# Sweet Egison

The Sweet Egison is a shallow embedding implementation of non-linear pattern matching with extensible and polymorphic patterns [1].
This library desguars the Egison pattern-match expressions into Haskell programs that use non-deterministic monads.
This library provides a base of the pattern-match-oriented (PMO) programming style [2] for Haskell users at a practical level of efficiency.

## Getting started

We code the equivalent pattern match of `case [1, 2, 3] of x : xs -> (x, xs)`

in this library as follows:

```
> matchAll dfs [1, 2, 3] (List Something) [[mc| $x : $xs -> (x, xs) |]]
[(1,[2,3])]
```

Here, we can only observe the small syntactic difference in pattern expressions: the variable bindings are prefixed with `$`

. (We'll come back to `List Something`

later.)
You may notice that `matchAll`

returns a list.
In our library, pattern matching can return many results.
See the following example that doubles all elements in a list:

```
> take 10 $ matchAll dfs [1 ..] (List Something) [[mc| _ ++ $x : _ -> x * 2 |]]
[2,4,6,8,10,12,14,16,18,20]
```

`++`

is the *join* operator that decomposes a list into an initial prefix and the remaining suffix.
We can implement `map`

with pattern matching using this:

```
> map f xs = matchAll dfs xs (List Something) [[mc| _ ++ $x : _ -> f x |]]
> map (*2) [1,2,3]
[2,4,6]
```

Note that we don't see any recursions or `fold`

s in our `map`

definition! An intuition of `map`

function, that applies the function to all elements, are expressed directly in the pattern expression.

### Matchers

Because our pattern matching can return many results, we can use it to decompose *non-free data types* such as multisets and sets.
For example:

```
> matchAll dfs [1, 2, 3] (Multiset Something) [[mc| $x : $xs -> (x, xs) |]]
[(1,[2,3]),(2,[1,3]),(3,[1,2])]
```

We use `Multiset Something`

instead of `List Something`

here to match the target `[1, 2, 3]`

as a multiset.
These parameters such as `Multiset Something`

, `List (List Something)`

, and `Something`

are called *matchers* and specify pattern-matching methods.
Given a matcher `m`

, `Multiset m`

is a matcher for multisets that matches its elements with `m`

.
`Something`

is a matcher that provides simple matching methods for an arbitrary value.
Pattern constructors such as `:`

and `++`

are overloaded over matchers for collections to archive the ad-hoc polymorphism of patterns.

### Controlling matching strategy

Some pattern matching have infinitely many results and `matchAll bfs`

is designed to be able to enumerate all the results.
For this purpose, `matchAll bfs`

traverses a search tree for pattern matching in the breadth-first order.
The following example illustrates this:

```
> take 10 $ matchAll bfs [1 ..] (Set Something) [[mc| $x : $y : _ -> (x, y) |]]
[(1,1),(2,1),(1,2),(3,1),(1,3),(2,2),(1,4),(4,1),(1,5),(2,3)]
```

We can use the depth-first search with `matchAll dfs`

.

```
> take 10 $ matchAll dfs [1 ..] (Set Something) [[mc| $x : $y : _ -> (x, y) |]]
[(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10)]
```

In most cases, the depth-first search is faster than the default breadth-first search strategy.
It is recommended to always use `matchAll dfs`

if it is OK to do so.

With `matchAll dfs`

, we can define an intuitive pattern-matching version of `concat`

function on lists.

```
> concat xs = matchAll dfs xs (List (List Something)) [[mc| _ ++ (_ ++ $x : _) : _ -> x |]]
> concat [[1,2], [3,4,5]]
[1,2,3,4,5]
```

### Non-linear patterns

The non-linear pattern is another powerful pattern-matching feature.
It allows us to refer the value bound to variables appear in the left side of the pattern.
We provide a pattern syntax named value patterns in the form of `#e`

.
The `Eql`

matcher enables value patterns to match with targets that are equal to the corresponding expression.
For example, the following example enumerates (p, p+2) pairs of primes:

```
> import Data.Numbers.Primes ( primes )
> take 10 $ matchAll bfs primes (List Eql) [[mc| _ ++ $p : #(p + 2) : _ -> (p, p+2) |]]
[(3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61),(71,73),(101,103),(107,109)]
```

We can implement a pattern-matching version of set functions such as `member`

and `intersect`

in a declarative way using non-linear patterns.
Match clauses are monoids and can be concatenated using `<>`

.

```
> member x xs = match dfs xs (Multiset Eql) [[mc| #x : _ -> True |], [mc| _ -> False |]]
> member 1 [3,4,1,4]
True
> intersect xs ys = matchAll dfs (xs, ys) (Pair (Set Eql) (Set Eql)) [[mc| ($x : _, #x : _) -> x |]]
> intersect [1,2,3] [4,5,3,2]
[2,3]
```

### Further readings

Some practical applications of PMO such as a SAT solver are placed under example/.
Detailed information of Egison, the original PMO language implementation, can be found on https://www.egison.org/ or in [1].
You can learn more about pattern-match-oriented programming style in [2].

## Implementation

Sweet Egison transform patterns into a program that uses non-deterministic monads.
Our quasi-quoter `mc`

translates match clauses into functions that take a target and return a non-deterministic computation as `MonadPlus`

-like monadic expression.
As `MonadPlus`

can express backtracking computation, we can perform efficient backtracking pattern matching.
For example, the match clause `[mc| $x : #(x + 10) : _ -> (x, x + 10) |]`

is transformed as follows:

```
\ (mat_a5sV, tgt_a5sW)
-> let (tmpM_a5sX, tmpM_a5sY) = (consM mat_a5sV) tgt_a5sW
in
((fromList (((cons (GP, GP)) mat_a5sV) tgt_a5sW))
>>=
(\ (tmpT_a5sZ, tmpT_a5t0)
-> let x = tmpT_a5sZ in
let (tmpM_a5t1, tmpM_a5t2) = (consM tmpM_a5sY) tmpT_a5t0
in
((fromList (((cons (GP, WC)) tmpM_a5sY) tmpT_a5t0))
>>=
(\ (tmpT_a5t3, tmpT_a5t4)
-> ((fromList ((((value (x + 10)) ()) tmpM_a5t1) tmpT_a5t3))
>>= (\ () -> pure (x, x + 10)))))))
```

The infix operators `:`

and `++`

are synonyms of `cons`

and `join`

, respectively, and desugared in that way during translation.

The `matchAll`

function is defined as a function that creates and passes the argument for this non-deterministic monads.

```
matchAll strategy target matcher =
concatMap (\b -> toList (strategy (matcher, target) >>= b))
```

Consequently, the pattern-match expression

```
matchAll dfs [1, 2, 3, 12] (Multiset Eql)
[[mc| $x : #(x + 10) : _ -> (x, x + 10) |]]
-- [(2, 12)]
```

is transformed into a program that is equivalent to the following:

```
concatMap (\b -> toList (dfs (Multiset Eql, [1, 2, 3, 12]) >>= b))
[\ (mat_a5sV, tgt_a5sW)
-> let (tmpM_a5sX, tmpM_a5sY) = (consM mat_a5sV) tgt_a5sW
in
((fromList (((cons (GP, GP)) mat_a5sV) tgt_a5sW))
>>=
(\ (tmpT_a5sZ, tmpT_a5t0)
-> let x = tmpT_a5sZ in
let (tmpM_a5t1, tmpM_a5t2) = (consM tmpM_a5sY) tmpT_a5t0
in
((fromList (((cons (GP, WC)) tmpM_a5sY) tmpT_a5t0))
>>=
(\ (tmpT_a5t3, tmpT_a5t4)
-> ((fromList ((((value (x + 10)) ()) tmpM_a5t1) tmpT_a5t3))
>>= (\ () -> pure (x, x + 10)))))))]
```

### MiniEgison (Deep Embedding) vs. Sweet Egison (Shallow Embedding)

miniEgison is also a Haskell library that implements Egison pattern matching.
The main difference between miniEgison and Sweet Egison is that Sweet Egison translates pattern matching into Haskell control expressions (shallow embedding), whereas miniEgison translates it into Haskell data expressions (deep embedding).
As a result, Sweet Egison is faster than miniEgison.
The following benchmark is taken using MacBook Pro (2017, 2.3 GHz Intel Core i5).

| | comb2 (n = 15000) | perm2 (n = 5000) | CDCL (50 vars) |
|--------------|-------------------|------------------|----------------|
| miniEgison | 13.029 sec | 3.854 sec | 1.025 sec |
| Sweet Egison | 0.303 sec | 0.462 sec | 0.097 sec |

There is almost no execution performance differences between programs written using list comprehensions and Sweet Egison.

| | comb2 (n = 15000) | comb2 (n = 15000) | perm2 (n = 5000) | perm2 (n = 10000) |
|---------------------|-------------------|-------------------|------------------|-------------------|
| List Comprehensions | 0.347 sec | 1.244 sec | 0.409 sec | 2.077 sec |
| Sweet Egison | 0.309 sec | 1.081 sec | 0.434 sec | 1.984 sec |

Programs used for the above benchmarks are follows:

## Bibliography

- [1] Satoshi Egi and Yuichi Nishiwaki: Functional Programming in Pattern-Match-Oriented Programming Style, The Art, Science, and Engineering of Programming, 2020, Vol. 4, Issue 3, Article 7, DOI: 10.22152/programming-journal.org/2020/4/7
- [2] Satoshi Egi and Yuichi Nishiwaki: Non-linear Pattern Matching with Backtracking for Non-free Data Types, APLAS 2018 - Asian Symposium on Programming Languages and Systems, DOI: 11.1007/978-3-030-02768-1_1