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Can two fixpoint types in Mendler style unite?

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Ki Yung Ahn
Ki Yung Ahn

This document discusses Mendler-style recursion schemes for mixed-variant datatypes. It introduces Mendler-style iteration (mit) and iteration with a syntactic inverse (msfit). msfit allows formatting of untyped and simply-typed higher-order abstract syntax (HOAS) by evaluating at different kinds. A new recursion scheme is proposed to define a function for evaluating simply-typed HOAS into a user-defined value domain. The relationship between Mu and Mu' fixed point types is also discussed.

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Mendler-style Recursion Schemes for Mixed-Variant Datatypes Ki Yung Ahn Tim Sheard Marcelo P. Fiore kya@cs.pdx.edu sheard@cs.pdx.edu Marcelo.Fiore@cl.cam.ac.uk TYPES 2013
Outline Mendler-style Recursion Schemes mit (iteration) msfit (iteration with syntactic Inverse) Untyped HOAS (Regular Mixed-Variant Datatype) Formatting Untyped HOAS (using msfit at kind *) Simply-Typed HOAS (Indexed Mixed-Variant Datatype) Formatting Untyped HOAS (using msfit at kind * β†’ *) The Nax language Need for yet another Mendler-style Recursion Scheme
already inconsistent as a logic (via Curry--Howard correspondence) without even having to introduce any term-level recursion since we can already define general recursion using this (iso-) Recursive Types in Functional Languages
Having both unrestricted formation and unrestricted elimination of ΞΌ leads to inconsistency data T a r = C (r β†’ a) w : ΞΌ (T a) β†’ a -- encoding of (Ξ»x. x x) in the untyped Ξ»-calc w = Ξ»v. case (unIn v) of (C x) β†’ f (C x) -- w (C w) amounts to a well-known diverging term in Ξ»-calculus fix : (a β†’ a) β†’ a -- the Y combinator (Ξ»f.(Ξ»x.f(xx))(Ξ»x.f(xx)))) fix = Ξ»f. (Ξ»x. f (w x)) (In (C (Ξ»x. f (w x)))) To recover consistency, one could either restrict formation (conventional approach) or restrict elimination (Mendler-style approach)
Conventional Iteration over Recursive Types allow formation of only positive recursive types (i.e., when F has a map) freely eliminate (i.e. use unIn) recursive values
Mendler-style Iteration over Recursive Types elimination is possible only through mit freely form ANY recursive type!! note, no (ΞΌ-elim) rule
Mendler-style Iteration -- Mu at different kinds (there are many more, one at each kind) data Mu0 (f :: * β†’ *) = In0 (f (Mu0 f )) data Mu1 (f :: (*β†’*)β†’(*β†’*)) i= In1 (f (Mu1 f ) i) -- Mendler-style iterators at different kinds mit0 :: Phi0 f a β†’ Mu f β†’ a mit0 phi (In0 x) = phi (mit0 phi) x mit1 :: Phi1 f a i β†’ Mu f i β†’ a i mit1 phi (In1 x) = phi (mit1 phi) x type Phi0 f = βˆ€r. (r β†’ a ) β†’ (f r β†’ a ) type Phi1 f = βˆ€r. (βˆ€i. r i β†’ a i) β†’ (βˆ€i. f r i β†’ a i) βœ” Uniformly defined at different kinds (can handle non-regular datatypes quite the same way as handling regular datatypes) βœ” Terminates for ANY datatype (can embed Mu and mit into Fw. see Abel, Matthes and Uustalu TCS'04) βœ” However, not very useful for mixed-variant datatypes eliminating In0 or In1 by pattern matching is only allowed in Mendler-style iterators recursive call
Mendler-style Iteration with a syntactic inverse -- Mu' at different kinds (we only use two of them in this talk) data Mu'0 (f :: * β†’ *) a = In'0 (f (Mu'0 f a)) | Inverse0 a data Mu'1 (f :: (*β†’*)β†’(*β†’*)) a i = In'1 (f (Mu'1 f a) i)| Inverse1 (a i) -- msfit at different kinds msfit0 :: Phi'0 f a β†’ (βˆ€a. Mu'0 f a) β†’ a msfit0 phi (In'0 x) = phi Inverse0 (msfit0 phi) x msfit1 :: Phi'1 f a i β†’ (βˆ€a. Mu'1 f a i) β†’ a i msfit1 phi (In'1 x) = phi Inverse1 (msfit1 phi) x type Phi'0 f a = βˆ€r. (a β†’ r a ) β†’ (r a β†’ a ) β†’ (f r a β†’ a ) type Phi'1 f a = βˆ€r. (βˆ€i. a i β†’ r a i) β†’ (βˆ€i. r a i β†’ a i) β†’ (βˆ€i. f (r a) i β†’ a i) βœ” Terminates for ANY datatype (can embed Mu and msfit into Fw. see Ahn and Sheard ICFP'11) βœ” msfit is quite useful for mixed-variant datatypes, due to "inverse" operation in addition to "recursive call" eliminating In'0 or In'1 by pattern matching is only allowed in Mendler-style iterators recursive callinverse
-- using general recursion at type level data Exp = Lam (Exp β†’ Exp) | App Exp Exp -- using fixpoint (Mu'0) over non-recursive base structure (ExpF) data ExpF r = Lam (r β†’ r) | App r r type Exp' a = Mu'0 ExpF a -- (Exp' a) may contain Inverse type Exp = βˆ€a . Exp' a -- Exp does not contain Inverse lam :: (βˆ€a. Exp' a β†’ Exp' a) -> Exp -- it's not (Exp β†’ Exp) β†’ Exp lam f = In'0 (Lam f ) -- f can handle Inverse containing values -- but can never examine its content app :: Exp β†’ Exp β†’ Exp app e1 e2 = In'0 (App e1 e2) Untyped HOAS (a regular mixed-variant datatype)
Formatting Untyped HOAS using msfit0 at kind * showExp :: Exp β†’ String showExp e = msfit0 phi e vars where phi :: Phi'0 ExpF ([String] β†’ String) phi inv showE (Lam z) = Ξ»(v:vs) β†’ "(Ξ»"++v++"β†’"++ showE (z (inv (const v))) vs ++")" phi inv showE (App x y) = Ξ»vs β†’ "("++ showE x vs ++" "++ showE y vs ++")" type Phi'0 f a = βˆ€ r. (a β†’ r a) β†’ (r a β†’ a) β†’ (f r a β†’ a) vars = [ "x"++show n | n<-[0..] ] :: [String] recursive callinverse
Simply-Typed HOAS -- using general recursion at type level data Exp t where -- Exp :: * β†’ * Lam :: (Exp t1 β†’ Exp t2) β†’ Exp (t1 β†’ t2) App :: Exp (t1 β†’ t2) β†’ Exp t1 β†’ Exp t2 -- using fixpoint (Mu'1) over non-recursive base structure (ExpF) data ExpF r t where -- ExpF :: (* β†’ *) β†’ (* β†’ *) Lam :: (r t1 β†’ r t2) β†’ ExpF r (t1 β†’ t2) App :: r (t1 β†’ t2) β†’ r t1 β†’ ExpF r t2 type Exp' a t = Mu'1 ExpF a t -- (Exp' a t) might contain Inverse type Exp t = βˆ€a . Exp' a t -- (Exp t) does not contain Inverse lam :: (βˆ€a . Exp' a t1 β†’ Exp' a t2) β†’ Exp (t1 β†’ t2) lam f = In'1 (Lam f ) -- f can handle Inverse containing values -- but can never examine its content app :: Exp (t1 β†’ t2) β†’ Exp t1 β†’ Exp t2 app e1 e2 = In'1 (App e1 e2) (a non-regular mixed-variant datatype)
newtype Id a = MkId { unId :: a } evalHOAS :: Exp t β†’ Id t evalHOAS e = msfit1 phi e where phi :: Phi'1 ExpF Id {- v :: t1 -} {- f :: r Id t1 -> r Id t2 -} phi inv ev (Lam f ) = MkId(Ξ»v β†’ unId(ev (f (inv (MkId v))))) phi inv ev (App e1 e2) = MkId(unId(ev e1) (unId(ev e2))) type Phi'1 f a = βˆ€r. (βˆ€i. a i β†’ r a i) β†’ (βˆ€i. r a i β†’ a i) β†’ (βˆ€i. f (r a) i β†’ a i) This is example is a super awesome cooooool discovery that System Fw can express Simply-Typed HOAS evaluation!!! Evaluating Simply-Typed HOAS using msfit1 at kind *β†’* recursive callinverse
The Nax language βœ”Built upon the idea of Mendler-style recursion schemes βœ”Supports an extension of Hindley--Milner type inference to make it easier to use Mendler-style recursion schemes and indexed datatypes βœ”Handling different notions of recursive type operators (Mu, Mu') still needs more rigorous clarification in the theory, but intuitively, Mu0 f = βˆ€a. Mu'0 f a Mu1 f = βˆ€a. Mu'1 f a i ... So, we hide Mu' from users as if there is one kind of Mu and Mu' is only used during the computation of msfit βœ”Supports term-indexed types as well as type-indexed types. Our term indexed calculus, System Fi (to appear in TLCA 2013), extends System Fw with erasable term indices.
Need for yet another Mendler-style recursion scheme There are more recursion schemes other than mit and msfit E.g., Mendler-style primitive recursion (mpr) can cast from abstract recursive type (r) to concrete recursive type (Mu f ). Phi0 f a = βˆ€r. (r β†’ Mu f ) β†’ (r β†’ a ) β†’ (f r β†’ a ) Phi1 f a = βˆ€r. (βˆ€i. r i β†’ Mu f i) β†’ (βˆ€i. r i β†’ a i) β†’ (βˆ€i. f r i β†’ a i) With mpr, one can write constant time predecessor for natural numbers and tail function for lists by using the cast operation. Next example motivates an extension to mpr that can uncast from concrete recursive type (Mu f i) to abstract recursive type (r i) Phi1 f a = βˆ€r. (βˆ€i. Mu f i β†’ r i) β†’ (βˆ€i. r i β†’ Mu f i) β†’ (βˆ€i. r i β†’ a i) β†’ (βˆ€i. f r i β†’ a i) Recursion scheme with above Phi1 type does not terminate for mixed- variant datatypes -- needs some additional restriction on uncast.
data V r t where V :: (r t1 β†’ r t2) β†’ V r (t1 β†’ t2) type Val t = Mu V t val f = In1 (V f ) evalHOAS :: Exp t β†’ Val t evalHOAS e = msfit1 phi e where phi :: Phi'1 ExpF (Mu1 V) -- f :: r Val t1 -> r Val t2 , v :: Val t1 phi inv ev (Lam f ) = val(Ξ»v β†’ ev (f (inv v))) phi inv ev (App e1 e2) = unVal (ev e1) (ev e2) Only if we had unVal :: Val (t1 β†’ t2) β†’ (Val t1 β†’ Val t2) it would be possible to write this, but unVal does not seem to be definable using any known Mendler-style recursion scheme. Evaluating Simply-Typed HOAS into a user defined value domain
New recursion scheme to write unVal data V r t where V :: (r t1 β†’ r t2) β†’ V r (t1 β†’ t2) type Val t = Mu V t val f = In1 (V f ) unVal v = unId(mprsi phi v) where phi :: Phi1 V Id phi uncast cast (V f) = Id(Ξ»v. cast (f (uncast v))) mprsi :: Phi1 f a β†’ Mu f i β†’ a i mprsi phi (In1 x) = phi id id (mrpsi phi) x -- size restriction over indices in both uncast and cast type Phi1 f a = βˆ€ r j. (βˆ€i. (i < j) β‡’ Mu f i β†’ r i) β†’ -- uncast (βˆ€i. (i < j) β‡’ r i β†’ Mu f i) β†’ -- cast (βˆ€i. r i β†’ a i) β†’ (r j β†’ a j) -- or, maybe without the size restriction over indices in cast type Phi1 f a = βˆ€ r j. (βˆ€i. (i < j) β‡’ Mu f i β†’ r i) β†’ -- uncast (βˆ€i. r i β†’ Mu f i) β†’ -- cast (βˆ€i. r i β†’ a i) β†’ (f r j β†’ a j) Preliminary idea that still need further studies for the termination proof
How are Mu and Mu' related? data Mu0 (f :: * β†’ *) = In0 (f (Mu0 f )) data Mu'0 (f :: * β†’ *) a = In'0 (f (Mu'0 f a)) | Inverse0 a type Phi0 f = βˆ€r. (r β†’ a ) β†’ (f r β†’ a ) mit0 :: Phi0 f a β†’ Mu f β†’ a mit0 phi (In0 x) = phi (mit0 phi) x type Phi'0 f a = βˆ€ r. (a β†’ r a) β†’ (r a β†’ a) β†’ (f (r a) β†’ a) msfit0 :: Phi'0 f a β†’ (βˆ€a. Mu'0 f a) β†’ a msfit0 phi (In'0 x) = phi Inverse0 (msfit0 phi) x Can we coerece from (βˆ€a. Mu'0 f a) to Mu f ? Yes!!! Can we coerece from Mu f to (βˆ€a. Mu'0 f a) ? Not exactly...
Corecion from Mu' to Mu data E r = Lam (r -> r) | App r r type Expr = Mu0 E type Exp' a = Mu'0 E a type Exp = (forall a. Exp' a) -- .i.e (forall a. Rec0 E a) exp2expr :: Exp -> Expr -- i.e., (forall a. Rec0 E a) -> Mu0 E exp2expr = msfit0 phi where phi inv p2r (Lam f) = In0(Lam((x -> p2r (f (inv x))))) phi inv p2r (App e1 e2) = In0(App (p2r e1) (p2r e2))
Attempt from Mu to Mu' msfit' :: (forall r. (a -> r a) -> (r a -> a) -> f (r a) -> a) -> Mu'0 f a -> a msfit' phi (In'0 x) = phi Inverse0 (msfit' phi) x msfit' phi (Inverse0 z) = z exp'2expr :: Exp' Expr -> Expr -- i.e., Mu'0 E (Mu0 E) -> Mu0 E exp'2expr = msfcata' phi where phi inv p2r (Lam f) = In0(Lam((x -> p2r (f (inv x))))) phi inv p2r (App e1 e2) = In0(App (p2r e1) (p2r e2)) -- using general recursion and pattern matching against In0 expr2exp' :: Expr -> Exp' Expr -- i.e., Mu0 E -> Mu'0 E (Mu0 E) expr2exp' (In0(Lam f)) = In'0(Lam (x -> expr2exp' (f (exp'2expr x)))) expr2exp' (In0(App e1 e2)) = In'0(App (expr2exp' e1) (expr2exp' e2))
Why from Mu' to Mu possible but not the other way? data E r = Lam (r β†’ r) | App r r type Exp' a = Mu'0 ExpF a -- (Exp' a) may contain Inverse type Exp = βˆ€a . Exp' a -- Exp does not contain Inverse lam :: (βˆ€a. Exp' a β†’ Exp' a) -> Exp -- it's not (Exp β†’ Exp) β†’ Exp lam f = In'0 (Lam f ) -- f can handle Inverse containing values -- but can never examine its content type Expr = Mu0 E lam :: Expr β†’ Expr lam f = In0 (Lam f) Assume that Expr = Exp = (βˆ€a . Exp' a) There are less functions with type (βˆ€a. Exp' a) β†’ (βˆ€a. Exp' a) than functions with type (βˆ€a. Exp' a β†’ Exp' a)
Thanks for listening. Questions or Suggestions? ===============

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Can two fixpoint types in Mendler style unite?

  • 1. Mendler-style Recursion Schemes for Mixed-Variant Datatypes Ki Yung Ahn Tim Sheard Marcelo P. Fiore kya@cs.pdx.edu sheard@cs.pdx.edu Marcelo.Fiore@cl.cam.ac.uk TYPES 2013
  • 2. Outline Mendler-style Recursion Schemes mit (iteration) msfit (iteration with syntactic Inverse) Untyped HOAS (Regular Mixed-Variant Datatype) Formatting Untyped HOAS (using msfit at kind *) Simply-Typed HOAS (Indexed Mixed-Variant Datatype) Formatting Untyped HOAS (using msfit at kind * β†’ *) The Nax language Need for yet another Mendler-style Recursion Scheme
  • 3. already inconsistent as a logic (via Curry--Howard correspondence) without even having to introduce any term-level recursion since we can already define general recursion using this (iso-) Recursive Types in Functional Languages
  • 4. Having both unrestricted formation and unrestricted elimination of ΞΌ leads to inconsistency data T a r = C (r β†’ a) w : ΞΌ (T a) β†’ a -- encoding of (Ξ»x. x x) in the untyped Ξ»-calc w = Ξ»v. case (unIn v) of (C x) β†’ f (C x) -- w (C w) amounts to a well-known diverging term in Ξ»-calculus fix : (a β†’ a) β†’ a -- the Y combinator (Ξ»f.(Ξ»x.f(xx))(Ξ»x.f(xx)))) fix = Ξ»f. (Ξ»x. f (w x)) (In (C (Ξ»x. f (w x)))) To recover consistency, one could either restrict formation (conventional approach) or restrict elimination (Mendler-style approach)
  • 5. Conventional Iteration over Recursive Types allow formation of only positive recursive types (i.e., when F has a map) freely eliminate (i.e. use unIn) recursive values
  • 6. Mendler-style Iteration over Recursive Types elimination is possible only through mit freely form ANY recursive type!! note, no (ΞΌ-elim) rule
  • 7. Mendler-style Iteration -- Mu at different kinds (there are many more, one at each kind) data Mu0 (f :: * β†’ *) = In0 (f (Mu0 f )) data Mu1 (f :: (*β†’*)β†’(*β†’*)) i= In1 (f (Mu1 f ) i) -- Mendler-style iterators at different kinds mit0 :: Phi0 f a β†’ Mu f β†’ a mit0 phi (In0 x) = phi (mit0 phi) x mit1 :: Phi1 f a i β†’ Mu f i β†’ a i mit1 phi (In1 x) = phi (mit1 phi) x type Phi0 f = βˆ€r. (r β†’ a ) β†’ (f r β†’ a ) type Phi1 f = βˆ€r. (βˆ€i. r i β†’ a i) β†’ (βˆ€i. f r i β†’ a i) βœ” Uniformly defined at different kinds (can handle non-regular datatypes quite the same way as handling regular datatypes) βœ” Terminates for ANY datatype (can embed Mu and mit into Fw. see Abel, Matthes and Uustalu TCS'04) βœ” However, not very useful for mixed-variant datatypes eliminating In0 or In1 by pattern matching is only allowed in Mendler-style iterators recursive call
  • 8. Mendler-style Iteration with a syntactic inverse -- Mu' at different kinds (we only use two of them in this talk) data Mu'0 (f :: * β†’ *) a = In'0 (f (Mu'0 f a)) | Inverse0 a data Mu'1 (f :: (*β†’*)β†’(*β†’*)) a i = In'1 (f (Mu'1 f a) i)| Inverse1 (a i) -- msfit at different kinds msfit0 :: Phi'0 f a β†’ (βˆ€a. Mu'0 f a) β†’ a msfit0 phi (In'0 x) = phi Inverse0 (msfit0 phi) x msfit1 :: Phi'1 f a i β†’ (βˆ€a. Mu'1 f a i) β†’ a i msfit1 phi (In'1 x) = phi Inverse1 (msfit1 phi) x type Phi'0 f a = βˆ€r. (a β†’ r a ) β†’ (r a β†’ a ) β†’ (f r a β†’ a ) type Phi'1 f a = βˆ€r. (βˆ€i. a i β†’ r a i) β†’ (βˆ€i. r a i β†’ a i) β†’ (βˆ€i. f (r a) i β†’ a i) βœ” Terminates for ANY datatype (can embed Mu and msfit into Fw. see Ahn and Sheard ICFP'11) βœ” msfit is quite useful for mixed-variant datatypes, due to "inverse" operation in addition to "recursive call" eliminating In'0 or In'1 by pattern matching is only allowed in Mendler-style iterators recursive callinverse
  • 9. -- using general recursion at type level data Exp = Lam (Exp β†’ Exp) | App Exp Exp -- using fixpoint (Mu'0) over non-recursive base structure (ExpF) data ExpF r = Lam (r β†’ r) | App r r type Exp' a = Mu'0 ExpF a -- (Exp' a) may contain Inverse type Exp = βˆ€a . Exp' a -- Exp does not contain Inverse lam :: (βˆ€a. Exp' a β†’ Exp' a) -> Exp -- it's not (Exp β†’ Exp) β†’ Exp lam f = In'0 (Lam f ) -- f can handle Inverse containing values -- but can never examine its content app :: Exp β†’ Exp β†’ Exp app e1 e2 = In'0 (App e1 e2) Untyped HOAS (a regular mixed-variant datatype)
  • 10. Formatting Untyped HOAS using msfit0 at kind * showExp :: Exp β†’ String showExp e = msfit0 phi e vars where phi :: Phi'0 ExpF ([String] β†’ String) phi inv showE (Lam z) = Ξ»(v:vs) β†’ "(Ξ»"++v++"β†’"++ showE (z (inv (const v))) vs ++")" phi inv showE (App x y) = Ξ»vs β†’ "("++ showE x vs ++" "++ showE y vs ++")" type Phi'0 f a = βˆ€ r. (a β†’ r a) β†’ (r a β†’ a) β†’ (f r a β†’ a) vars = [ "x"++show n | n<-[0..] ] :: [String] recursive callinverse
  • 11. Simply-Typed HOAS -- using general recursion at type level data Exp t where -- Exp :: * β†’ * Lam :: (Exp t1 β†’ Exp t2) β†’ Exp (t1 β†’ t2) App :: Exp (t1 β†’ t2) β†’ Exp t1 β†’ Exp t2 -- using fixpoint (Mu'1) over non-recursive base structure (ExpF) data ExpF r t where -- ExpF :: (* β†’ *) β†’ (* β†’ *) Lam :: (r t1 β†’ r t2) β†’ ExpF r (t1 β†’ t2) App :: r (t1 β†’ t2) β†’ r t1 β†’ ExpF r t2 type Exp' a t = Mu'1 ExpF a t -- (Exp' a t) might contain Inverse type Exp t = βˆ€a . Exp' a t -- (Exp t) does not contain Inverse lam :: (βˆ€a . Exp' a t1 β†’ Exp' a t2) β†’ Exp (t1 β†’ t2) lam f = In'1 (Lam f ) -- f can handle Inverse containing values -- but can never examine its content app :: Exp (t1 β†’ t2) β†’ Exp t1 β†’ Exp t2 app e1 e2 = In'1 (App e1 e2) (a non-regular mixed-variant datatype)
  • 12. newtype Id a = MkId { unId :: a } evalHOAS :: Exp t β†’ Id t evalHOAS e = msfit1 phi e where phi :: Phi'1 ExpF Id {- v :: t1 -} {- f :: r Id t1 -> r Id t2 -} phi inv ev (Lam f ) = MkId(Ξ»v β†’ unId(ev (f (inv (MkId v))))) phi inv ev (App e1 e2) = MkId(unId(ev e1) (unId(ev e2))) type Phi'1 f a = βˆ€r. (βˆ€i. a i β†’ r a i) β†’ (βˆ€i. r a i β†’ a i) β†’ (βˆ€i. f (r a) i β†’ a i) This is example is a super awesome cooooool discovery that System Fw can express Simply-Typed HOAS evaluation!!! Evaluating Simply-Typed HOAS using msfit1 at kind *β†’* recursive callinverse
  • 13. The Nax language βœ”Built upon the idea of Mendler-style recursion schemes βœ”Supports an extension of Hindley--Milner type inference to make it easier to use Mendler-style recursion schemes and indexed datatypes βœ”Handling different notions of recursive type operators (Mu, Mu') still needs more rigorous clarification in the theory, but intuitively, Mu0 f = βˆ€a. Mu'0 f a Mu1 f = βˆ€a. Mu'1 f a i ... So, we hide Mu' from users as if there is one kind of Mu and Mu' is only used during the computation of msfit βœ”Supports term-indexed types as well as type-indexed types. Our term indexed calculus, System Fi (to appear in TLCA 2013), extends System Fw with erasable term indices.
  • 14. Need for yet another Mendler-style recursion scheme There are more recursion schemes other than mit and msfit E.g., Mendler-style primitive recursion (mpr) can cast from abstract recursive type (r) to concrete recursive type (Mu f ). Phi0 f a = βˆ€r. (r β†’ Mu f ) β†’ (r β†’ a ) β†’ (f r β†’ a ) Phi1 f a = βˆ€r. (βˆ€i. r i β†’ Mu f i) β†’ (βˆ€i. r i β†’ a i) β†’ (βˆ€i. f r i β†’ a i) With mpr, one can write constant time predecessor for natural numbers and tail function for lists by using the cast operation. Next example motivates an extension to mpr that can uncast from concrete recursive type (Mu f i) to abstract recursive type (r i) Phi1 f a = βˆ€r. (βˆ€i. Mu f i β†’ r i) β†’ (βˆ€i. r i β†’ Mu f i) β†’ (βˆ€i. r i β†’ a i) β†’ (βˆ€i. f r i β†’ a i) Recursion scheme with above Phi1 type does not terminate for mixed- variant datatypes -- needs some additional restriction on uncast.
  • 15. data V r t where V :: (r t1 β†’ r t2) β†’ V r (t1 β†’ t2) type Val t = Mu V t val f = In1 (V f ) evalHOAS :: Exp t β†’ Val t evalHOAS e = msfit1 phi e where phi :: Phi'1 ExpF (Mu1 V) -- f :: r Val t1 -> r Val t2 , v :: Val t1 phi inv ev (Lam f ) = val(Ξ»v β†’ ev (f (inv v))) phi inv ev (App e1 e2) = unVal (ev e1) (ev e2) Only if we had unVal :: Val (t1 β†’ t2) β†’ (Val t1 β†’ Val t2) it would be possible to write this, but unVal does not seem to be definable using any known Mendler-style recursion scheme. Evaluating Simply-Typed HOAS into a user defined value domain
  • 16. New recursion scheme to write unVal data V r t where V :: (r t1 β†’ r t2) β†’ V r (t1 β†’ t2) type Val t = Mu V t val f = In1 (V f ) unVal v = unId(mprsi phi v) where phi :: Phi1 V Id phi uncast cast (V f) = Id(Ξ»v. cast (f (uncast v))) mprsi :: Phi1 f a β†’ Mu f i β†’ a i mprsi phi (In1 x) = phi id id (mrpsi phi) x -- size restriction over indices in both uncast and cast type Phi1 f a = βˆ€ r j. (βˆ€i. (i < j) β‡’ Mu f i β†’ r i) β†’ -- uncast (βˆ€i. (i < j) β‡’ r i β†’ Mu f i) β†’ -- cast (βˆ€i. r i β†’ a i) β†’ (r j β†’ a j) -- or, maybe without the size restriction over indices in cast type Phi1 f a = βˆ€ r j. (βˆ€i. (i < j) β‡’ Mu f i β†’ r i) β†’ -- uncast (βˆ€i. r i β†’ Mu f i) β†’ -- cast (βˆ€i. r i β†’ a i) β†’ (f r j β†’ a j) Preliminary idea that still need further studies for the termination proof
  • 17. How are Mu and Mu' related? data Mu0 (f :: * β†’ *) = In0 (f (Mu0 f )) data Mu'0 (f :: * β†’ *) a = In'0 (f (Mu'0 f a)) | Inverse0 a type Phi0 f = βˆ€r. (r β†’ a ) β†’ (f r β†’ a ) mit0 :: Phi0 f a β†’ Mu f β†’ a mit0 phi (In0 x) = phi (mit0 phi) x type Phi'0 f a = βˆ€ r. (a β†’ r a) β†’ (r a β†’ a) β†’ (f (r a) β†’ a) msfit0 :: Phi'0 f a β†’ (βˆ€a. Mu'0 f a) β†’ a msfit0 phi (In'0 x) = phi Inverse0 (msfit0 phi) x Can we coerece from (βˆ€a. Mu'0 f a) to Mu f ? Yes!!! Can we coerece from Mu f to (βˆ€a. Mu'0 f a) ? Not exactly...
  • 18. Corecion from Mu' to Mu data E r = Lam (r -> r) | App r r type Expr = Mu0 E type Exp' a = Mu'0 E a type Exp = (forall a. Exp' a) -- .i.e (forall a. Rec0 E a) exp2expr :: Exp -> Expr -- i.e., (forall a. Rec0 E a) -> Mu0 E exp2expr = msfit0 phi where phi inv p2r (Lam f) = In0(Lam((x -> p2r (f (inv x))))) phi inv p2r (App e1 e2) = In0(App (p2r e1) (p2r e2))
  • 19. Attempt from Mu to Mu' msfit' :: (forall r. (a -> r a) -> (r a -> a) -> f (r a) -> a) -> Mu'0 f a -> a msfit' phi (In'0 x) = phi Inverse0 (msfit' phi) x msfit' phi (Inverse0 z) = z exp'2expr :: Exp' Expr -> Expr -- i.e., Mu'0 E (Mu0 E) -> Mu0 E exp'2expr = msfcata' phi where phi inv p2r (Lam f) = In0(Lam((x -> p2r (f (inv x))))) phi inv p2r (App e1 e2) = In0(App (p2r e1) (p2r e2)) -- using general recursion and pattern matching against In0 expr2exp' :: Expr -> Exp' Expr -- i.e., Mu0 E -> Mu'0 E (Mu0 E) expr2exp' (In0(Lam f)) = In'0(Lam (x -> expr2exp' (f (exp'2expr x)))) expr2exp' (In0(App e1 e2)) = In'0(App (expr2exp' e1) (expr2exp' e2))
  • 20. Why from Mu' to Mu possible but not the other way? data E r = Lam (r β†’ r) | App r r type Exp' a = Mu'0 ExpF a -- (Exp' a) may contain Inverse type Exp = βˆ€a . Exp' a -- Exp does not contain Inverse lam :: (βˆ€a. Exp' a β†’ Exp' a) -> Exp -- it's not (Exp β†’ Exp) β†’ Exp lam f = In'0 (Lam f ) -- f can handle Inverse containing values -- but can never examine its content type Expr = Mu0 E lam :: Expr β†’ Expr lam f = In0 (Lam f) Assume that Expr = Exp = (βˆ€a . Exp' a) There are less functions with type (βˆ€a. Exp' a) β†’ (βˆ€a. Exp' a) than functions with type (βˆ€a. Exp' a β†’ Exp' a)
  • 21. Thanks for listening. Questions or Suggestions? ===============