A more elegant specification for FRP

Transcription

1 A more elegant specification for FRP Conal Elliott LambdaJam 2015 Conal Elliott A more elegant specification for FRP LambdaJam / 20

2 The story so far Conal Elliott A more elegant specification for FRP LambdaJam / 20

3 FRP s two fundamental properties Precise, simple denotation. (Elegant & rigorous.) Continuous time. (Natural & composable.) FRP is not about: Conal Elliott A more elegant specification for FRP LambdaJam / 20

4 FRP s two fundamental properties Precise, simple denotation. (Elegant & rigorous.) Continuous time. (Natural & composable.) FRP is not about: graphs, updates and propagation, streams, doing Conal Elliott A more elegant specification for FRP LambdaJam / 20

5 Semantics Central abstract type: Behavior a a flow of values. Conal Elliott A more elegant specification for FRP LambdaJam / 20

6 Semantics Central abstract type: Behavior a a flow of values. Precise & simple semantics: µ :: Behavior a Ñ pt Ñ aq where T R (reals). Conal Elliott A more elegant specification for FRP LambdaJam / 20

7 Semantics Central abstract type: Behavior a a flow of values. Precise & simple semantics: µ :: Behavior a Ñ pt Ñ aq where T R (reals). Much of API and its specification can follow from this one choice. Conal Elliott A more elegant specification for FRP LambdaJam / 20

8 Original formulation Conal Elliott A more elegant specification for FRP LambdaJam / 20

9 API time :: Behavior T lift 0 :: a Ñ Behavior a lift 1 :: pa Ñ bq Ñ Behavior a Ñ Behavior b lift 2 :: pa Ñ b Ñ cq Ñ Behavior a Ñ Behavior b Ñ Behavior c timetrans :: Behavior a Ñ Behavior T Ñ Behavior a integral :: VS a ñ Behavior a Ñ T Ñ Behavior a... instance Num a ñ Num pbehavior aq where Reactivity later. Conal Elliott A more elegant specification for FRP LambdaJam / 20

10 Semantics µ time λt Ñ t µ plift 0 aq λt Ñ a µ plift 1 f xsq λt Ñ f pµ xs tq µ plift 2 f xs ysq λt Ñ f pµ xs tq pµ ys tq µ ptimetrans xs ttq λt Ñ µ xs pµ tt tq instance Num a ñ Num pbehavior aq where frominteger lift 0 frominteger p`q lift 2 p`q... Conal Elliott A more elegant specification for FRP LambdaJam / 20

11 Semantics µ time id µ plift 0 aq const a µ plift 1 f xsq f µ xs µ plift 2 f xs ysq lifta 2 f pµ xsq pµ ysq µ ptimetrans xs ttq µ xs µ tt instance Num a ñ Num pbehavior aq where frominteger lift 0 frominteger p`q lift 2 p`q... Conal Elliott A more elegant specification for FRP LambdaJam / 20

12 Events Secondary type: µ :: Event a Ñ rpt, aqs -- non-decreasing times never :: Event a once :: T Ñ a Ñ Event a p..q :: Event a Ñ Event a Ñ Event a pùñq :: Event a Ñ pa Ñ bq Ñ Event b predicate :: Behavior Bool Ñ Event pq snapshot :: Event a Ñ Behavior b Ñ Event pa, bq Exercise: define semantics of these operations. Conal Elliott A more elegant specification for FRP LambdaJam / 20

13 A more elegant specification Conal Elliott A more elegant specification for FRP LambdaJam / 20

14 API Replace several operations with standard abstractions: instance Functor Behavior where... instance Applicative Behavior where... instance Monoid a ñ Monoid pbehavior aq where... Why? instance Functor Event where... instance Monoid a ñ Monoid pevent aq where... Conal Elliott A more elegant specification for FRP LambdaJam / 20

15 API Replace several operations with standard abstractions: instance Functor Behavior where... instance Applicative Behavior where... instance Monoid a ñ Monoid pbehavior aq where... instance Functor Event where... instance Monoid a ñ Monoid pevent aq where... Why? Less learning, more leverage. Specifications and laws for free. Conal Elliott A more elegant specification for FRP LambdaJam / 20

16 Semantic instances instance Functor ppñq z q where... instance Applicative ppñq z q where... instance Monoid a ñ Monoid pz Ñ aq where... instance Num a ñ Num pz Ñ aq where The Behavior instances follow in precise analogy to denotation. Conal Elliott A more elegant specification for FRP LambdaJam / 20

17 Homomorphisms A homomorphism h is a function that preserves (distributes over) an algebraic structure. For instance, for Monoid: h ε ε h pas bsq h as h bs Conal Elliott A more elegant specification for FRP LambdaJam / 20

18 Homomorphisms A homomorphism h is a function that preserves (distributes over) an algebraic structure. For instance, for Monoid: h ε ε h pas bsq h as h bs Some monoid homomorphisms: length 1 :: ra s Ñ Sum Int length 1 Sum length log 1 :: Product R Ñ Sum R log 1 Sum log getproduct Conal Elliott A more elegant specification for FRP LambdaJam / 20

19 More homomorphism properties Functor: h pfmap f xsq fmap f ph xsq Applicative: h ppure aq pure a h pfs ă ą xsq h fs ă ą h xs Monad: h pm ą kq h m ą h k Conal Elliott A more elegant specification for FRP LambdaJam / 20

20 Specification by semantic homomorphism Specification: µ as homomorphism. For instance, µ pfmap f asq fmap f pµ asq µ ppure aq pure a µ pfs ă ą xsq µ fs ă ą µ xs Conal Elliott A more elegant specification for FRP LambdaJam / 20

21 Semantic instances instance Monoid a ñ Monoid pz Ñ aq where ε λz Ñ ε f g λz Ñ f z g z instance Functor ppñq z q where fmap g f g f instance Applicative ppñq z q where pure a λz Ñ a ff ă ą fx λz Ñ pff zq pfx zq Conal Elliott A more elegant specification for FRP LambdaJam / 20

22 Semantic homomorphisms Put the pieces together: µ ppure aq pure a µ pfs ă ą xsq µ fs ă ą µ xs λt Ñ a λt Ñ pµ fs tq pµ xs tq Likewise for Functor, Monoid, Num, etc. Conal Elliott A more elegant specification for FRP LambdaJam / 20

23 Semantic homomorphisms Put the pieces together: µ ppure aq pure a µ pfs ă ą xsq µ fs ă ą µ xs λt Ñ a λt Ñ pµ fs tq pµ xs tq Likewise for Functor, Monoid, Num, etc. Notes: Corresponds exactly to the original FRP denotation. Follows inevitably from semantic homomorphism principle. Laws hold for free (already paid for). Conal Elliott A more elegant specification for FRP LambdaJam / 20

24 Laws for free µ ε ε µ pa bq µ a µ b ñ a ε a ε b b a pb cq pa bq c where equality is semantic. Conal Elliott A more elegant specification for FRP LambdaJam / 20

25 Laws for free µ ε ε µ pa bq µ a µ b ñ a ε a ε b b a pb cq pa bq c where equality is semantic. Proofs: µ pa εq µ a µ ε µ a ε µ a µ pε bq µ ε µ b ε µ b µ b µ pa pb cqq µ a pµ b µ cq pµ a µ bq µ c µ ppa bq cq Works for other classes as well. Conal Elliott A more elegant specification for FRP LambdaJam / 20

26 Events newtype Event a Event pbehavior ra sq deriving pmonoid, Functor q -- discretely non-empty Conal Elliott A more elegant specification for FRP LambdaJam / 20

27 Events newtype Event a Event pbehavior ra sq deriving pmonoid, Functor q -- discretely non-empty Derived instances: instance Monoid a ñ Monoid pevent aq where ε Event ppure εq Event u Event v Event plifta 2 p q u vq instance Functor Event where fmap f pevent bq Event pfmap pfmap f q bq Conal Elliott A more elegant specification for FRP LambdaJam / 20

28 Events newtype Event a Event pbehavior ra sq deriving pmonoid, Functor q -- discretely non-empty Derived instances: instance Monoid a ñ Monoid pevent aq where ε Event ppure εq Event u Event v Event plifta 2 p q u vq instance Functor Event where fmap f pevent bq Event pfmap pfmap f q bq Alternatively, type Event Behavior r s Conal Elliott A more elegant specification for FRP LambdaJam / 20

29 Conclusion Two fundamental properties: Precise, simple denotation. (Elegant & rigorous.) Continuous time. (Natural & composable.) Warning: most recent FRP systems lack both. Conal Elliott A more elegant specification for FRP LambdaJam / 20

30 Conclusion Two fundamental properties: Precise, simple denotation. (Elegant & rigorous.) Continuous time. (Natural & composable.) Warning: most recent FRP systems lack both. Semantic homomorphisms: Mine semantic model for API. Inevitable API semantics (minimize invention). Laws hold for free (already paid for). No abstraction leaks. Matches original FRP semantics. Generally useful principle for library design. Conal Elliott A more elegant specification for FRP LambdaJam / 20