1 / 107

Relating Static and Dynamic Semantics

Relating Static and Dynamic Semantics. COS 441 Princeton University Fall 2004. Motivations. We want to know that when evaluating certain well-formed programs certain errors never occur Example

varick
Download Presentation

Relating Static and Dynamic Semantics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Relating Static and Dynamic Semantics COS 441 Princeton University Fall 2004

  2. Motivations • We want to know that when evaluating certain well-formed programs certain errors never occur • Example • Transition semantics for -calculus is “stuck” when applied to expressions with free variables in it • So if {} `Eok then E should never be “stuck”

  3. Formal Statement isFinal(e) = e 2 F steps(e) = 9 e’. e  e’ stuck(e) = :(steps(e) or isFinal(e)) Soundness Theorem: If {} `Eokand E*E’then:stuck(E’)

  4. Formal Statement isFinal(e) = e 2 F steps(e) = 9 e’. e  e’ stuck(e) = :(steps(e) or isFinal(e)) Soundness Theorem: If {} `Eokand E*E’then (steps(E’) or isFinal(E’))

  5. Proof: Soundness Theorem By induction on derivations of * with Preservation and Progress Lemmas Preservation Lemma: If {} `Eokand EE’then {} `E’ok Progress Lemma: If {} `Eokthen (steps(E) or isFinal(E))

  6. Warning!! • The remainder of the lecture consists of a series of tedious proofs • Take that swig of coffee now • Slides will be on web-site • Last set of tedious proofs in lecture • I’ll assign them as homework from now on! ;) • What we discuss today is a template for Assignment 3

  7. S  S’ S’ * S’’ Z* S* S * S S * S’’ Proof by Induction over * To show 8 e,e’ P(e,e’) we must show case Z*: IH(E,E) case S*: IfEE’andIH(E’,E’’) then IH(E,E’’) IH(e,e’) = If {} ` e okand e * e’ then (steps(e’) or isFinal(e’))

  8. Proof: Soundness Theorem case Z*: IH(E,E)

  9. Proof: Soundness Theorem case Z*: If {} `Eokand E*Ethen (steps(E) or isFinal(E))

  10. Proof: Soundness Theorem case Z*: (steps(E) or isFinal(E)) • {} `Eokand E*E by assumption

  11. Proof: Soundness Theorem case Z*: • {} `Eokand E*E by assumption 2. (steps(E) or isFinal(E)) by ??

  12. Proof: Soundness Theorem case Z*: • {} `Eokand E*E by assumption 2. (steps(E) or isFinal(E)) by Progress Lemma with (1)

  13. Proof: Soundness Theorem case S*: IfEE’andIH(E’,E’’) then IH(E,E’’)

  14. Proof: Soundness Theorem case S*: IH(E,E’’) 1. EE’andIH(E’,E’’) by assumption

  15. Proof: Soundness Theorem case S*: If {} `Eokand E*E’’then (steps(E’’) or isFinal(E’’)) • EE’andIH(E’,E’’) by assumption

  16. Proof: Soundness Theorem case S*: (steps(E’’) or isFinal(E’’)) • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • ` E’ ok by Preservation with (2,1) • E’ * E’’ by inversion of S* and (2) • (steps(E) or isFinal(E’’)) by IH with (3, 4)

  17. Proof: Soundness Theorem case S*: (steps(E’’) or isFinal(E’’)) • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • {} `E’ok by ?? E’ * E’’ by inversion of S* and (2) • (steps(E) or isFinal(E’’)) by IH with (3, 4)

  18. Proof: Soundness Theorem case S*: (steps(E’’) or isFinal(E’’)) • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • {} `E’ok by Preservation with (2,1) • E’ * E’’ by inversion of S* and (2) • (steps(E) or isFinal(E’’)) by IH with (3, 4)

  19. Proof: Soundness Theorem case S*: (steps(E’’) or isFinal(E’’)) • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • {} `E’ok by Preservation with (2,1) • E’*E’’ by ?? • (steps(E’’) or isFinal(E’’)) by IH with (3, 4)

  20. Proof: Soundness Theorem case S*: (steps(E’’) or isFinal(E’’)) • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • {} `E’ok by Preservation with (2,1) • E’*E’’ by inversion of S* and (2) • (steps(E’’) or isFinal(E’’)) by IH with (3, 4)

  21. Proof: Soundness Theorem case S*: • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • {} `E’ok by Preservation with (2,1) • E’*E’’ by inversion of S* and (2) • (steps(E’’) or isFinal(E’’)) by ??

  22. Proof: Soundness Theorem case S*: • EE’andIH(E’,E’’) by assumption • {} `Eokand E*E’’ by assumption • {} `E’ok by Preservation with (2,1) • E’*E’’ by inversion of S* and (2) • (steps(E’’) or isFinal(E’’)) by IH(E’,E’’) with (3, 4)

  23. Notes About our Proof • Note our Proof works for any single step relation () • Specific details of step function factored into Progress and Preservation lemmas • Need to refer to the static and dynamic semantics of the step relation to prove Progress and Preservation Lemmas

  24. ok-V X2  ` X ok  ` E1ok  ` E2ok  [{X}` Eok X  ok-A ok-L  ` apply(E1,E2)ok  ` lam(X.E)ok Static Semantics for -calculus

  25. e2 e’2 e1 e’1 A2 A1 A3 ((x.e1) e2)  ((x.e1) e’2) ((x.e1) (y.e2))  [xÃ(y.e2)] e1 (e1 e2)  (e’1 e2) Dynamic Semantics for -calculus

  26. Proof: Preservation Lemma Proof by induction on the derivations of EE’ case A1: IH(((X.E1) (Y.E2)),[XÃ (Y.E2)] E1) case A2: IfIH(E2,E’2) then IH(((X.E1) E2)),((X.E1) E’2)) case A3: IfIH(E1,E’1) then IH((E1E2)),(E’1E2)) IH(e,e’) =If {} ` e okand e  e’ then {} ` e’ ok

  27. Proof: Preservation Lemma case A1: If {} ` ((X.E1) (Y.E1))okand ((X.E1) (Y.E1))[XÃ (Y.E2)] E1then {} `[XÃ (Y.E2)] E1ok

  28. Proof: Preservation Lemma case A1: {} `[XÃ (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by inversion of ok-A and (1) • {} [ {X} ` E1ok by inversion of ok-L and (2) • {} `[X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

  29. Proof: Preservation Lemma case A1: {} `[XÃ (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by ?? • {} [ {X} ` E1ok by inversion of ok-L and (2) • {} `[X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

  30. Proof: Preservation Lemma case A1: {} `[XÃ (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by inversion of ok-A and (1) • {} [ {X} ` E1ok by inversion of ok-L and (2) • {} `[X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

  31. Proof: Preservation Lemma case A1: {} `[XÃ (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by inversion of ok-A and (1) • {} [ {X} `E1ok by ?? • {} `[X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

  32. Proof: Preservation Lemma case A1: {} `[XÃ (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by inversion of ok-A and (1) • {} [ {X} `E1ok by inversion of ok-L and (2) • {} `[X Ã (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

  33. Proof: Preservation Lemma case A1: {} `[X Ã (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by inversion of ok-A and (1) • {} [ {X} `E1ok by inversion of ok-L and (2) • {} `[XÃ (Y.E2)] E1 ok by ??

  34. Proof: Preservation Lemma case A1: {} `[X Ã (Y.E2)] E1 ok • {} ` ((X.E1) (Y.E2))okand ((X.E1) (Y.E2))[XÃ (Y.E2)] E1 by assumption • {} `(X.E1)ok and {} `(Y.E2)ok by inversion of ok-A and (1) • {} [ {X} `E1ok by inversion of ok-L and (2) • {} `[XÃ (Y.E2)] E1 ok by Substitution Lemma with (3) and (2)

  35. Substitution Lemma Proof by induction on the derivations of `E ok If[ {X} `E ok and {} `E’ ok then ` [XÃE’]E ok case ok-V: … case ok-L: … case ok-A: … IH(env,e) =If env [ {X} ` eok and {} `E’ ok then env ` [XÃE’]eok

  36. Substitution Proof by induction on the derivations of `E ok If[ {X} `E ok and {} `E’ ok then ` [XÃE’]E ok case ok-V: If X2 then IH(,X) case ok-L: If IH( [ {X}, E) and X   then IH(,(X.E)) case ok-A: If IH(,E1) and IH(,E2) then IH(,(E1E2)) IH(env,e) =If env [ {X} ` eok and {} `E’ ok then env ` [XÃE’]eok

  37. Proof: Substitution case ok-V: 1. X2 by assumption 2. [ {Y} `X ok and {} `E’ ok by assumption 3. ` [YÃE’]X ok by cases case X = Y: 3.1. [YÃE’]X = E’ by def of subst. 3.2. ` E’ok by (2) 3.3. ` [YÃE’]X ok by (3.1) and (3.2) case XY: 3.1. [YÃE’]X = X by def of subst. 3.2. `X ok by ok-V and (1) 3.3. ` [YÃE’]X ok by (3.1) and (3.2)

  38. Proof: Substitution case ok-L: If IH( [ {X}, E) and X   then IH(,(X.E)) …

  39. Proof: Substitution case ok-A: If IH(,E1) and IH(,E2) then IH(,(E1E2)) …

  40. Proof: Preservation Lemma case A2: IfIH(E2,E’2) then IH(((X.E1) E2)),((X.E1) E’2))

  41. Proof: Preservation Lemma case A2: IH(((X.E1) E2)),((X.E1) E’2)) • IH(E2,E’2) by assumption

  42. Proof: Preservation Lemma case A2: If {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)then {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption

  43. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} ` E2ok by inversion of ok-A and (2) • E2 E’2 by inversion of A2 • {} ` E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  44. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by ?? • E2 E’2 by inversion of A2 • {} ` E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  45. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by inversion of ok-A and (2) • E2 E’2 by inversion of A2 • {} ` E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  46. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by inversion of ok-A and (2) • E2E’2 by ?? • {} ` E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  47. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by inversion of ok-A and (2) • E2E’2 by inversion of A2 and (2) • {} ` E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  48. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by inversion of ok-A and (2) • E2E’2 by inversion of A2 and (2) • {} `E’2ok by ?? • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  49. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by inversion of ok-A and (2) • E2E’2 by inversion of A2 and (2) • {} `E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ok-A with (3) and (5)

  50. Proof: Preservation Lemma case A2: {} `((X.E1) E’2)ok • IH(E2,E’2) by assumption • {} ` ((X.E1) E2))okand ((X.E1) E2))((X.E1) E’2)by assumption • {} `(X.E1) ok and {} `E2ok by inversion of ok-A and (2) • E2E’2 by inversion of A2 and (2) • {} `E’2ok by IH(E2,E’2) with (3) and (4) • {} `((X.E1) E’2)ok by ??

More Related