# code-conjure

synthesize Haskell functions out of partial definitions

https://github.com/rudymatela/conjure#readme

LTS Haskell 20.9: | 0.5.2 |

Stackage Nightly 2023-01-29: | 0.5.2 |

Latest on Hackage: | 0.5.2 |

**Rudy Matela**

**Rudy Matela**

`code-conjure-0.5.2@sha256:4383616fbfa6151b4be28007f479aa964494b29f8c7671266784aa7eb83446d1,4135`

#### Module documentation for 0.5.2

*(full list with versions)*:

# 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 install code-conjure
```

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.

Starting from Cabal v3.0, you need to pass `--lib`

as an argument to
`cabal install`

to install packages globally on the `default`

user environment:

```
$ cabal install code-conjure --lib
```

If you already have Conjure installed Cabal may refuse to update to the latest version. To update, you need to reset your user’s cabal installation with:

```
rm -rf ~/.cabal/{bin,lib,logs,share,store} ~/.ghc/*/
```

WARNING: the above command will erase all user-local packages.

## 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.

## Related work

**Conjure’s dependencies**.
Internally, Conjure uses LeanCheck, Speculate and Express.
LeanCheck does testing similarly to QuickCheck, SmallCheck or Feat.
Speculate discovers equations similarly to QuickSpec.
Express encodes expressions involving Dynamic types.

**Program synthesis within Haskell.**

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

function.
Igor II (2010) is able to synthesize Haskell
programs as well.

Hoogle (2004) is a search engine for Haskell functions. It is not able to synthesize expressions but it can find functions that match a type. Hoogle+ (2020) is similar to Hoogle but is able to search for small expressions. In addition to the type, Hoogle+ allows users to provide tests that the function should pass.

**Program synthesis beyond Haskell.**

PushGP (2002) and G3P (2017) are genetic programming systems that are able to synthesize programs in Push and Python respectively. Differently from Conjure or MagicHaskeller, they require around a hundred tests for traning instead of just about half a dozen.

Barliman (2016) for Lisp is another tool that does program synthesis.

## 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.

## Changes

# Changelog for (Code) Conjure

## 0.5.2

- show number of tested candidates
- complete
`Conjurable`

derivation functions - reference related work on README
- add switch to unique-modulo-testing candidates (slow) to allow computing the near upper/lower limit on pruning

## v0.5.0

- allow synthesizing/conjuring from properties with
`conjureFromSpec`

; - complete Haddock documentation;
- remove several unused functions;
- add stub
`conjurableDerive`

functions; - Makefile: add targets to run GPS(2) and TerpreT benches.

## v0.4.4

- remove need for explicit deconstructions:
- use
`-`

and`1`

instead of`dec`

; - allow
`mod`

and`div`

as deconstructions;

- use
- bump Express requirement to v1.0.6 (bugfix);
- complete the GPS1 benchmark;
- add GPS2 and TerpreT benchmarks;
- minor fixes in the README.

## v0.4.2

- default to using top-level patterns on generated functions;
- memoize function evaluation;
- double-check theory at the end and report warning on incorrect properties;
- add
`prif`

to`Conjure`

; - simplify deconstructor discovery and add
`conjureSize`

to`Conjurable`

; - add
`cevaluate`

,`ceval`

and`cvl`

to`Conjure.Conjurable`

; - add
`bench/gps`

and`bench/lowtests`

; - improve tests and benchmarks.

## v0.4.0

- background primitives are now provided with
`pr`

and`prim`

. - report number of rules used in pruning
- require Express v1.0.4 and Speculate v0.4.12
- allow
`..`

notation - add benchmarks, replicate, subset, p12, p30 and candidates
- add and use the
`Defn`

type and`conjureDefns`

- minor changes in benchmarks
- cleanup unused code

## v0.3.6

- add switch for descending recursions
to allow generation of
`gcd`

- refactor recursion generation (replace a hole later)
- change
`conjpureWith`

to take`Args`

- rename two args fields to
`maxBodyRecursions`

and`maxEvalRecursions`

at this point, the old names were misnomers.

## v0.3.4

- reallow recursions under
`&&`

and`||`

(simplifies the generated`or`

,`and`

,`set`

and`elem`

functions) - only require deconstructions on a non-empty subset of arguments
(allows
`fib01`

to be produced) - limit number of terminal evaluations in
`recursiveToDynamic`

- fix bug in
`recursiveToDynamic`

(not counting some recursions) - add 4 new benchmarks:
`count`

,`gcd`

,`tree`

and`setelem`

## v0.3.2

- significant runtime reduction in several benchmarks, e.g.:
- take is now reachable in about 5 seconds

- improved candidate generation:
- faster runtime
- fewer redundant/invalid candidates

- limit recursive calls to use deconstructors
- test to find deconstructors automatically

- improve recursion evaluation method (
`revaluate`

replaces`recursexpr`

) - add fibonacci benchmark
- minor:
- record runtimes with one decimal place instead of two
- add longshot benchmark
- add intercalate to the list benchmark
- add stub
`Conjure.Constructors`

module

## v0.3.0

- only automatically include an
`if`

for the return type of the given function - add the
`take-drop`

benchmark - make bottom-up enumeration more type directed

## v0.2.8

- export the
`A`

,`B`

,`C`

,`D`

,`E`

and`F`

helper types

## v0.2.6

- require Express v0.1.10 due to
`hasHole`

being now exported there - require Eq result on
`conjure1`

,`conjure2`

and`conjure3`

- code cleanup and more tests

## v0.2.4

- allow conjuring from specifications in addition to partial definitions
(
`conjure1`

,`conjure2`

,`conjure3`

and related functions) - improve examples
- improve criteria for automatic primitive inclusion:
- only include
`if :: ... -> Bool`

if there are`Bool`

primitives - include
`False`

and`True`

automatically only on Speculate’s background

- only include
- add code-optional candidate nubbing and debug functions

## v0.2.2

- by default, search for 60 argument combinations among 100000 enumerated combinations

## v0.2.0

- search until 100% match is found and exit
- other misc changes

## v0.1.2

For the changelog of earlier versions, check the git commit history.