Monads for Relations. Jeremy G. Siek University of Colorado at Boulder SPLS, June Jeremy Siek Monads for Relations 1 / 20

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1 Monads for Relations Jeremy G. Siek University of Colorado at Boulder SPLS, June 2010 Jeremy Siek Monads for Relations 1 / 20

2 Natural (Big-Step) Semantics for CBV Lambda ρ e v Env Expr Value ρ x ρ x ρ λx.e λx.e, ρ ρ e 1 λx.e, ρ ρ e 2 v 2 ρ [x v 2 ] e v 3 ρ e 1 e 2 v 3 Jeremy Siek Monads for Relations 2 / 20

3 Natural (Big-Step) Semantics for CBV Lambda ρ e v Env Expr Value ρ x ρ x ρ λx.e λx.e, ρ ρ e 1 λx.e, ρ ρ e 2 v 2 ρ [x v 2 ] e v 3 ρ e 1 e 2 v 3 When adding features to this language, we d like to not change the above rules. Jeremy Siek Monads for Relations 2 / 20

4 CBV Lambda with References ρ; µ e v; µ Env Store Expr Value Store ρ; µ 0 e 1 λx.e, ρ ; µ 1 ρ; µ 1 e 2 v 2 ; µ 2 ρ [x v 2 ]; µ 2 e v 3 ; µ 3 ρ; µ x ρ x; µ ρ; µ λx.e λx.e, ρ ; µ ρ; µ 0 e 1 e 2 v 3 ; µ 3 ρ; µ e v; µ a / dom µ ρ; µ ref e a; µ [a v] ρ; µ e a; µ ρ; µ!e µ a; µ ρ; µ 0 e 1 a 1 ; µ 1 ρ; µ 1 e 2 v 2 ; µ 2 ρ; µ 0 e 1 := e 2 v 2 ; µ 2 [a 1 v 2 ] Jeremy Siek Monads for Relations 3 / 20

5 CBV Lambda with References and Abort ρ; µ e r Env Store Expr (Value Store + Abort) ρ; µ 0 e 1 λx.e, ρ ; µ 1 ρ; µ 1 e 2 v 2 ; µ 2 ρ [x v 2 ]; µ 2 e v 3 ; µ 3 ρ; µ x ρ x; µ ρ; µ λx.e λx.e, ρ ; µ ρ; µ 0 e 1 e 2 v 3 ; µ 3 ρ; µ e v; µ a / dom µ ρ; µ ref e a; µ [a v] ρ; µ e a; µ ρ; µ!e µ a; µ ρ; µ 0 e 1 a 1 ; µ 1 ρ; µ 1 e 2 v 2 ; µ 2 ρ; µ 0 e 1 :=e 2 v 2 ; µ 2 [a 1 v 2 ] ρ; µ e v; µ ρ; µ abort e abort v ρ; µ e abort v ρ; µ abort e abort v ρ; µ e 1 abort v ρ; µ e 1 e 2 abort v ρ; µ 0 e 1 λx.e, ρ ; µ 1 ρ; µ 1 e 2 abort v ρ; µ 0 e 1 e 2 abort v ρ; µ 0 e abort v ρ; µ 0 ref e abort v ρ; µ 0 e abort v ρ; µ 0!e abort v ρ; µ 0 e 1 abort v ρ; µ 0 e 1 :=e 2 abort v ρ; µ 0 e 1 a 1 ; µ 1 ρ; µ 1 e 2 abort v ρ; µ 0 e 1 :=e 2 abort v Jeremy Siek Monads for Relations 4 / 20

6 Denotational Semantics e ρ : Expr Env Value x ρ = ρ x λx.e ρ = λz. e ρ[x z] e 1 e 2 ρ = let f = e 1 ρ in let v = e 2 ρ in f v Jeremy Siek Monads for Relations 5 / 20

7 Make sequencing explicit with Monadic operators M is the type operator for the monad. Γ e : T Γ return e : MT Γ m : MT 1 Γ, x : T 1 body : MT 2 Γ let x = m in body : MT 2 x ρ = return ρ x λx.e ρ = return λz. e ρ[x z] e 1 e 2 ρ = let f = e 1 ρ in let v = e 2 ρ in f v (let x = m in e) = bind m (λx.e) Jeremy Siek Monads for Relations 6 / 20

8 CBV Lambda with References x ρ = return ρ x λx.e ρ = return λz. e ρ[x z] e 1 e 2 ρ = let f = e 1 ρ in let v = e 2 ρ in f v ref e ρ = let v = e ρ in alloc v!e ρ = let a = e ρ in get a e 1 := e 2 ρ = let a = e 1 ρ in let v = e 2 ρ in set a v Jeremy Siek Monads for Relations 7 / 20

9 CBV Lambda with References and Abort x ρ = return ρ x λx.e ρ = return λz. e ρ[x z] e 1 e 2 ρ = let f = e 1 ρ in let v = e 2 ρ in f v ref e ρ = let v = e ρ in alloc v!e ρ = let a = e ρ in get a e 1 := e 2 ρ = let a = e 1 ρ in let v = e 2 ρ in set a v abort e ρ = let v = e ρ in exit v Jeremy Siek Monads for Relations 8 / 20

10 The State and Error Monad Definition MT State (T State + Abort) return v λµ.inl(v, µ) let x = m in body λµ. case m µ of inl(v, µ ) (body[v/x])µ inr(abort v) inr(abort v) alloc v λµ.(a, µ[a v]) where a / dom(µ) get a λµ.(µ(a), µ) set a v λµ.(v, µ[a v]) Jeremy Siek Monads for Relations 9 / 20

11 Functions vs. Relations There s a standard correspondence between functions and relations: A B graph graph 1 A B graph(f ) = {(x, y) y = F (x)} graph 1 (R)(x) = y where y is the unique y s.t. (x, y) R Characteristic function of a relation: A B = A B bool Jeremy Siek Monads for Relations 10 / 20

12 From Functions to Relations R L B = B R L T 1 T 2 = R L T 1 R R T 2 R L T 1 T 2 = R L T 1 R L T 2 R R B = B bool R R T 1 T 2 = R L T 1 R R T 2 R R T 1 T 2 = R R T 1 R R T 2 Jeremy Siek Monads for Relations 11 / 20

13 Relational Return and Bind Functional version: return : T MT bind : MT 1 (T 1 MT 2 ) MT 2 Relational version: return : R T MT = T MT bool bind : R MT 1 (T 1 MT 2 ) MT 2 = MT 1 (T 1 MT 2 bool) MT 2 bool Jeremy Siek Monads for Relations 12 / 20

14 Notation for Relational Monadic Let bind : MT 1 (T 1 MT 2 bool) MT 2 bool m 12 { x m 1 m 2. body bind m 1 (λx m 2.body) m 12 Γ m 1 : MT 1 Γ, x : T 1, m 2 : MT 2 body : bool Γ m 12 : MT { 2 ) x m 1 Γ (m 12 : bool m 2. body Jeremy Siek Monads for Relations 13 / 20

15 Monadic CBV Lambda with References and Abort ρ e m Env Expr M Value return (ρ x) m ρ x m return λx.e, ρ m ρ λx.e m ρ { e 1 m 1 v m 1 m 1 12 m 2. v 1 = λx.e, ρ ρ e 2 m { 2 v m 2 m m 3. ρ [x v 2 ] e m 3 ρ e 1 e 2 m 13 ρ e m 1 m 12 { v m 1 m 2. alloc v m 2 ρ ref e m 12 ρ e m 1 m 12 { v m 1 m 2. a. v = a get a m 2 ρ!e m 12 ρ e 1 m 1 v 1 m 1 ρ e 2 m 2 m 13 m 23. v 2 m 2 m 23 m 3. a. v 1 = a set a v 2 m 3 ρ e 1 := e 2 m 13 ρ e m 1 m 12 { v m 1 m 2. exit v m 2 ρ abort e m 12 Jeremy Siek Monads for Relations 14 / 20

16 The State and Error Monad Definition MT State (T State + Abort) return v m µ. m = (µ, inl(v, µ)) m 12 { x m 1 m 2. body µ 0. (m 1 = (µ 0, inr(abort)) m 12 = (µ 0, inr(abort))) ( µ 1 v 1 m 2. m 1 = (µ 0, inl(v 1, µ 1 )) body[v 1 /x] ((m 2 = (µ 1, inr(abort)) m 12 = (µ 0, inr(abort))) ( v 2 µ 2.m 2 = (µ 1, inl(v 2, µ 2 )) m 12 = (µ 0, inl(v 2, µ 2 ))))) Jeremy Siek Monads for Relations 15 / 20

17 The State and Error Monad Definition, cont d alloc v m µa. m = (µ, inl(a, µ[a v])) a / dom(µ) get a m µ. m = (µ, inl(µ(a), µ)) set a v m µ. m = (µ, inl(v, µ[a v])) exit v m µ. m = (µ, inr(abort v)) Jeremy Siek Monads for Relations 16 / 20

18 Non-deterministic Choice e ::=... amb e e fail MT T set return v m (m = {v}) m 12 { x m 1 m 2. body { m 12 = v x m 2. x m 1 v m 2 body } Jeremy Siek Monads for Relations 17 / 20

19 First-Class Continuations (Untested) e ::=... letcc x.e throw e e MT (T Answer bool) Answer Answer Value return v 1 m ( c v 2.m = (c, v 2 ) c v 1 v 2 ) m 12 { x m 1 m 2. body c 1 v 2.m 12 = (c 1, v 2 ) m 1 = (λx v 1. m 2. m 2 = (c 1, v 1 ) body, v 2 ) Jeremy Siek Monads for Relations 18 / 20

20 Tool Support Isabelle state, error, and non-determinism work just fine. having trouble with continuations (need to convince Isabelle that the rules are monotonic). Kanren (logic programming in Scheme, from Dan Friedman et al.) state, error, and non-determinism work just fine. can t do continuations because it doesn t support higher-order unification On the to-do list: Agda λprolog Jeremy Siek Monads for Relations 19 / 20