µKanren: A Minimal Functional Core for Relational Programming

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µKanren A Minimal Functional Core For Relational Programming 1 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Brian Hicks [email protected] / @brianhicks STL Tech Slack: stltech.herokuapp.com 2 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Prolog(ue) mother_child(trude, sally). father_child(tom, sally). father_child(tom, erica). father_child(mike, tom). parent_child(X, Y) :- father_child(X, Y). parent_child(X, Y) :- mother_child(X, Y). sibling(X, Y) :- parent_child(Z, X), parent_child(Z, Y). 3 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Relational / Logic Programming! • sibling(X, Y) is a relation, not a function • unify over variables, which may be unified to values (like X and sally) • various strategies to get results 4 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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miniKanren (run 1 (out) (fresh (x) (== out x) (== 3 x))) Output: (3) 5 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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miniKanren (run 2 (out) (fresh (x) (== out x) (conde ((== 3 x)) ((== 4 x))))) Output: (3 4) 6 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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µKanren Hemann and Friedman, 2013 "We argue, though, that deeply buried within that 265- line miniKanren implementation is a small, beautiful, relational programming language seeking to get out. We believe µKanren is that language." 7 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Terms type alias Var = Int type Term a = LVar Var | LVal a 8 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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State "A µKanren program proceeds through the application of a goal to a state." type alias Substitution a = Dict Var (Term a) type alias State a = { subs : Substitution a , next : Var } init = { subs = Dict.empty, next = 0 } 9 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Goal "Goals are often understood by analogy to predicates. [...] a goal pursued to a given state can either succeed or fail." type alias Goal a = State a -> Stream a 10 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Stream "A goal's success may result in a sequence of (enlarged) states, which we term a stream." type Stream a = Empty | Mature (State a) (Stream a) 11 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Stream Constructors zero : Stream a zero = Empty singleton : State a -> Stream a singleton state = Mature state zero 12 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Goal Constructors • ≡ (spelled identical) - unify two terms • call/fresh - introduce a new term • disj - takes two goals, either of whom may succeed • conj - takes two goals, both of whom must succeed 13 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: unify unify : Term a -> Term a -> Substitution a -> Maybe (Substitution a) unify leftVar rightVar subs = let left = walk leftVar subs right = walk rightVar subs in case (left, right) of -- next slides! 14 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: unify (two vars) (LVar leftRef, LVar rightRef) -> if leftRef == rightRef then Just sub else Just <| Dict.insert leftRef right subs 15 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: unify (two vals) (LVal leftVal, LVal rightVal) -> if leftVal == rightVal then Just subs else Nothing 16 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: unify (mismatch) (LVar ref, _) -> Just <| Dict.insert ref right subs (_, LVar ref) -> Just <| Dict.insert ref left subs 17 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: unify unify (LVar 1) (LVar 2) Dict.empty == Just (Dict.fromList [ (2, LVar 1) ]) unify (LVar 1) (LVal "foo") Dict.empty == Just (Dict.fromList [ (1, LVal "foo") ]) unify (LVal "foo") (LVal "bar") Dict.empty == Nothing 18 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Goals: identical identical : Term a -> Term a -> Goal a identical left right = \state -> case unify left right state.subs of Just uniﬁed -> singleton { state | subs = uniﬁed } Nothing -> zero 19 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Goals: callFresh callFresh : (Term a -> Goal a) -> Goal a callFresh termToGoal = \state -> termToGoal (LVar state.next) { state | next = state.next + 1 } 20 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using callFresh and identical callFresh (\out -> identical out (LVal 1)) init == Mature { subs = Dict.fromList [ (0, LVal 1) ] } , next = 1 } Empty 21 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Goals: disjoin disjoin : Goal a -> Goal a -> Goal a disjoin g1 g2 = \state -> mplus (g1 state) (g2 state) 22 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utils: mplus mplus : Stream a -> Stream a -> Stream a mplus s1 s2 = case s1 of Empty -> s2 Mature state stream -> Mature state (mplus s2 stream) 23 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using disjoin callFresh (\out -> disjoin (identical out (LVal 1)) (identical out (LVal 2)) ) init 24 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using disjoin (result) Mature { substitutions = Dict.fromList [ (0, LVal 1) ] , next = 1 } (Mature { substitutions = Dict.fromList [ (0, LVal 2) ] , next = 1 } Empty) 25 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Goals: conjoin conjoin : Goal a -> Goal a -> Goal a conjoin g1 g2 = \state -> bind (g1 state) g2 26 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: bind bind : Stream a -> Goal a -> Stream a bind stream goal = case stream of Empty -> zero Mature state next -> mplus (goal state) (bind next goal) 27 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using conjoin callFresh (\out -> conjoin (identical out (LVal 1)) (identical out (Lval 2)) ) init This fails (1 ≠ 2), so we get Empty 28 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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What About Infinite Streams? type Stream a = Empty | Immature (() -> Stream a) | Mature (State a) (Stream a) zzz : Goal a -> Goal a zzz goal = \stream -> Immature <| \_ -> goal stream 29 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Infinite Streams With mplus mplus : Stream a -> Stream a -> Stream a mplus s1 s2 = case s1 of Empty -> s2 Immature next -> Immature <| \_ -> mplus s2 (next ()) Mature state stream -> Mature state (mplus s2 stream) 30 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Infinite Streams With bind bind : Stream a -> Goal a -> Stream a bind stream goal = case stream of Empty -> zero Immature next -> Immature <| \_ -> bind (next ()) goal Mature state next -> mplus (goal state) (bind next goal) 31 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using Infinite Streams (naïve) fives : Term number -> Goal number fives term = disjoin (identical term (LVal 5)) (fives term) This recursive definition works, but causes a stack overflow. fives has to be evaluated in order to evaluate fives. 32 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using Infinite Streams (still naïve) fives : Term number -> Goal number fives term = disjoin (identical term (LVal 5)) (zzz <| fives term) Still overflows because fives has to be evaluated to send to zzz, in which fives has to be evaluated to send to zzz and so on. 33 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: lazy lazy : (() -> Goal a) -> Goal a lazy goal = \state -> (goal ()) state 34 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Utilities: lazy lazy : (() -> Goal a) -> Goal a lazy goal = \state -> (zzz <| goal ()) state 35 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using Infinite Streams fives : Term number -> Goal number fives term = disjoin (identical term (LVal 5)) (lazy <| \_ -> fives term) 36 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using Infinite Streams natStartingWith : Int -> Term Int -> Goal Int natStartingWith n term = disjoin (identical term (LVal n)) (lazy <| \_ natStartingWith (n + 1) term) nat = natStartingWith 0 37 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using Infinite Streams callFresh nat init == Mature { subs = Dict.fromList [ ( 0, LVal 0) ] , next = 1 } (Immature ) 38 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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That's all of µKanren but not the whole paper Questions? 39 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: disjoinAll disjoinAll : List (Goal a) -> Goal a disjoinAll goals = \state -> case goals of g :: rest -> disjoin (g state) (disjoinAll rest) [] -> zero 40 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: conjoinAll conjoinAll : List (Goal a) -> Goal a conjoinAll goals = \state -> case goals of g :: rest -> conjoin (g state) (conjoinAll rest) [] -> zero 41 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: conde conde : List ( Goal a, List (Goal a) ) -> Goal a conde = List.map (\( condition, body ) -> condition :: body) >> List.map conjoinAll >> disjoinAll 42 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: fresh fresh1 : (Term a -> Goal a) -> Goal a fresh1 fn = callFresh fn fresh2 : (Term a -> Term a -> Goal a) -> Goal a fresh2 fn = fresh1 (\v1 -> callFresh <| fn v1) fresh3 : (Term a -> Term a -> Term a -> Goal a) -> Goal a fresh3 fn = fresh2 (\v1 v2 -> callFresh <| fn v1 v2) 43 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: pull pull : Stream a -> Stream a pull stream = case stream of Immature next -> pull <| next () _ -> stream 44 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: takeAll takeAll : Stream a -> Stream a takeAll = case stream of Empty -> Empty Immature _ -> takeAll <| pull stream Mature state stream -> Mature state (takeAll stream) 45 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: take take : Int -> Stream a -> Stream a take n = if n == 0 then Empty else case stream of Empty -> Empty Immature _ -> take n <| pull stream Mature state stream -> Mature state (take (n - 1) stream) 46 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: reify reify : State a -> Term a reify state = walk (LVar 0) state.subs 47 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Recovering miniKanren: run and run* runAll : (Term a -> Stream a) -> List (Term a) runAll = ﬂip fresh1 init >> takeAll >> toList >> List.map reify run : Int -> (Term a -> Stream a) -> Stream a -> List (Term a) run n = ﬂip fresh1 init >> take n >> toList >> List.map reify 48 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using miniKanren! (run 1 (out) (fresh (x) (== out x) (== 3 x))) 49 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using miniKanren! run 1 <| \out -> fresh1 <| \x -> conjoin (identical out x) (identical x (LVal 3)) -- output: [3] 50 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Using miniKanren! run 2 <| \out -> fresh1 <| \x -> conjoin (identical out x) (disjoin (identical x (LVal 3)) (identical x (LVal 4))) -- output: [3, 4] 51 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Next: My Enhancements Questions? 52 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Renames From To Why? mplus interleave a b a b a b a b a b bind andThen Elm convention disjoin / disjoinAll either / any either/any condition can succeed conjoin / conjoinAll both / all both/all conditions must succeed zzz inﬁnite really only useful for infinite lists 53 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Error Handling Error Handling! type Error a = CouldNotUnify (Term a) (Term a) type alias State a = Result (Error a) { substitutions : Substitution a , nextVar : Var } 54 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Error Handling 55 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Error Handling 56 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Our First Example, Again relation : List ( a, a ) -> Term a -> Term a -> Goal a relation tuples left right = tuples |> List.map (\( a, b ) -> both (identical (LVal a) left) (identical (LVal b) right) ) |> any 57 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Our First Example, Again motherChild : Term String -> Term String -> Goal String motherChild = relation [ ( "trude", "sally" ) ] fatherChild : Term String -> Term String -> Goal String fatherChild = relation [ ( "tom", "sally" ) , ( "tom", "erica" ) , ( "mike", "tom" ) ] 58 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Our First Example, Again parentChild : Term String -> Term String -> Goal String parentChild parent child = either (motherChild parent child) (fatherChild parent child) sibling : Term String -> Term String -> Goal String sibling x y = fresh1 <| \parent -> both (parentChild parent x) (parentChild parent y) 59 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Our First Example, Again run 1 <| \child -> sibling (LVal "erica") child -- [ LVal "sally" ] 60 Brian Hicks for Papers we Love St. Louis, as presented May 2017

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Thanks! 61 Brian Hicks for Papers we Love St. Louis, as presented May 2017