Differential geometry: tensor analysis
7: Differential geometry: differential forms
==============================
Choose a section to learn.
(1-7): 1
====================
We can do arithmetic operations with '+', '-', '*', '/', 'modulo' and 'power'.
Examples:
(+ 1 2)
(- 30 15)
(* 10 20)
(/ 20 5)
(modulo 17 4)
(power 2 10)
====================
>
We can try it also online. Enjoy!
Note for Developers
How to Run Test
% cabal test
How to Profile the Interpreter
% sudo apt-get install haskell-platform-doc haskell-platform-prof
% cabal sandbox init
% cabal install --enable-profiling
% egison +RTS -p -RTS -l sample/sequence.egi
% cat egison.prof
Community
We have a mailing list. Please join us!
We are on Twitter. Please follow us.
Acknowledgement
I thank Ryo Tanaka, Takahisa Watanabe, Takuya Kuwahara, Kentaro Honda, and Mayuko kori for their help to implement the interpreter.
License
Copyright (c) 2011-2018, Satoshi Egi
Egison is released under the MIT license.
I used husk-scheme by Justin Ethier as reference to implement the base part of the previous version of the interpreter.
Sponsors
Egison is sponsored by Rakuten, Inc. and Rakuten Institute of Technology.