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The Why/Krakatoa/Caduceus Platform for Deductive Program Verification Jean-Christophe Filli atre CNRS Universit e Paris Sud TYPES Summer School August 30th, 2007 TYPES Summer School August 30th, 2007 Jean-Christophe Filli

  1. Annotations assert { p } ; e e { p } Examples: assert { x > 0 } ; 1 / x x := 0 { ! x = 0 } if ! x > ! y then ! x else ! y { result ≥ ! x ∧ result ≥ ! y } x := ! x + 1 { ! x > old ( ! x ) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  2. Annotations assert { p } ; e e { p } Examples: assert { x > 0 } ; 1 / x x := 0 { ! x = 0 } if ! x > ! y then ! x else ! y { result ≥ ! x ∧ result ≥ ! y } x := ! x + 1 { ! x > old ( ! x ) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  3. Annotations assert { p } ; e e { p } Examples: assert { x > 0 } ; 1 / x x := 0 { ! x = 0 } if ! x > ! y then ! x else ! y { result ≥ ! x ∧ result ≥ ! y } x := ! x + 1 { ! x > old ( ! x ) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  4. Annotations (cont’d) Loop invariant and variant while e 1 do { invariant p variant t } e 2 done Example: while ! x < N do { invariant ! x ≤ N variant N − ! x } x := ! x + 1 done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  5. Annotations (cont’d) Loop invariant and variant while e 1 do { invariant p variant t } e 2 done Example: while ! x < N do { invariant ! x ≤ N variant N − ! x } x := ! x + 1 done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  6. Auxiliary variables Used to denote the intermediate values of variables Example: . . . { ! x = X } . . . { ! x > X } . . . We will use labels instead new construct L : e new annotation at ( t , L ) Example: . . . L : while . . . do { invariant ! x ≥ at ( ! x , L ) . . . } . . . done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  7. Auxiliary variables Used to denote the intermediate values of variables Example: . . . { ! x = X } . . . { ! x > X } . . . We will use labels instead new construct L : e new annotation at ( t , L ) Example: . . . L : while . . . do { invariant ! x ≥ at ( ! x , L ) . . . } . . . done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  8. Auxiliary variables Used to denote the intermediate values of variables Example: . . . { ! x = X } . . . { ! x > X } . . . We will use labels instead new construct L : e new annotation at ( t , L ) Example: . . . L : while . . . do { invariant ! x ≥ at ( ! x , L ) . . . } . . . done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  9. Auxiliary variables Used to denote the intermediate values of variables Example: . . . { ! x = X } . . . { ! x > X } . . . We will use labels instead new construct L : e new annotation at ( t , L ) Example: . . . L : while . . . do { invariant ! x ≥ at ( ! x , L ) . . . } . . . done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  10. Functions A function declaration introduces a precondition fun ( x : τ ) → { p } e rec f ( x 1 : τ 1 ) . . . ( x n : τ n ) : β { variant t } = { p } e Example: fun ( x : int ref) → { ! x > 0 } x := ! x − 1 { ! x ≥ 0 } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  11. Functions A function declaration introduces a precondition fun ( x : τ ) → { p } e rec f ( x 1 : τ 1 ) . . . ( x n : τ n ) : β { variant t } = { p } e Example: fun ( x : int ref) → { ! x > 0 } x := ! x − 1 { ! x ≥ 0 } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  12. Functions A function declaration introduces a precondition fun ( x : τ ) → { p } e rec f ( x 1 : τ 1 ) . . . ( x n : τ n ) : β { variant t } = { p } e Example: fun ( x : int ref) → { ! x > 0 } x := ! x − 1 { ! x ≥ 0 } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  13. Modularity A function declaration extends the ML function type with a precondition , an effect and a postcondition f : x : τ 1 → { p } τ 2 reads x 1 , . . . , x n writes y 1 , . . . , y m { q } Example: swap : x : int ref → y : int ref → {} unit writes x , y { ! x = old ( ! y ) ∧ ! y = old ( ! x ) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  14. Modularity A function declaration extends the ML function type with a precondition , an effect and a postcondition f : x : τ 1 → { p } τ 2 reads x 1 , . . . , x n writes y 1 , . . . , y m { q } Example: swap : x : int ref → y : int ref → {} unit writes x , y { ! x = old ( ! y ) ∧ ! y = old ( ! x ) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  15. Exceptions Finally, we introduce exceptions in our language a more realistic ML fragment to interpret abrupt statements like return , break or continue new constructs raise ( E e ) : τ try e 1 with E x → e 2 end TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  16. Exceptions Finally, we introduce exceptions in our language a more realistic ML fragment to interpret abrupt statements like return , break or continue new constructs raise ( E e ) : τ try e 1 with E x → e 2 end TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  17. Exceptions Finally, we introduce exceptions in our language a more realistic ML fragment to interpret abrupt statements like return , break or continue new constructs raise ( E e ) : τ try e 1 with E x → e 2 end TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  18. Exceptions The notion of postcondition is extended if x < 0 then raise Negative else sqrt x { result ≥ 0 | Negative ⇒ x < 0 } So is the notion of effect div : x : int → y : int → { . . . } int raises Negative { . . . } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  19. Exceptions The notion of postcondition is extended if x < 0 then raise Negative else sqrt x { result ≥ 0 | Negative ⇒ x < 0 } So is the notion of effect div : x : int → y : int → { . . . } int raises Negative { . . . } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  20. Loops and exceptions We can replace the while loop by an infinite loop loop e { invariant p variant t } and simulate the while loop using an exception while e 1 do { invariant p variant t } e 2 done ≡ try loop if e 1 then e 2 else raise Exit { invariant p variant t } with Exit -> void end simpler constructs ⇒ simpler typing and proof rules TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  21. Loops and exceptions We can replace the while loop by an infinite loop loop e { invariant p variant t } and simulate the while loop using an exception while e 1 do { invariant p variant t } e 2 done ≡ try loop if e 1 then e 2 else raise Exit { invariant p variant t } with Exit -> void end simpler constructs ⇒ simpler typing and proof rules TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  22. Loops and exceptions We can replace the while loop by an infinite loop loop e { invariant p variant t } and simulate the while loop using an exception while e 1 do { invariant p variant t } e 2 done ≡ try loop if e 1 then e 2 else raise Exit { invariant p variant t } with Exit -> void end simpler constructs ⇒ simpler typing and proof rules TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  23. Summary Types τ ::= β | β ref | ( x : τ ) → κ { p } τ ǫ { q } κ ::= q ::= p ; E ⇒ p ; . . . ; E ⇒ p ǫ ::= reads x , . . . , x writes x , . . . , x raises E , . . . , E Annotations c | x | ! x | φ ( t , . . . , t ) | old ( t ) | at ( t , L ) t ::= p ::= True | False | P ( t , . . . , t ) | p ⇒ p | p ∧ p | p ∨ p | ¬ p | ∀ x : β. p | ∃ x : β. p TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  24. Summary Types τ ::= β | β ref | ( x : τ ) → κ { p } τ ǫ { q } κ ::= q ::= p ; E ⇒ p ; . . . ; E ⇒ p ǫ ::= reads x , . . . , x writes x , . . . , x raises E , . . . , E Annotations c | x | ! x | φ ( t , . . . , t ) | old ( t ) | at ( t , L ) t ::= p ::= True | False | P ( t , . . . , t ) | p ⇒ p | p ∧ p | p ∨ p | ¬ p | ∀ x : β. p | ∃ x : β. p TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  25. Programs c | x | ! x | φ ( u , . . . , u ) u ::= e ::= u | x := e | let x = e in e | let x = ref e in e | if e then e else e | loop e { invariant p variant t } | L : e | raise ( E e ) : τ | try e with E x → e end | assert { p } ; e | e { q } | fun ( x : τ ) → { p } e | rec x ( x : τ ) . . . ( x : τ ) : β { variant t } = { p } e | e e TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  26. Typing A typing judgment Γ ⊢ e : ( τ, ǫ ) Rules given in the notes (page 24) The main purpose is to exclude aliases In particular, references can’t escape their scopes TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  27. Typing A typing judgment Γ ⊢ e : ( τ, ǫ ) Rules given in the notes (page 24) The main purpose is to exclude aliases In particular, references can’t escape their scopes TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  28. Semantics Call-by-value semantics, with left to right evalutation Big-step operational semantics given in the notes (page 26) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  29. Proof Rules: Weakest Preconditions We define the predicate wp ( e , q ), called the weakest precondition for program e and postcondition q Property: If wp ( e , q ) holds, then e terminates and q holds at the end of execution (and all inner annotations are verified) The converse holds for the fragment without loops and functions The correctness of an annotated program e is thus wp ( e , True) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  30. Proof Rules: Weakest Preconditions We define the predicate wp ( e , q ), called the weakest precondition for program e and postcondition q Property: If wp ( e , q ) holds, then e terminates and q holds at the end of execution (and all inner annotations are verified) The converse holds for the fragment without loops and functions The correctness of an annotated program e is thus wp ( e , True) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  31. Proof Rules: Weakest Preconditions We define the predicate wp ( e , q ), called the weakest precondition for program e and postcondition q Property: If wp ( e , q ) holds, then e terminates and q holds at the end of execution (and all inner annotations are verified) The converse holds for the fragment without loops and functions The correctness of an annotated program e is thus wp ( e , True) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  32. Proof Rules: Weakest Preconditions We define the predicate wp ( e , q ), called the weakest precondition for program e and postcondition q Property: If wp ( e , q ) holds, then e terminates and q holds at the end of execution (and all inner annotations are verified) The converse holds for the fragment without loops and functions The correctness of an annotated program e is thus wp ( e , True) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  33. Definition of wp ( e , q ) We actually define wp ( e , q ; r ) where q is the “normal” postcondition r ≡ E 1 ⇒ q 1 ; . . . ; E n ⇒ q n is the set of “exceptional” post. TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  34. Basic constructs wp ( u , q ; r ) = q [ result ← u ] wp ( x := e , q ; r ) = wp ( e , q [ result ← void ; ! x ← result ]; r ) wp ( let x = e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ x ← result ]; r ) wp ( let x = ref e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ ! x ← result ]; r ) wp ( if e 1 then e 2 else e 3 , q ; r ) = wp ( e 1 , if result then wp ( e 2 , q ; r ) else wp ( e 3 , q ; r ); r ) wp ( L : e , q ; r ) = wp ( e , q ; r )[ at ( t , L ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  35. Basic constructs wp ( u , q ; r ) = q [ result ← u ] wp ( x := e , q ; r ) = wp ( e , q [ result ← void ; ! x ← result ]; r ) wp ( let x = e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ x ← result ]; r ) wp ( let x = ref e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ ! x ← result ]; r ) wp ( if e 1 then e 2 else e 3 , q ; r ) = wp ( e 1 , if result then wp ( e 2 , q ; r ) else wp ( e 3 , q ; r ); r ) wp ( L : e , q ; r ) = wp ( e , q ; r )[ at ( t , L ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  36. Basic constructs wp ( u , q ; r ) = q [ result ← u ] wp ( x := e , q ; r ) = wp ( e , q [ result ← void ; ! x ← result ]; r ) wp ( let x = e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ x ← result ]; r ) wp ( let x = ref e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ ! x ← result ]; r ) wp ( if e 1 then e 2 else e 3 , q ; r ) = wp ( e 1 , if result then wp ( e 2 , q ; r ) else wp ( e 3 , q ; r ); r ) wp ( L : e , q ; r ) = wp ( e , q ; r )[ at ( t , L ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  37. Basic constructs wp ( u , q ; r ) = q [ result ← u ] wp ( x := e , q ; r ) = wp ( e , q [ result ← void ; ! x ← result ]; r ) wp ( let x = e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ x ← result ]; r ) wp ( let x = ref e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ ! x ← result ]; r ) wp ( if e 1 then e 2 else e 3 , q ; r ) = wp ( e 1 , if result then wp ( e 2 , q ; r ) else wp ( e 3 , q ; r ); r ) wp ( L : e , q ; r ) = wp ( e , q ; r )[ at ( t , L ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  38. Basic constructs wp ( u , q ; r ) = q [ result ← u ] wp ( x := e , q ; r ) = wp ( e , q [ result ← void ; ! x ← result ]; r ) wp ( let x = e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ x ← result ]; r ) wp ( let x = ref e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ ! x ← result ]; r ) wp ( if e 1 then e 2 else e 3 , q ; r ) = wp ( e 1 , if result then wp ( e 2 , q ; r ) else wp ( e 3 , q ; r ); r ) wp ( L : e , q ; r ) = wp ( e , q ; r )[ at ( t , L ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  39. Basic constructs wp ( u , q ; r ) = q [ result ← u ] wp ( x := e , q ; r ) = wp ( e , q [ result ← void ; ! x ← result ]; r ) wp ( let x = e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ x ← result ]; r ) wp ( let x = ref e 1 in e 2 , q ; r ) = wp ( e 1 , wp ( e 2 , q ; r )[ ! x ← result ]; r ) wp ( if e 1 then e 2 else e 3 , q ; r ) = wp ( e 1 , if result then wp ( e 2 , q ; r ) else wp ( e 3 , q ; r ); r ) wp ( L : e , q ; r ) = wp ( e , q ; r )[ at ( t , L ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  40. Traditional rules Assignment of a side-effects free expression wp ( x := u , q ) = q [ ! x ← u ] Exception-free sequence wp ( e 1 ; e 2 , q ) = wp ( e 1 , wp ( e 2 , q )) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  41. Traditional rules Assignment of a side-effects free expression wp ( x := u , q ) = q [ ! x ← u ] Exception-free sequence wp ( e 1 ; e 2 , q ) = wp ( e 1 , wp ( e 2 , q )) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  42. Exceptions wp ( raise ( E e ) : τ, q ; r ) = wp ( e , r E ; r ) wp ( try e 1 with E x → e 2 end , q ; r ) = wp ( e 1 , q ; E ⇒ wp ( e 2 , q ; r )[ x ← result ]; r ) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  43. Exceptions wp ( raise ( E e ) : τ, q ; r ) = wp ( e , r E ; r ) wp ( try e 1 with E x → e 2 end , q ; r ) = wp ( e 1 , q ; E ⇒ wp ( e 2 , q ; r )[ x ← result ]; r ) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  44. Annotations wp ( assert { p } ; e , q ; r ) = p ∧ wp ( e , q ; r ) = wp ( e , q ′ ∧ q ; r ′ ∧ r ) wp ( e { q ′ ; r ′ } , q ; r ) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  45. Annotations wp ( assert { p } ; e , q ; r ) = p ∧ wp ( e , q ; r ) = wp ( e , q ′ ∧ q ; r ′ ∧ r ) wp ( e { q ′ ; r ′ } , q ; r ) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  46. Loops wp ( loop e { invariant p variant t } , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L : e , p ∧ t < at ( t , L ); r ) where ω = the variables (possibly) modified by e Usual while loop wp ( while e 1 do { invariant p variant t } e 2 done , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L :if e 1 then e 2 else raise E , p ∧ t < at ( t , L ) , E ⇒ q ; r ) = p ∧ ∀ ω. p ⇒ wp ( e 1 , if result then wp ( e 2 , p ∧ t < at ( t , L )) else q , r )[ at ( x , L ) ← x ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  47. Loops wp ( loop e { invariant p variant t } , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L : e , p ∧ t < at ( t , L ); r ) where ω = the variables (possibly) modified by e Usual while loop wp ( while e 1 do { invariant p variant t } e 2 done , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L :if e 1 then e 2 else raise E , p ∧ t < at ( t , L ) , E ⇒ q ; r ) = p ∧ ∀ ω. p ⇒ wp ( e 1 , if result then wp ( e 2 , p ∧ t < at ( t , L )) else q , r )[ at ( x , L ) ← x ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  48. Loops wp ( loop e { invariant p variant t } , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L : e , p ∧ t < at ( t , L ); r ) where ω = the variables (possibly) modified by e Usual while loop wp ( while e 1 do { invariant p variant t } e 2 done , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L :if e 1 then e 2 else raise E , p ∧ t < at ( t , L ) , E ⇒ q ; r ) = p ∧ ∀ ω. p ⇒ wp ( e 1 , if result then wp ( e 2 , p ∧ t < at ( t , L )) else q , r )[ at ( x , L ) ← x ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  49. Loops wp ( loop e { invariant p variant t } , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L : e , p ∧ t < at ( t , L ); r ) where ω = the variables (possibly) modified by e Usual while loop wp ( while e 1 do { invariant p variant t } e 2 done , q ; r ) = p ∧ ∀ ω. p ⇒ wp ( L :if e 1 then e 2 else raise E , p ∧ t < at ( t , L ) , E ⇒ q ; r ) = p ∧ ∀ ω. p ⇒ wp ( e 1 , if result then wp ( e 2 , p ∧ t < at ( t , L )) else q , r )[ at ( x , L ) ← x ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  50. Functions wp ( fun ( x : τ ) → { p } e , q ; r ) = q ∧ ∀ x . ∀ ρ. p ⇒ wp ( e , True) wp ( rec f ( x 1 : τ 1 ) . . . ( x n : τ n ) : τ { variant t } = { p } e , q ; r ) = q ∧ ∀ x 1 . . . . ∀ x n . ∀ ρ. p ⇒ wp ( L : e , True) when computing wp ( L : e , True), f is assumed to have type ( x 1 : τ 1 ) → · · · → ( x n : τ n ) → { p ∧ t < at ( t , L ) } τ ǫ { q } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  51. Functions wp ( fun ( x : τ ) → { p } e , q ; r ) = q ∧ ∀ x . ∀ ρ. p ⇒ wp ( e , True) wp ( rec f ( x 1 : τ 1 ) . . . ( x n : τ n ) : τ { variant t } = { p } e , q ; r ) = q ∧ ∀ x 1 . . . . ∀ x n . ∀ ρ. p ⇒ wp ( L : e , True) when computing wp ( L : e , True), f is assumed to have type ( x 1 : τ 1 ) → · · · → ( x n : τ n ) → { p ∧ t < at ( t , L ) } τ ǫ { q } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  52. Function call Simplified using e 1 e 2 ≡ let x 1 = e 1 in let x 2 = e 2 in x 1 x 2 Assuming x 1 : ( x : τ ) → { p ′ } τ ′ ǫ { q ′ } we define wp ( x 1 x 2 , q ) = p ′ [ x ← x 2 ] ∧ ∀ ω. ∀ result . ( q ′ [ x ← x 2 ] ⇒ q )[ old ( t ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  53. Function call Simplified using e 1 e 2 ≡ let x 1 = e 1 in let x 2 = e 2 in x 1 x 2 Assuming x 1 : ( x : τ ) → { p ′ } τ ′ ǫ { q ′ } we define wp ( x 1 x 2 , q ) = p ′ [ x ← x 2 ] ∧ ∀ ω. ∀ result . ( q ′ [ x ← x 2 ] ⇒ q )[ old ( t ) ← t ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  54. Outline 1 An intermediate language for program verification syntax, typing, semantics, proof rules 1 the Why tool 2 multi-prover approach 3 2 Verifying C and Java programs specification languages 1 models of program execution 2 3 A challenging case study TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  55. The Why Tool This intermediate language is implemented in the Why tool input = polymorphic first-order logic declarations + programs output = logical declarations + goals, in the syntax of the selected prover TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  56. Logical Declarations type t logic zero : t logic succ : t -> t logic le : t, t -> prop axiom a : forall x:t. le(zero,x) goal g : le(zero, succ(zero)) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  57. Programs parameter x : int ref parameter g : b:t -> { x>=0 } t writes x { result=succ(b) and x=x@+1 } let h (a:int) (b:t) = { x>=0 } if !x = a then x := 0; g (succ b) { result=succ(succ(b)) } exception E exception F of int TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  58. Usage it is a compiler: why --coq f.why to produce a re-editable Coq file f why.v why --simplify f.why to produce a Simplify script f why.sx etc. the following provers/formats are supported: Coq, PVS, Isabelle/HOL, HOL-light, HOL4, Mizar Simplify, Ergo, SMT (Yices, CVC3, etc.), CVC-Lite, haRVey, Zenon there is a graphical user interface, gwhy TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  59. Example: Dijkstra’s Dutch national flag Goal: to sort an array where elements only have three different values (blue, white and red) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  60. Algorithm 0 b i r n . . . to do. . . BLUE WHITE RED flag ( t , n ) ≡ b ← 0 i ← 0 r ← n while i < r case t [ i ] BLUE : swap t [ b ] and t [ i ]; b ← b + 1; i ← i + 1 WHITE : i ← i + 1 RED : r ← r − 1; swap t [ r ] and t [ i ] TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  61. Correctness proof we want to prove termination absence of runtime error = no array access out of bounds behavioral correctness = the final array is sorted and contains the same elements as the initial array TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  62. Modelization We model colors using an abstract datatype arrays using references containing functional arrays TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  63. An abstract type for colors type color logic blue : color logic white : color logic red : color predicate is color(c:color) = c=blue or c=white or c=red parameter eq color : c1:color -> c2:color -> {} bool { if result then c1=c2 else c1<>c2 } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  64. Functional arrays type color array logic acc : color array, int -> color logic upd : color array, int, color -> color array axiom acc upd eq : forall a:color array. forall i:int. forall c:color. acc(upd(a,i,c),i) = c axiom acc upd neq : forall a:color array. forall i,j:int. forall c:color. i <> j -> acc(upd(a,j,c),i) = acc(a,i) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  65. Array bounds logic length : color array -> int axiom length update : forall a:color array. forall i:int. forall c:color. length(upd(a,i,c)) = length(a) parameter get : t:color array ref -> i:int -> { 0<=i<length(t) } color reads t { result=acc(t,i) } parameter set : t:color array ref -> i:int -> c:color -> { 0<=i<length(t) } unit writes t { t=upd(t@,i,c) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  66. Array bounds logic length : color array -> int axiom length update : forall a:color array. forall i:int. forall c:color. length(upd(a,i,c)) = length(a) parameter get : t:color array ref -> i:int -> { 0<=i<length(t) } color reads t { result=acc(t,i) } parameter set : t:color array ref -> i:int -> c:color -> { 0<=i<length(t) } unit writes t { t=upd(t@,i,c) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  67. The swap function let swap (t:color array ref) (i:int) (j:int) = { 0 <= i < length(t) and 0 <= j < length(t) } let u = get t i in set t i (get t j); set t j u { t = upd(upd(t@,i,acc(t@,j)), j, acc(t@,i)) } 5 proofs obligations 3 automatically discharged by Why 2 left to the user (and automatically discharged by Simplify) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  68. The swap function let swap (t:color array ref) (i:int) (j:int) = { 0 <= i < length(t) and 0 <= j < length(t) } let u = get t i in set t i (get t j); set t j u { t = upd(upd(t@,i,acc(t@,j)), j, acc(t@,i)) } 5 proofs obligations 3 automatically discharged by Why 2 left to the user (and automatically discharged by Simplify) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  69. Function code let dutch flag (t:color array ref) (n:int) = let b = ref 0 in let i = ref 0 in let r = ref n in while !i < !r do if eq color (get t !i) blue then begin swap t !b !i; b := !b + 1; i := !i + 1 end else if eq color (get t !i) white then i := !i + 1 else begin r := !r - 1; swap t !r !i end done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  70. Function specification let dutch flag (t:color array ref) (n:int) = { 0 <= n and length(t) = n and forall k:int. 0 <= k < n -> is color(acc(t,k)) } . . . { (exists b:int. exists r:int. monochrome(t,0,b,blue) and monochrome(t,b,r,white) and monochrome(t,r,n,red)) and permutation(t,t@,0,n-1) } TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  71. The monochrome property predicate monochrome(t:color array,i:int,j:int,c:color) = forall k:int. i<=k<j -> acc(t,k)=c TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  72. The permutation property logic permutation : color array, color array, int, int -> prop axiom permut refl : forall t:color array. forall l,r:int. permutation(t,t,l,r) axiom permut sym : forall t1,t2:color array. forall l,r:int. permutation(t1,t2,l,r) -> permutation(t2,t1,l,r) axiom permut trans : forall t1,t2,t3:color array. forall l,r:int. permutation(t1,t2,l,r) -> permutation(t2,t3,l,r) -> permutation(t1,t3,l,r) axiom permut swap : forall t:color array. forall l,r,i,j:int. l <= i <= r -> l <= j <= r -> permutation(t, upd(upd(t,i,acc(t,j)), j, acc(t,i)), l, r) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  73. The permutation property logic permutation : color array, color array, int, int -> prop axiom permut refl : forall t:color array. forall l,r:int. permutation(t,t,l,r) axiom permut sym : forall t1,t2:color array. forall l,r:int. permutation(t1,t2,l,r) -> permutation(t2,t1,l,r) axiom permut trans : forall t1,t2,t3:color array. forall l,r:int. permutation(t1,t2,l,r) -> permutation(t2,t3,l,r) -> permutation(t1,t3,l,r) axiom permut swap : forall t:color array. forall l,r,i,j:int. l <= i <= r -> l <= j <= r -> permutation(t, upd(upd(t,i,acc(t,j)), j, acc(t,i)), l, r) TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  74. Loop invariant . . . init: while !i < !r do { invariant 0 <= b <= i and i <= r <= n and monochrome(t,0,b,blue) and monochrome(t,b,i,white) and monochrome(t,r,n,red) and length(t) = n and (forall k:int. 0 <= k < n -> is color(acc(t,k))) and permutation(t,t@init,0,n-1) variant r - i } . . . done TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

  75. Proof obligations 11 proof obligations loop invariant holds initially loop invariant is preserved and variant decreases (3 cases) swap precondition (twice) array access within bounds (twice) postcondition holds at the end of function execution All automatically discharged by Simplify! TYPES Summer School – August 30th, 2007 Jean-Christophe Filliˆ atre Why/Krakatoa/Caduceus

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