Haskell's type system and logic programming - how to port Prolog programs to type level

Question

I'm trying to understand the relation between a logic programming language(Prolog in my case) and Haskell's type system.

I know both use unification and variables to find values(or types, in Haskell's type system) depending on relations. As an exercise to understand similarities and differences between them better, I tried to rewrite some simple prolog programs in Haskell's type level, but I'm having trouble with some parts.

First, I rewrote this simple prolog program:

numeral(0). numeral(succ(X)) :- numeral(X).  add(0,Y,Y). add(succ(X),Y,succ(Z)) :- add(X,Y,Z). 

as:

class Numeral a where     numeral :: a     numeral = u  data Zero data Succ a  instance Numeral Zero instance (Numeral a) => Numeral (Succ a)  class (Numeral a, Numeral b, Numeral c) => Add a b c | b c -> a where     add :: b -> c -> a     add = u  instance (Numeral a) => Add a Zero a instance (Add x y z) => Add (Succ x) (Succ y) z 

it works fine, but I couldn't extend it with this Prolog:

greater_than(succ(_),0). greater_than(succ(X),succ(Y)) :- greater_than(X,Y). 

What I tried was this:

class Boolean a data BTrue data BFalse instance Boolean BTrue instance Boolean BFalse  class (Numeral a, Numeral b, Boolean r) => Greaterthan a b r | a b -> r where     greaterthan :: a -> b -> r     greaterthan = u  instance Greaterthan Zero Zero BFalse instance (Numeral a) => Greaterthan (Succ a) Zero BTrue instance (Numeral a) => Greaterthan Zero (Succ a) BFalse instance (Greaterthan a b BTrue)  => Greaterthan (Succ a) (Succ b) BTrue instance (Greaterthan a b BFalse) => Greaterthan (Succ a) (Succ b) BFalse 

The problem with this code is that last two instances are causing fundep conflict. I can understand why, but it seems to me that it shouldn't be a problem since their guard parts(or whatever it's called it, I mean the (Greaterthan a b c) => part) are different, so that as and bs in last two insance declarations are actually different values and there are no conflicts.

---

Another program I tried to rewrite was this:

child(anne,bridget). child(bridget,caroline). child(caroline,donna). child(donna,emily).  descend(X,Y) :- child(X,Y). descend(X,Y) :- child(X,Z),                 descend(Z,Y). 

(btw, examples are from Learn Prolog Now book)

data Anne data Bridget data Caroline data Donna data Emily  class Child a b | a -> b where     child :: a -> b     child = u  instance Child Anne Bridget instance Child Bridget Caroline instance Child Caroline Donna instance Child Donna Emily  class Descend a b | b -> a where     descend :: b -> a     descend = u  instance (Child a b) => Descend a b instance (Child a c, Descend c b) => Descend a b 

And I'm getting "duplicate instances" error in last line. I think it's a similar problem, even if I have a different guard parts, I'm getting errors because body parts(I mean Descend a b parts) are the same.

So I'm looking for ways to port that Prolog programs to Haskell's type level, if possible. Any help will be appreciated.

EDIT:

Ed'ka's solution works but in a completely different way. I'm still trying to understand when we can run a Prolog program in type-system, when/why we need to write a different algorithm to make it work(like in Ed'ka's solution), and when/why there's no way to implemenet a program in Haskell's type-system.

Maybe I can find some pointers about this after reading "Fun With Functional Dependencies".

Answer 1

As @stephen tetley has already pointed out when GHC tries to match instance declaration it considers only instance head (the stuff after =>) completely ignoring instance context (stuff before =>), once the unambiguous instance is found it tries to match instance context. Your first problematic example clearly has duplication in instance head but it can easily be fixed by replacing two conflicting instances with one more general instance:

instance (Greaterthan a b r)  => Greaterthan (Succ a) (Succ b) r 

The second example though is much harder one. I suspect that to make it work in Haskell we need a type-level function which could produce two different result depending on whether a specific instance is defined or not for a particular type arguments (i.e. if there is an instance Child Name1 Name2 - recursively do something with Name2 otherwise return BFalse). I am not sure if it is possible to code this with GHC types (I suspect it is not).

However, I can propose a "solution" which works for slightly different type of input: instead of implying absence of parent->child relation (when no instance is defined for such pair) we could explicitly encode all existing relationship using type-level lists. Then we can define Descend type-level function although it have to rely on much-dreaded OverlappingInstances extension:

{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies,   FlexibleInstances, FlexibleContexts, TypeOperators,   UndecidableInstances, OverlappingInstances #-}  data Anne data Bridget data Caroline data Donna data Emily data Fred data George  -- Type-level list data Nil infixr 5 ::: data x ::: xs  -- `bs` are children of `a` class Children a bs | a -> bs  instance Children Anne (Bridget ::: Nil) instance Children Bridget (Caroline ::: Donna ::: Nil) instance Children Caroline (Emily ::: Nil) -- Note that we have to specify children list for everyone -- (`Nil` instead of missing instance declaration) instance Children Donna Nil instance Children Emily Nil instance Children Fred (George ::: Nil) instance Children George Nil  -- `or` operation for type-level booleans  class OR a b bool | a b -> bool instance OR BTrue b BTrue instance OR BFalse BTrue BTrue instance OR BFalse BFalse BFalse  -- Is `a` a descendant of `b`? class Descend  a b bool | a b -> bool instance (Children b cs, PathExists cs a r) => Descend a b r  -- auxiliary function which checks if there is a transitive relation -- to `to` using all possible paths passing `children` class PathExists children to bool | children to -> bool  instance PathExists Nil a BFalse instance PathExists (c ::: cs) c BTrue instance (PathExists cs a r1, Children c cs1, PathExists cs1 a r2, OR r1 r2 r)     => PathExists (c ::: cs) a r  -- Some tests instance Show BTrue where     show _ = "BTrue"  instance Show BFalse where     show _ = "BFalse"  t1 :: Descend Donna Anne r => r t1 = undefined -- outputs `BTrue`  t2 :: Descend Fred Anne r => r t2 = undefined -- outputs `BFalse` 

OverlappingInstances is necessary here since 2nd and 3rd instances of PathExists can both match the case when children is not empty list but GHC can determine more specific one in our case depending whether the head of the list is equal to to argument (and if it is it means we have found the path i.e. descendant).