# A taste of Curry

Curry is a programming language that integrates functional and logic programming. Last week, Denis Firsov and I had a look at Curry, and Thursday, I gave an introductory talk about Curry in the Theory Lunch. This blog post is mostly a write-up of my talk.

Like Haskell, Curry has support for literate programming. So I wrote this blog post as a literate Curry file, which is available for download. If you want to try out the code, you have to install the Curry system KiCS2. The code uses the functional patterns language extension, which is only supported by KiCS2, as far as I know.

## Functional programming

The functional fragment of Curry is very similar to Haskell. The only fundamental difference is that Curry does not support type classes.

Let us do some functional programming in Curry. First, we define a type whose values denote me and some of my relatives.

``````data Person = Paul
| Joachim
| Rita
| Wolfgang
| Veronika
| Johanna
| Jonathan
| Jaromir``````

Now we define a function that yields the father of a given person if this father is covered by the `Person` type.

``````father :: Person -> Person
father Joachim  = Paul
father Rita     = Joachim
father Wolfgang = Joachim
father Veronika = Joachim
father Johanna  = Wolfgang
father Jonathan = Wolfgang
father Jaromir  = Wolfgang``````

Based on `father`, we define a function for computing grandfathers. To keep things simple, we only consider fathers of fathers to be grandfathers, not fathers of mothers.

``````grandfather :: Person -> Person
grandfather = father . father``````

## Combining functional and logic programming

Logic programming languages like Prolog are able to search for variable assignments that make a given proposition true. Curry, on the other hand, can search for variable assignments that make a certain expression defined.

For example, we can search for all persons that have a grandfather according to the above data. We just enter

`grandfather person where person free`

at the KiCS2 prompt. KiCS2 then outputs all assignments to the `person` variable for which `grandfather person` is defined. For each of these assignments, it additionally prints the result of the expression `grandfather person`.

## Nondeterminism

Functions in Curry can actually be non-deterministic, that is, they can return multiple results. For example, we can define a function `element` that returns any element of a given list. To achieve this, we use overlapping patterns in our function definition. If several equations of a function definition match a particular function application, Curry takes all of them, not only the first one, as Haskell does.

``````element :: [el] -> el
element (el : _)   = el
element (_  : els) = element els``````

Now we can enter

`element "Hello!"`

at the KiCS2 prompt, and the system outputs six different results.

## Logic programming

We have already seen how to combine functional and logic programming with Curry. Now we want to do pure logic programming. This means that we only want to search for variable assignments, but are not interested in expression results. If you are not interested in results, you typically use a result type with only a single value. Curry provides the type `Success` with the single value `success` for doing logic programming.

Let us write some example code about routes between countries. We first introduce a type of some European and American countries.

``````data Country = Canada
| Estonia
| Germany
| Latvia
| Lithuania
| Mexico
| Poland
| Russia
| USA``````

Now we want to define a relation called `borders` that tells us which country borders which other country. We implement this relation as a function of type

`Country -> Country -> Success`

that has the trivial result `success` if the first country borders the second one, and has no result otherwise.

Note that this approach of implementing a relation is different from what we do in functional programming. In functional programming, we use `Bool` as the result type and signal falsity by the result `False`. In Curry, however, we signal falsity by the absence of a result.

Our `borders` relation only relates countries with those neighbouring countries whose names come later in alphabetical order. We will soon compute the symmetric closure of `borders` to also get the opposite relationships.

``````borders :: Country -> Country -> Success
Estonia   `borders` Latvia    = success
Estonia   `borders` Russia    = success
Germany   `borders` Poland    = success
Latvia    `borders` Lithuania = success
Latvia    `borders` Russia    = success
Lithuania `borders` Poland    = success
Mexico    `borders` USA       = success``````

Now we want to define a relation `isConnected` that tells whether two countries can be reached from each other via a land route. Clearly, `isConnected` is the equivalence relation that is generated by `borders`. In Prolog, we would write clauses that directly express this relationship between `borders` and `isConnected`. In Curry, on the other hand, we can write a function that generates an equivalence relation from any given relation and therefore does not only work with `borders`.

We first define a type alias `Relation` for the sake of convenience.

``type Relation val = val -> val -> Success``

Now we define what reflexive, symmetric, and transitive closures are.

``````reflClosure :: Relation val -> Relation val
reflClosure rel val1 val2 = rel val1 val2
reflClosure rel val  val  = success

symClosure :: Relation val -> Relation val
symClosure rel val1 val2 = rel val1 val2
symClosure rel val2 val1 = rel val1 val2

transClosure :: Relation val -> Relation val
transClosure rel val1 val2 = rel val1 val2
transClosure rel val1 val3 = rel val1 val2 &
transClosure rel val2 val3

where val2 free``````

The operator `&` used in the definition of `transClosure` has type

`Success -> Success -> Success`

and denotes conjunction.

We define the function for generating equivalence relations as a composition of the above closure operators. Note that it is crucial that the transitive closure operator is applied after the symmetric closure operator, since the symmetric closure of a transitive relation is not necessarily transitive.

``````equivalence :: Relation val -> Relation val
equivalence = reflClosure . transClosure . symClosure``````

The implementation of `isConnected` is now trivial.

``````isConnected :: Country -> Country -> Success
isConnected = equivalence borders``````

Now we let KiCS2 compute which countries I can reach from Estonia without a ship or plane. We do so by entering

`Estonia `isConnected` country where country free`

at the prompt.

We can also implement a nondeterministic function that turns a country into the countries connected to it. For this, we use a guard that is of type `Success`. Such a guard succeeds if it has a result at all, which can only be `success`, of course.

``````connected :: Country -> Country
connected country1
| country1 `isConnected` country2 = country2

where country2 free``````

## Equational constraints

Curry has a predefined operator

`=:= :: val -> val -> Success`

that stands for equality.

We can use this operator, for example, to define a nondeterministic function that yields the grandchildren of a given person. Again, we keep things simple by only considering relationships that solely go via fathers.

``````grandchild :: Person -> Person
grandchild person
| grandfather grandkid =:= person = grandkid

where grandkid free``````

Note that `grandchild` is the inverse of `grandfather`.

## Functional patterns

Functional patterns are a language extension that allows us to use ordinary functions in patterns, not just data constructors. Functional patterns are implemented by KiCS2.

Let us look at an example again. We want to define a function `split` that nondeterministically splits a list into two parts.1 Without functional patterns, we can implement splitting as follows.

``````split' :: [el] -> ([el],[el])
split' list | front ++ rear =:= list = (front,rear)

where front, rear free``````

With functional patterns, we can implement splitting in a much simpler way.

``````split :: [el] -> ([el],[el])
split (front ++ rear) = (front,rear)``````

As a second example, let us define a function `sublist` that yields the sublists of a given list.

``````sublist :: [el] -> [el]
sublist (_ ++ sub ++ _) = sub``````

## Inverting functions

In the `grandchild` example, we showed how we can define the inverse of a particular function. We can go further and implement a generic function inversion operator.

``````inverse :: (val -> val') -> (val' -> val)
inverse fun val' | fun val =:= val' = val where val free``````

With this operator, we could also implement `grandchild` as `inverse grandfather`.

Inverting functions can make our lives a lot easier. Consider the example of parsing. A parser takes a string and returns a syntax tree. Writing a parser directly is a non-trivial task. However, generating a string from a syntax tree is just a simple functional programming exercise. So we can implement a parser in a simple way by writing a converter from syntax trees to strings and inverting it.

We show this for the language of all arithmetic expressions that can be built from addition, multiplication, and integer constants. We first define types for representing abstract syntax trees. These types resemble a grammar that takes precedence into account.

``````type Expr = Sum

data Sum     = Sum Product [Product]
data Product = Product Atom [Atom]
data Atom    = Num Int | Para Sum``````

Now we implement the conversion from abstract syntax trees to strings.

``````toString :: Expr -> String
toString = sumToString

sumToString :: Sum -> String
sumToString (Sum product products)
= productToString product                           ++
concatMap ((" + " ++) . productToString) products

productToString :: Product -> String
productToString (Product atom atoms)
= atomToString atom                           ++
concatMap ((" * " ++) . atomToString) atoms

atomToString :: Atom -> String
atomToString (Num num)  = show num
atomToString (Para sum) = "(" ++ sumToString sum ++ ")"``````

Implementing the parser is now extremely simple.

``````parse :: String -> Expr
parse = inverse toString``````

KiCS2 uses a depth-first search strategy by default. However, our parser implementation does not work with depth-first search. So we switch to breadth-first search by entering

`:set bfs`

at the KiCS2 prompt. Now we can try out the parser by entering

`parse "2 * (3 + 4)"` .

1. Note that our `split` function is not the same as the `split` function in Curry’s `List` module.

## 8 thoughts on “A taste of Curry”

1. Pingback: A taste of Curry | Theory Lunch

2. igor-d

Wait … Isn’t this definition going to trigger some non-deterministic behaviour thus risking testing `rel val1 val2` and failing if `rel` is not reflexive?

```reflClosure :: Relation val -> Relation val
reflClosure rel val1 val2 = rel val1 val2
reflClosure rel val  val  = success```

Or, because failure corresponds to partiality, it will fall back to the next branch?

Like

1. Wolfgang Jeltsch Post author

In fact, this definition introduces non-determinism. If the second and third argument of `reflClosure` are the same, both equations of `reflClosure` are considered. What exactly happens depends on the search strategy.

With depth-first search, Curry first evaluates `rel val1 val2` from the first equation, and if this terminates, it takes the `success` from the second equation. So if `rel val1 val2` terminates with `success`, then `reflClosure` yields the value `success` twice; if `rel val1 val2` terminates with failure (meaning no result), then `reflClosure` yields `success` once. Only if `rel val1 val2` does not terminate, `reflClosure` does not terminate and yields no result.

With breadth-first search, both equations are processed concurrently. So the `success` from the second equation will always be returned. Whether there will be another `success` and whether `reflClosure` will terminate depends on the behavior of `rel val1 val2`.

Note that in the declarative semantics of Curry, there is nothing like falling back to the second equation, as there is in Haskell. If multiple equations of a function definition match, then all these equations are taken, and all are treated equally. It is just a specific search strategy, like depth-first search, that can make program execution not conform to this ideal semantics.

Like

3. Wolfgang Jeltsch Post author

Contrary to what I originally thought, functional patterns are also implemented by the Curry systems PAKCS and MCC.

Like

1. Григорий Ярёменко

I just tried the equivalence thing on PAKCS 2.0.0. When executing `Estonia `isConnected` country where country free`, it gets stuck in infinite recursion. It’s kind of disappointing, to be fair. I was fascinated by the power and elegance of those code samples, but it turned out that the language itself is not even properly implemented. And it’s already been like 5 years since you’ve made the post.

Like

1. Wolfgang Jeltsch Post author

This is strange. When I prepared this code, I tested it with the then current version of KiCS2, and it worked just fine.

Unfortunately, I cannot help you with solving this problem, as I am not a Curry user (I just looked a bit at Curry again around the time I wrote the above code). I think it would be good if you would contact the KiCS developers.

Like

4. Pingback: MIU in Curry « Wolfgang Jeltsch