Operational Semantics Using the Partiality Monad

Transcription

1 page.1 Operational Semantics Using the Partiality Monad Nils Anders Danielsson (Göteborg) Shonan Meeting 026: Coinduction for computation structures and programming languages The research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme (FP7/ )/ERC grant agreement n This presentation does not necessarily reflect the views of the ERC or the EU. The EU is not liable for any use of the presented information.

2 page.2 Total Definitional Interpreters Nils Anders Danielsson (Göteborg) Shonan Meeting 026: Coinduction for computation structures and programming languages The research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme (FP7/ )/ERC grant agreement n This presentation does not necessarily reflect the views of the ERC or the EU. The EU is not liable for any use of the presented information.

3 page.3 Outline Using partiality monad to: Define semantics of partial language: total definitional interpreter. Prove type soundness. Prove compiler correctness.

4 page.4 A partial language A language with two effects: non-termination and crashes. t = c x λx.t t t Represented using well-scoped de Bruijn indices: data Tm (n N) Set where con N Tm n var Fin n Tm n lam Tm (1 + n) Tm n app Tm n Tm n Tm n

5 page.5 A partial language A language with two effects: non-termination and crashes. t = c x λx.t t t Represented using well-scoped de Bruijn indices: data Tm (n N) Set where con N Tm n var Fin n Tm n lam Tm (1 + n) Tm n app Tm n Tm n Tm n

6 page.6 Environments and values Based on closures: Env N Set Env n = Vec Value n data Value Set where con N Value clo Tm (1 + n) Env n Value

7 page.7 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

8 page.8 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

9 page.9 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

10 page.10 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

11 page.11 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

12 page.12 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

13 page.13 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ) Does not work in Agda, Coq Call-by-value? Call-by-name?

14 page.14 Relational, big-step semantics Can define inductive big-step semantics: ρ t v Very similar to definitional interpreter, but we cannot run the semantics. Does not distinguish between non-termination and crashes.

15 page.15 Relational, big-step semantics Can add coinductive big-step semantics: ρ t Problem: Duplication of rules. Problem: Have we forgotten a rule?

16 page.16 Alternative Definitional interpreter + partiality monad functional, big-step semantics.

17 page.17 Partiality monad

18 page.18 Partiality monad codata (A Set) Set where now A A later A A Constructive partiality monad, not maybe monad or exception monad. A ν C. A + C.

19 page.19 Partiality monad codata (A Set) Set where now A A later A A Constructive partiality monad, not maybe monad or exception monad. A ν C. A + C.

20 page.20 Partiality monad codata (A Set) Set where now A A later A A Constructive partiality monad, not maybe monad or exception monad. A ν C. A + C.

21 page.21 Partiality monad codata (A Set) Set where now A A later A A never A never = later never >= A (A B ) B now x >= f = f x later x >= f = later (x >= f)

22 page.22 Partiality monad codata (A Set) Set where now A A later A A What is the right notion of equality for A? later (later (now 5)) now 5 later never never now 5 never Equality up to finite differences in the number of later constructors.

23 page.23 Total definitional interpreter

24 page.24 Definitional interpreter Tm n Env n Value con i ρ = con i var x ρ = lookup x ρ lam t ρ = clo t ρ app t 1 t 2 ρ = t 1 ρ t 2 ρ Value Value Value clo t 1 ρ v 2 = t 1 (v 2 ρ)

25 page.25 Definitional interpreter With Maybe monad: Tm n Env n Maybe Value con i ρ = return (con i) var x ρ = return (lookup x ρ) lam t ρ = return (clo t ρ) app t 1 t 2 ρ = t 1 ρ >= λ v 1 t 2 ρ >= λ v 2 v 1 v 2 Value Value Maybe Value clo t 1 ρ v 2 = t 1 (v 2 ρ) con i 1 v 2 = fail

26 page.26 Definitional interpreter With Maybe monad: Tm n Env n Maybe Value con i ρ = return (con i) var x ρ = return (lookup x ρ) lam t ρ = return (clo t ρ) app t 1 t 2 ρ = t 1 ρ >= λ v 1 t 2 ρ >= λ v 2 v 1 v 2 Value Value Maybe Value clo t 1 ρ v 2 = t 1 (v 2 ρ) con i 1 v 2 = fail

27 page.27 Total definitional interpreter With Maybe + : Tm n Env n (Maybe Value) con i ρ = return (con i) var x ρ = return (lookup x ρ) lam t ρ = return (clo t ρ) app t 1 t 2 ρ = t 1 ρ >= λ v 1 t 2 ρ >= λ v 2 v 1 v 2 Value Value (Maybe Value) clo t 1 ρ v 2 = later ( t 1 (v 2 ρ)) con i 1 v 2 = fail

28 page.28 Total definitional interpreter With Maybe + : Tm n Env n (Maybe Value) con i ρ = return (con i) var x ρ = return (lookup x ρ) lam t ρ = return (clo t ρ) app t 1 t 2 ρ = t 1 ρ >= λ v 1 t 2 ρ >= λ v 2 v 1 v 2 Value Value (Maybe Value) clo t 1 ρ v 2 = later ( t 1 (v 2 ρ)) con i 1 v 2 = fail

29 page.29 Total definitional interpreter Tm n Env n (Maybe Value) Type signature deterministic. Type signature computable. No duplication of rules. Impossible to forget a case. Classically equivalent to ρ t v plus ρ t.

30 page.30 Type soundness

31 page.31 Types Coinductive simple types (to allow non-termination): codata Ty Set where nat Ty Ty Ty Ty τ Ty τ = τ nat Typing relation: Γ t σ

32 page.32 Type soundness Statement of type soundness: [ ] t σ t [ ] fail Proof: Generalise, then nested corecursion/ structural recursion.

33 page.33 Compiler correctness

34 page.34 Virtual machine data State Set where data Result Set where continue State Result done VM-Value Result crash Result step State Result step = step = crash

35 page.35 Virtual machine Small-step, total, functional semantics: exec State (Maybe VM-Value) exec s = case step s of continue s later (exec s ) done v return v crash fail Type signature deterministic. Type signature computable. Possible to forget a case.

36 page.36 Compiler correctness Small-step, total, functional semantics: exec State (Maybe VM-Value) Compilers: comp comp v Tm 0 State Value VM-Value

37 page.37 Compiler correctness Small-step, total, functional semantics: exec State (Maybe VM-Value) Compilers: comp comp v Tm 0 State Value VM-Value Compiler correctness: (t Tm 0) exec (comp t) ( t [ ] >= λ v return (comp v v))

38 page.38 Conclusions Definitional interpreter + partiality monad functional, total, big-step semantics. Can mechanise (some) meta-theory. See paper for non-determinism, contextual equivalence, applicative bisimilarity.

39 page.39 Bonus slides

40 page.40 Operational semantics? Very close to usual, relational big-step operational semantics. Not compositional. Values contain closures.

41 page.41 Typing rules Γ con i nat Γ var x lookup x Γ σ Γ t τ Γ lam t σ τ Γ t 1 σ τ Γ t 2 σ Γ app t 1 t 2 τ

42 page.42 Applicative bisimilarity, part 1 data (Maybe Value) (Maybe Value) Set where now u MV v now u now v later (x y) later x later y later l x y later x y later r x y x later y data MV Maybe Value Maybe Value Set where just u V v just u MV just v nothing nothing MV nothing

43 page.43 Applicative bisimilarity, part 2 data V Value Value Set where con con i V con i clo ( v ( t 1 (v ρ 1 ) t 2 (v ρ 2 ))) clo t 1 ρ 1 V clo t 2 ρ 2 T Tm n Tm n Set t 1 T t 2 = ρ t 1 ρ t 2 ρ