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
Canada    `borders` USA       = 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)" .

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