Liquid Has iquid Haskel ell: l: Refj efjned, ned, refm fmectiv - - PowerPoint PPT Presentation

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Liquid Has iquid Haskel ell: l: Refj efjned, ned, refm fmectiv - - PowerPoint PPT Presentation

Oct 02, 2023 542 likes •865 views

Liquid Has iquid Haskel ell: l: Refj efjned, ned, refm fmectiv ective, and cl and clas assy Ryan Scott PL Wonks March 23, 2018 Refj fjnem nements ts divide :: Int -> {v:Int | v /= 0} -> Int divide n d = n `div` d Refj

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Ryan Scott

PL Wonks March 23, 2018

Liquid Has iquid Haskel ell: l:

Refj efjned, ned, refm fmectiv ective, and cl and clas assy

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Refj fjnem nements ts

divide :: Int

  • > {v:Int | v /= 0}
  • > Int

divide n d = n `div` d

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Refj fjnem nement r ent refm efmection ction

{-@ reflect fib @-} fib :: Int -> Int fib i | i == 0 = 0 | i == 1 = 1 | otherwise = fib (i-1) + fib (i-2) fibOne :: {fib 1 == 1} fibOne = trivial *** QED

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Refj fjnem nement r t refm efmection + type ction + type clas lasses? ses?

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Refj fjnem nement r t refm efmection + type ction + type clas lasses? ses?

class Semigroup a where (<>) :: a -> a -> a

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Refj fjnem nement r t refm efmection + type ction + type clas lasses? ses?

class Semigroup a where (<>) :: a -> a -> a class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }
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SLIDE 8

Refj fjnem nement r t refm efmection + type ction + type clas lasses? ses?

class Semigroup a where (<>) :: a -> a -> a class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }
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SLIDE 9

Refj fjnem nement r t refm efmection + type ction + type clas lasses? ses?

class Semigroup a where (<>) :: a -> a -> a class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

A t l e a s t , n

  • t

t

  • d

a y . . .

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Why Why not?

  • t?
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Wh Why y not not? Desugari Desugaring ng

class Semigroup a where (<>) :: a -> a -> a

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Wh Why y not not? Desugari Desugaring ng

class Semigroup a where -- Surface syntax (<>) :: a -> a -> a

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Wh Why y not not? Desugari Desugaring ng

class Semigroup a where -- Surface syntax (<>) :: a -> a -> a data Semigroup a {

  • - GHC core syntax

(<>) :: a -> a -> a }

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De Desugar aring ng i ins nstances es

instance Semigroup Unit where Unit <> Unit = Unit

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De Desugar aring ng i ins nstances es

instance Semigroup Unit where Unit <> Unit = Unit semigroupUnit :: Semigroup Unit semigroupUnit = Semigroup { (<>) = appendUnit } appendUnit :: Unit -> Unit -> Unit appendUnit Unit Unit = Unit

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Desugar Desugaring functions ing functions

smashList :: Semigroup a => a -> [a] -> a smashList x [] = x smashList x (y:ys) = smashList (x <> y) ys

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Desugar Desugaring functions ing functions

smashList :: Semigroup a => a -> [a] -> a smashList x [] = x smashList x (y:ys) = smashList (x <> y) ys smashList :: Semigroup a -> a -> [a] -> a smashList _ x [] = x smashList dSemigroup x (y:ys) = smashList dSemigroup ((<>) dSemigroup x y) ys

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Key i y ins nsight ht

Any refjned type involving type classes must be able to survive the translation to GHC core.

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Fi First (naï (naïve) a e) attem empt pt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }
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Fi First (naï (naïve) a e) attem empt pt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

data VerifiedSemigroup a { semigroupSuperClass :: Semigroup a , appendAssoc :: x:a -> y:a -> z:a

  • > { (<>) d x ((<>) d y z)

== (<>) d ((<>) d x y) z } }

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Fi First (naï (naïve) a e) attem empt pt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

data VerifiedSemigroup a { semigroupSuperClass :: Semigroup a , appendAssoc :: x:a -> y:a -> z:a

  • > { (<>) d x ((<>) d y z)

== (<>) d ((<>) d x y) z } }

?

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Fi First (naï (naïve) a e) attem empt pt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

data VerifiedSemigroup a { semigroupSuperClass :: Semigroup a , appendAssoc :: x:a -> y:a -> z:a ->

  • > { forall d:VerifiedSemigroup a.

(<>) d x ((<>) d y z) == (<>) d ((<>) d x y) z } }

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Fi First (naï (naïve) a e) attem empt pt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

data VerifiedSemigroup a { semigroupSuperClass :: Semigroup a , appendAssoc :: x:a -> y:a -> z:a ->

  • > { forall d:VerifiedSemigroup a.

(<>) d x ((<>) d y z) == (<>) d ((<>) d x y) z } }

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We e can’t sh can’t shove forall foralls withi hin pr predic dicat ates wil willy ly-n

  • nilly

ly.

Liquid Haskell is based on the quantifjer-free logic of linear arithmetic and uninterpreted functions (QF-ULIA).

{ forall d:VerifiedSemigroup a. ... } { forall d:VerifiedSemigroup a. ... }

Can’t be expressed in this system.

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Ob Obser servation ation

We can dictate the behavior of type classes in Liquid Haskell by their instances.

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Bett Better r at attem empt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }
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Bett Better r at attem empt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

instance VerifiedSemigroup Unit where ...

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Bett Better r at attem empt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

$dVSUnit :: VerifiedSemigroup Unit $dVSUnit = VerifiedSemigroup { ... } (appendAssoc $dVSUnit) :: x:a -> y:a -> z:a

  • > { (<>) $dVSUnit x ((<>) $dVSUnit y z)

== (<>) $dVSUnit ((<>) $dVSUnit x y) z && ... }

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Bett Better r at attem empt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

instance VerifiedSemigroup Int where ...

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Bett Better r at attem empt

class Semigroup a => VerifiedSemigroup a where appendAssoc :: x:a -> y:a -> z:a

  • > { x <> (y <> z) == (x <> y) <> z }

$dVSInt :: VerifiedSemigroup Int $dVSInt = VerifiedSemigroup { ... } (appendAssoc $dVSInt) :: x:a -> y:a -> z:a

  • > { (<>) $dVSInt x ((<>) $dVSInt y z)

== (<>) $dVSInt ((<>) $dVSInt x y) z && ... }

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Too l

  • o lon
  • ng; di

g; didn’t w n’t wat atch ch

We begin to extend Liquid Haskell towards supporting refjnement refmection + type classes:

  • Accommodate typing rules to be instance-

aware (not as simple as it looks!)

  • Desugar refjnements involving type classes into

refjnements involving dictionaries