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.
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
father :: Person -> Person father Joachim = Paul father Rita = Joachim father Wolfgang = Joachim father Veronika = Joachim father Johanna = Wolfgang father Jonathan = Wolfgang father Jaromir = Wolfgang
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
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
at the KiCS2 prompt, and the system outputs six different results.
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.
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
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
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
& 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
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
grandchild is the inverse of
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
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
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
at the KiCS2 prompt. Now we can try out the parser by entering
parse "2 * (3 + 4)" .
Note that our
splitfunction is not the same as the
splitfunction in Curry’s