# Conjure

Conjure is a tool that synthesizes Haskell functions out of partial definitions.

## Installing

To install the latest Conjure version from Hackage, just run:

```
$ cabal update
$ cabal v1-install code-conjure
```

If you are using Cabal v3.0 or later,
avoid using `cabal install`

for the time being
and use `v1-install`

instead.

Prerequisites are express, leancheck and speculate.
They should be automatically resolved and installed by Cabal.

NOTE: the name of the Hackage package is `code-conjure`

-- not to be confused with Conjure the BitTorrent client.

## Synthesizing functions

To use Conjure, import the library with:

```
import Conjure
```

Then, declare a partial definition of a function to be synthesized.
For example,
here is a partial implementation of a function that squares a number:

```
square :: Int -> Int
square 0 = 0
square 1 = 1
square 2 = 4
```

Next, declare a list of primitives that seem like interesting pieces
in the final fully-defined implementation.
For example,
here is a list of primitives including
addition, multiplication and their neutral elements:

```
primitives :: [Prim]
primitives = [ pr (0::Int)
, pr (1::Int)
, prim "+" ((+) :: Int -> Int -> Int)
, prim "*" ((*) :: Int -> Int -> Int)
]
```

Finally, call the `conjure`

function,
passing the function name, the partial definition and the list of primitives:

```
> conjure "square" square primitives
square :: Int -> Int
-- testing 3 combinations of argument values
-- pruning with 14/25 rules
-- looking through 3 candidates of size 1
-- looking through 4 candidates of size 2
-- looking through 9 candidates of size 3
square x = x * x
```

Conjure is able to synthesize the above implementation in less than a second.

For more information, see the `eg/arith.hs`

example and
the Haddock documentation for the `conjure`

and `conjureWith`

functions.

## Synthesizing recursive functions

Conjure supports synthetization of recursive functions.

Take for example the following partial implementation of a function
that computes the factorial of a number:

```
factorial :: Int -> Int
factorial 1 = 1
factorial 2 = 2
factorial 3 = 6
factorial 4 = 24
```

Here is a list of primitives:

```
primitives :: [Prim]
primitives = [ pr (0::Int)
, pr (1::Int)
, prim "+" ((+) :: Int -> Int -> Int)
, prim "*" ((*) :: Int -> Int -> Int)
, prim "-" ((-) :: Int -> Int -> Int)
]
```

And here is what Conjure produces
with the above partial definition and list of primitives:

```
> conjure "factorial" factorial primitives
factorial :: Int -> Int
-- testing 4 combinations of argument values
-- pruning with 27/65 rules
-- looking through 3 candidates of size 1
-- looking through 4 candidates of size 2
-- looking through 13 candidates of size 3
-- looking through 34 candidates of size 4
-- looking through 75 candidates of size 5
-- looking through 183 candidates of size 6
-- looking through 577 candidates of size 7
factorial 0 = 1
factorial x = x * factorial (x - 1)
```

The above synthetization takes less than a second.

It is also possible to generate a folding implementation
like the following:

```
factorial x = foldr (*) 1 [1..x]
```

by including `enumFromTo`

and `foldr`

in the background.

For more information, see the `eg/factorial.hs`

example and
the Haddock documentation for the `conjure`

and `conjureWith`

functions.

## Synthesizing from specifications (for advanced users)

Conjure also supports synthesizing from a functional specification
with the functions `conjureFromSpec`

and `conjureFromSpecWith`

as, in some cases,
a partial definition may not be appropriate
for one of two reasons:

- Conjure may fail to "hit" the appropriate data points;
- specifying argument-result bindings may not be easy.

Take for example a function `duplicates :: Eq a => [a] -> [a]`

that should return the duplicate elements in a list without repetitions.

Let's start with the primitives:

```
primitives :: [Prim]
primitives = [ pr ([] :: [Int])
, prim "not" not
, prim "&&" (&&)
, prim ":" ((:) :: Int -> [Int] -> [Int])
, prim "elem" (elem :: Int -> [Int] -> Bool)
, prif (undefined :: [Int])
]
```

Now here's a first attempt at a partial definition:

```
duplicates' :: [Int] -> [Int]
duplicates' [] = []
duplicates' [1,2,3,4,5] = []
duplicates' [1,2,2,3,4] = [2]
duplicates' [1,2,3,3,3] = [3]
duplicates' [1,2,2,3,3] = [2,3]
```

Here is what `conjureWith`

prints:

```
> conjureWith args{maxSize=18} "duplicates" duplicates primitives
duplicates :: [Int] -> [Int]
-- testing 1 combinations of argument values
-- pruning with 21/26 rules
-- looking through 2 candidates of size 1
duplicates xs = xs
```

The generated function clearly does not follow our specification.
But if we look at the reported number of tests,
we see that only *one* of the argument-result bindings
of our partial definition was used.
Conjure failed to hit any of the argument values with five elements.
(Since Conjure uses enumeration to test functions these values have to be kept "small").

Here is a second attempt:

```
duplicates :: [Int] -> [Int]
duplicates [0,0] = [0]
duplicates [0,1] = []
duplicates [1,0,1] = [1]
```

Here is what `conjureWith`

now prints:

```
> conjureWith args{maxSize=18} "duplicates" duplicates primitives
duplicates :: [Int] -> [Int]
-- testing 3 combinations of argument values
-- pruning with 21/26 rules
-- ...
-- looking through 16 candidates of size 9
duplicates [] = []
duplicates (x:xs) = if elem x xs then [x] else []
```

The `duplicates`

function that Conjure generated is still not correct.
Nevertheless, it does follow our partial definition. We have to refine it.
Here is a third attempt with more argument-result bindings:

```
duplicates :: [Int] -> [Int]
duplicates [0,0] = [0]
duplicates [0,1] = []
duplicates [1,0,1] = [1]
duplicates [0,1,0,1] = [0,1]
```

Here is what Conjure prints:

```
duplicates [] = []
duplicates (x:xs) = if elem x xs then x:duplicates xs else []
```

This implementation follows our partial definition, but may return duplicate duplicates,
see:

```
duplicates [1,0,1,0,1] = [1,0,1]
```

Here is a fourth and final refinement:

```
duplicates :: [Int] -> [Int]
duplicates [0,0] = [0]
duplicates [0,1] = []
duplicates [1,0,1] = [1]
duplicates [0,1,0,1] = [0,1]
duplicates [1,0,1,0,1] = [0,1]
duplicates [0,1,2,1] = [1]
```

Now Conjure prints a correct implementation:

```
> conjureWith args{maxSize=18} "duplicates" duplicates primitives
duplicates :: [Int] -> [Int]
-- testing 6 combinations of argument values
-- ...
-- looking through 2189 candidates of size 17
duplicates [] = []
duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) then x:duplicates xs else duplicates xs
(in 1.5s)
```

In this case,
specifying the function with specific argument-result bindings
is perhaps not the best approach.
It took us four refinements of the partial definition to get a result.
Specifying test properties perhaps better describes what we want.
Again, we would like `duplicates`

to return all duplicate elements
without repetitions.
This can be encoded in a function using `holds`

from LeanCheck:

```
import Test.LeanCheck (holds)
duplicatesSpec :: ([Int] -> [Int]) -> Bool
duplicatesSpec duplicates = and
[ holds 360 $ \x xs -> (count (x ==) xs > 1) == elem x (duplicates xs)
, holds 360 $ \x xs -> count (x ==) (duplicates xs) <= 1
] where count p = length . filter p
```

This function takes as argument a candidate implementation of `duplicates`

and returns whether it is valid.
The first property states that all duplicates must be listed.
The second property states that duplicates themselves must not repeat.

Now, we can use the function `conjureFromSpecWith`

to generate the same duplicates function
passing our `duplicatesSpec`

as argument:

```
> conjureFromSpecWith args{maxSize=18} "duplicates" duplicatesSpec primitives
duplicates :: [Int] -> [Int]
duplicates [] = []
duplicates (x:xs) = if elem x xs && not (elem x (duplicates xs)) then x:duplicates xs else duplicates xs
(in 1.5s)
```

For more information see the `eg/dupos.hs`

example and
the Haddock documentation for the `conjureFromSpec`

and `conjureFromSpecWith`

functions.

The functions `conjureFromSpec`

and `conjureFromSpecWith`

also accept specifications
that bind specific arguments to results.
Just use `==`

and `&&`

accordingly:

```
duplicatesSpec :: ([Int] -> [Int]) -> Bool
duplicatesSpec duplicates = duplicates [0,0] == [0]
&& duplicates [0,1] == []
&& duplicates [1,0,1] == [1]
&& duplicates [0,1,0,1] == [0,1]
&& duplicates [1,0,1,0,1] == [0,1]
&& duplicates [0,1,2,1] == [1]
```

With this, there is no way for Conjure to miss argument-result bindings.

MagicHaskeller (2007) is another tool
that is able to generate Haskell code automatically.
It supports recursion through
catamorphisms, paramorphisms and the `fix`

function.

Barliman for Lisp is another tool that does program synthesis.

There are hundreds of others,
I'll add the most closely related here when I have the time.

## Further reading

For a detailed documentation of each function, see
Conjure's Haddock documentation.

The `eg`

folder in the source distribution
contains more than 60 examples of use.

Conjure, Copyright 2021 Rudy Matela,
distribued under the 3-clause BSD license.