**Monthly Downloads**: 4,174

**Programming language**: Haskell

**License**: MIT License

**Latest version**: v4.0.0

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## README

## The Egison Programming Language

Egison is a functional programming language featuring its expressive pattern-matching facility. Egison allows users to define efficient and expressive pattern-matching methods for arbitrary user-defined data types including non-free data types such as lists, multisets, sets, trees, graphs, and mathematical expressions. This is the repository of the interpreter of Egison.

For more information, visit our website.

## Refereed Papers

### Pattern Matching

- Satoshi Egi, Yuichi Nishiwaki: Non-linear Pattern Matching with Backtracking for Non-free Data Types (APLAS 2018)
- Satoshi Egi, Yuichi Nishiwaki: Functional Programming in Pattern-Match-Oriented Programming Style ( 2020)

### Tensor Index Notation

- Satoshi Egi: Scalar and Tensor Parameters for Importing Tensor Index Notation including Einstein Summation Notation (Scheme Workshop 2017)

## Non-Linear Pattern Matching for Non-Free Data Types

We can describe non-linear pattern matching for non-free data types in Egison. A non-free data type is a data type whose data have no canonical form, a standard way to represent that object. For example, multisets are non-free data types because the multiset {a,b,b} has two other equivalent but literally different forms {b,a,b} and {b,b,a}. Expressive pattern matching for these data types 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!

```
twinPrimes :=
matchAll primes as list integer with
| _ ++ $p :: #(p + 2) :: _ -> (p, p + 2)
take 8 twinPrimes
-- [(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)]
```

### Poker Hands

The following code is the program that determines poker-hands written in Egison. All hands are expressed in a single pattern.

```
poker cs :=
match cs as multiset card with
| card $s $n :: card #s #(n-1) :: card #s #(n-2) :: card #s #(n-3) :: card #s #(n-4) :: _
-> "Straight flush"
| card _ $n :: card _ #n :: card _ #n :: card _ #n :: _ :: []
-> "Four of a kind"
| card _ $m :: card _ #m :: card _ #m :: card _ $n :: card _ #n :: []
-> "Full house"
| card $s _ :: card #s _ :: card #s _ :: card #s _ :: card #s _ :: []
-> "Flush"
| card _ $n :: card _ #(n-1) :: card _ #(n-2) :: card _ #(n-3) :: card _ #(n-4) :: []
-> "Straight"
| card _ $n :: card _ #n :: card _ #n :: _ :: _ :: []
-> "Three of a kind"
| card _ $m :: card _ #m :: card _ $n :: card _ #n :: _ :: []
-> "Two pair"
| card _ $n :: card _ #n :: _ :: _ :: _ :: []
-> "One pair"
| _ :: _ :: _ :: _ :: _ :: [] -> "Nothing"
```

### Graphs

We can pattern-match against graphs. We can write program to solve the travelling salesman problem in a single pattern-matching expression.

```
graph := multiset (string, multiset (string, integer))
graphData :=
[("Berlin", [("New York", 14), ("London", 2), ("Tokyo", 14), ("Vancouver", 13)]),
("New York", [("Berlin", 14), ("London", 12), ("Tokyo", 18), ("Vancouver", 6)]),
("London", [("Berlin", 2), ("New York", 12), ("Tokyo", 15), ("Vancouver", 10)]),
("Tokyo", [("Berlin", 14), ("New York", 18), ("London", 15), ("Vancouver", 12)]),
("Vancouver", [("Berlin", 13), ("New York", 6), ("London", 10), ("Tokyo", 12)])]
trips :=
let n := length graphData in
matchAll graphData as graph with
| (#"Berlin", (($s_1,$p_1) : _)) ::
loop $i (2, n - 1)
((#s_(i - 1), ($s_i, $p_i) :: _) :: ...)
((#s_(n - 1), (#"Berlin" & $s_n, $p_n) :: _) :: [])
-> sum (map (\i -> p_i) [1..n]), map (\i -> s_i) [1..n]
car (sortBy (\(_, x), (_, y) -> compare x y)) trips)
-- (["London", "New York", "Vancouver", "Tokyo"," Berlin"], 46)
```

## Egison as a Computer Algebra System

As an application of Egison pattern matching, we have implemented a computer algebra system on Egison. The most part of this computer algebra system is written in Egison and extensible using Egison.

### Symbolic Algebra

Egison treats unbound variables as symbols.

```
> x
x
> (x + y)^2
x^2 + 2 * x * y + y^2
> (x + y)^4
x^4 + 4 * x^3 * y + 6 * x^2 * y^2 + 4 * x * y^3 + y^4
```

We can handle algebraic numbers, too.

```
> sqrt x
sqrt x
> sqrt 2
sqrt 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 + i) * (1 + i)
2 * i
> (x + y * i) * (x + y * i)
x^2 + 2 * x * y * i - 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 * 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.

```
> qF' 1 1 (-1)
((-1 + sqrt 5) / 2, (-1 - sqrt 5) / 2)
> t := fst (qF' 1 1 (-1))
> qF' 1 (-t) 1
((-1 + sqrt 5 + sqrt 2 * sqrt (-5 - sqrt 5)) / 4, (-1 + sqrt 5 - sqrt 2 * sqrt (-5 - sqrt 5)) / 4)
> z := fst (qF' 1 (-t) 1)
> z
(-1 + sqrt 5 + sqrt 2 * sqrt (-5 - 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
exp (x * i) * i
> d/d (d/d (log x) x) x
-1 / x^2
> d/d (cos x * sin x) x
-2 * (sin x)^2 + 1
```

### Taylor Expansion

The following sample executes Taylor expansion on Egison. We verify Euler's formula in the following sample.

```
> take 8 (taylorExpansion (exp (i * x)) x 0)
[1, x * i, - x^2 / 2, - x^3 * i / 6, x^4 / 24, x^5 * i / 120, - x^6 / 720, - x^7 * i / 5040]
> take 8 (taylorExpansion (cos x) x 0)
[1, 0, - x^2 / 2, 0, x^4 / 24, 0, - x^6 / 720, 0]
> take 8 (taylorExpansion (i * sin x) x 0)
[0, x * i, 0, - x^3 * i / 6, 0, x^5 * i / 120, 0, - x^7 * i / 5040]
> take 8 (map2 (+) (taylorExpansion (cos x) x 0) (taylorExpansion (i * sin x) x 0))
[1, x * i, - x^2 / 2, - x^3 * i / 6, x^4 / 24, x^5 * i / 120, - x^6 / 720, - x^7 * i / 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
x := [| θ, φ |]
X := [| r * (sin θ) * (cos φ) -- x
, r * (sin θ) * (sin φ) -- y
, r * (cos θ) -- z
|]
e_i_j := (∂/∂ X_j x~i)
-- Metric tensors
g_i_j := generateTensor (\x y -> V.* e_x_# e_y_#) [2, 2]
g~i~j := 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
Γ_i_j_k := (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)
Γ_1_#_# -- [| [| 0, 0 |], [| 0, -1 * r^2 * (sin θ) * (cos θ) |] |]_#_#
Γ_2_#_# -- [| [| 0, r^2 * (sin θ) * (cos θ) |], [| r^2 * (sin θ) * (cos θ), 0 |] |]_#_#
Γ~i_j_k := withSymbols [m]
g~i~m . Γ_m_j_k
Γ~1_#_# -- [| [| 0, 0 |], [| 0, -1 * (sin θ) * (cos θ) |] |]_#_#
Γ~2_#_# -- [| [| 0, (cos θ) / (sin θ) |], [| (cos θ) / (sin θ), 0 |] |]_#_#
-- Riemann curvature
R~i_j_k_l := withSymbols [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
x := [| θ, φ |]
g_i_j := [| [| r^2, 0 |], [| 0, r^2 * (sin θ)^2 |] |]_i_j
g~i~j := [| [| 1 / r^2, 0 |], [| 0, 1 / (r^2 * (sin θ)^2) |] |]~i~j
-- Christoffel symbols
Γ_j_l_k := (1 / 2) * (∂/∂ g_j_l x~k + ∂/∂ g_j_k x~l - ∂/∂ g_k_l x~j)
Γ~i_k_l := withSymbols [j] g~i~j . Γ_j_l_k
-- Exterior derivative
d %t := !(flip ∂/∂) x t
-- Wedge product
infixl expression 7 ∧
(∧) %x %y := x !. y
-- Connection form
ω~i_j := Γ~i_j_#
-- Curvature form
Ω~i_j := withSymbols [k]
antisymmetrize (d ω~i_j + ω~i_k ∧ ω~k_j)
Ω~#_#_1_1 -- [| [| 0, 0 |], [| 0, 0 |] |]~#_#
Ω~#_#_1_2 -- [| [| 0, (sin θ)^2 / 2|], [| -1 / 2, 0 |] |]~#_#
Ω~#_#_2_1 -- [| [| 0, -1 * (sin θ)^2 / 2 |], [| 1 / 2, 0 |] |]~#_#
Ω~#_#_2_2 -- [| [| 0, 0 |], [| 0, 0 |] |]~#_#
```

### 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 fulfills the following two requirements at the same time.

- Efficient backtracking algorithm for non-linear pattern matching.
- Extensibility of patterns.

Additionally, it fulfills the following requirements.

- Polymorphism of patterns.
- Pattern matching with infinitely many results.

Please read our paper for details.

## Installation

Installation guide for MacOS, Linux and Windows are available on our website.

If you are a beginner of Egison, it would be better to install `egison-tutorial`

.

We also have online interpreter and online tutorial. Enjoy!

## Notes for Developers

### How to Build

```
$ stack init
$ stack build --fast
```

### How to Run Test

```
$ stack init
$ stack test
```

## Community

We have a mailing list. Please join us!

We are on Twitter. Please follow us.

## License

Egison is released under the MIT license.

We 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.

*
*Note that all licence references and agreements mentioned in the egison README section above
are relevant to that project's source code only.
*