Download presentation
Presentation is loading. Please wait.
Published byPatrick Edwards Modified over 7 years ago
1
Program Analysis and Verification Spring 2015 Program Analysis and Verification Lecture 6: Axiomatic Semantics III Roman Manevich Ben-Gurion University
2
Tentative syllabus Semantics Natural Semantics Structural semantics Axiomatic Verification Static Analysis Automating Hoare Logic Control Flow Graphs Equation Systems Collecting Semantics Abstract Interpretation fundamentals LatticesFixed-Points Chaotic Iteration Galois Connections Domain constructors Widening/ Narrowing Analysis Techniques Numerical Domains Alias analysis Interprocedural Analysis Shape Analysis CEGAR Crafting your own Soot From proofs to abstractions Systematically developing transformers 2
3
Previously Hoare logic – Inference system – Annotated programs – Soundness and completeness Weakest precondition calculus Strongest postcondition calculus 3
4
Weakest (liberal) precondition rules 1.wlp( skip, Q) = Q 2.wlp( x := a, Q) = Q[a/ x ] 3.wlp(S 1 ; S 2, Q) = wlp(S 1, wlp(S 2, Q)) 4.wlp( if b then S 1 else S 2, Q) = (b wlp(S 1, Q)) ( b wlp(S 2, Q)) 5.wlp( while b do { } S, Q) = where {b } S { } and b Q 4 Use consequence rule here
5
Strongest postcondition rules 1.sp( skip, P) = P 2.sp( x := a, P) = v. x=a[v/x] P[v/x] 3.sp(S 1 ; S 2, P) = sp(S 2, sp(S 1, P)) 4.sp( if b then S 1 else S 2, P) = sp(S 1, b P) sp(S 2, b P) 5.sp( while b do { } S, P) = b where {b } S { } and P b 5 Use consequence rule here
6
Warm-up 6
7
Proof of swap by WP 7 { y=b x=a } t := x { y=b t=a } x := y { x=b t=a } y := t { x=b y=a }
8
Prove swap via SP 8 { y=b x=a } t := x { } x := y {} y := t { x=b y=a }
9
Prove swap via SP 9 { y=b x=a } t := x { v. t=x y=b x=a } x := y { v. x=y t=v y=b v=a } { x=y y=b v. t=v v=a } { x=y y=b t=a } y := t { v. y=t x=v v=b t=a } { y=t t=a v. x=v v=b } { y=t t=a x=b } { x=b y=a } Quantifier elimination (a very trivial one) Quantifier elimination
10
Proof of absolute value via WP 10 { x=v } { (-x=|v| x<0) (x=|v| x 0) } if x<0 then { -x=|v| } x := -x { x=|v| } else { x=|v| } skip { x=|v| } { x=|v| }
11
Prove absolute value by SP 11 { x=v } {} if x<0 then {} x := -x {} else { } skip {} { } { x=|v| }
12
Proof of absolute value via WP 12 { x=v } if x<0 then { x=v x<0 } x := -x { w. x=-w w=v w<0 } { x=-v v<0 } else { x=v x 0 } skip { x=v x 0 } { x=-v v<0 x=v x 0 } { x=|v| }
13
Agenda Some useful rules Extension for memory Proving termination 13
14
Making the proof system more practical 14
15
Conjunction rule 15 Allows breaking up proofs into smaller, easier to manage, sub-proofs
16
More useful rules 16 { P } C { Q } { P’ } C { Q’ } { P P’ } C {Q Q’ } [disj p ] { P } C { Q } { v. P } C { v. Q } [exist p ] v FV( C) { P } C { Q } { v. P } C { v. Q } [univ p ] v FV(C ) { F } C { F } Mod(C) FV(F)={} [Inv p ] Mod(C) = set of variables assigned to in sub-statements of C FV(F) = free variables of F Breaks if C is non- deterministic
17
Invariance + Conjunction = Constancy 17 Mod(C) = set of variables assigned to in sub-statements of C FV(F) = free variables of F { P } C { Q } { F P } C { F Q } [constancy p ] Mod(C) FV(F)={}
18
Strongest postcondition calculus practice 18 By Vadim Plessky (http://svgicons.sourceforge.net/) [see page for license], via Wikimedia Commons
19
Floyd’s strongest postcondition rule Example { z=x } x:=x+1 { ?} 19 { P } x := a { v. x=a[v/x] P[v/x] } where v is a fresh variable [ass Floyd ] The value of x in the pre-state
20
Floyd’s strongest postcondition rule Example { z=x } x:=x+1 { v. x=v+1 z=v } This rule is often considered problematic because it introduces a quantifier – needs to be eliminated further on We will now see a variant of this rule 20 { P } x := a { v. x=a[v/x] P[v/x] } where v is a fresh variable [ass Floyd ] meaning: {x=z+1}
21
“Small” assignment axiom Examples: {x=n} x:=5*y {x=5*y} {x=n} x:=x+1 {x=n+1} {x=n} x:=y+1 {x=y+1} [exist p ] { n. x=n} x:=y+1 { n. x=y+1} therefore {true} x:=y+1 {x=y+1} [constancy p ] {z=9} x:=y+1 {z=9 x=y+1} 21 { x=v } x:=a { x=a[v/x] } where v FV(a) [ass floyd ] First evaluate a in the precondition state (as a may access x) Then assign the resulting value to x Create an explicit Skolem variable in precondition
22
“Small” assignment axiom Examples: {x=n} x:=5*y {x=5*y} {x=n} x:=x+1 {x=n+1} {x=n} x:=y+1 {x=y+1} [exist p ] { n. x=n} x:=y+1 { n. x=y+1} therefore {true} x:=y+1 {x=y+1} [constancy p ] {z=9} x:=y+1 {z=9 x=y+1} 22 { x=v } x:=a { x=a[v/x] } where v FV(a) [ass floyd ]
23
“Small” assignment axiom Examples: {x=n} x:=5*y {x=5*y} {x=n} x:=x+1 {x=n+1} {x=n} x:=y+1 {x=y+1} [exist p ] { n. x=n} x:=y+1 { n. x=y+1} therefore {true} x:=y+1 {x=y+1} [constancy p ] {z=9} x:=y+1 {z=9 x=y+1} 23 { x=v } x:=a { x=a[v/x] } where v FV(a) [ass floyd ]
24
“Small” assignment axiom Examples: {x=n} x:=5*y {x=5*y} {x=n} x:=x+1 {x=n+1} {x=n} x:=y+1 {x=y+1} [exist p ] { n. x=n} x:=y+1 { n. x=y+1} therefore {true} x:=y+1 {x=y+1} [constancy p ] {z=9} x:=y+1 {z=9 x=y+1} 24 { x=v } x:=a { x=a[v/x] } where v FV(a) [ass floyd ]
25
Prove using strongest postcondition 25 { x=a y=b } t := x x := y y := t { x=b y=a }
26
Prove using strongest postcondition 26 { x=a y=b } t := x { x=a y=b t=a } x := y y := t { x=b y=a }
27
Prove using strongest postcondition 27 { x=a y=b } t := x { x=a y=b t=a } x := y { x=b y=b t=a } y := t { x=b y=a }
28
Prove using strongest postcondition 28 { x=a y=b } t := x { x=a y=b t=a } x := y { x=b y=b t=a } y := t { x=b y=a t=a } { x=b y=a } // cons
29
Prove using strongest postcondition 29 { x=v } if x 0 } else { x=v x 0 } skip { x=v x 0 } { v<0 x=-v v 0 x=v } { x=|v| }
30
Prove using strongest postcondition 30 { x=v } if x 0 } else { x=v x 0 } skip { x=v x 0 } { v<0 x=-v v 0 x=v } { x=|v| }
31
Sum program – specify Define Sum(0, n) = 0+1+…+n 31 {?} x := 0 res := 0 while (x<y) do res := res+x x := x+1 {?} { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) } Background axiom
32
Sum program – specify Define Sum(0, n) = 0+1+…+n 32 { y 0 } x := 0 res := 0 while (x<y) do res := res+x x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) } Background axiom
33
Sum program – prove Define Sum(0, n) = 0+1+…+n 33 { y 0 } x := 0 res := 0 Inv = while (x<y) do res := res+x x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
34
Sum program – prove Define Sum(0, n) = 0+1+…+n 34 { y 0 } x := 0 { y 0 x=0 } res := 0 Inv = while (x<y) do res := res+x x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
35
Sum program – prove Define Sum(0, n) = 0+1+…+n 35 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = while (x<y) do res := res+x x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
36
Sum program – prove Define Sum(0, n) = 0+1+…+n 36 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) x y } while (x<y) do res := res+x x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
37
Sum program – prove Define Sum(0, n) = 0+1+…+n 37 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) x y } while (x<y) do { y 0 res=m x=n n y m=Sum(0, n) x<y } { y 0 res=m x=n m=Sum(0, n) n<y } res := res+x x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
38
Sum program – prove Define Sum(0, n) = 0+1+…+n 38 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) x y } while (x<y) do { y 0 res=m x=n n y m=Sum(0, n) x<y } { y 0 res=m x=n m=Sum(0, n) n<y } res := res+x { y 0 res=m+x x=n m=Sum(0, n) n<y } x := x+1 { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
39
Sum program – prove Define Sum(0, n) = 0+1+…+n 39 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) x y } while (x<y) do { y 0 res=m x=n n y m=Sum(0, n) x<y } { y 0 res=m x=n m=Sum(0, n) n<y } res := res+x { y 0 res=m+x x=n m=Sum(0, n) n<y } x := x+1 { y 0 res=m+x x=n+1 m=Sum(0, n) n<y } { y 0 res=Sum(0, x) x=n+1 n<y } // sum axiom { y 0 res=Sum(0, x) x y } // cons { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
40
Sum program – prove Define Sum(0, n) = 0+1+…+n 40 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) x y } while (x<y) do { y 0 res=m x=n n y m=Sum(0, n) x<y } { y 0 res=m x=n m=Sum(0, n) n<y } x := x+1 { y 0 res=m x=n+1 m=Sum(0, n) n<y } res := res+x { y 0 res=m+x x=n+1 m=Sum(0, n) n<y } { y 0 res=Sum(0, x) x=n+1 n<y } // sum axiom { y 0 res=Sum(0, x) x y } // cons { y 0 res=Sum(0, x) x y x y } { y 0 res=Sum(0, y) x=y } { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
41
Buggy sum program 1 Define Sum(0, n) = 0+1+…+n 41 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) x y } while (x<y) do { y 0 res=m x=n n y m=Sum(0, n) x<y } { y 0 res=m x=n m=Sum(0, n) n<y } res := res+x { y 0 res=m+x x=n m=Sum(0, n) n<y } x := x+1 { y 0 res=m+n x=n+1 m=Sum(0, n) n<y } { y 0 res=Sum(0, n)+n x=n+1 n<y } { y 0 res=Sum(0, x) x y } // cons { y 0 res=Sum(0, x) x y x y } { y 0 res=Sum(0, y) x=y } { res = Sum(0, y) } { x=Sum(0, n) } { y=n+1 } { x+y=Sum(0, n+1) }
42
Buggy sum program 2 42 { y 0 } x := 0 { y 0 x=0 } res := 0 { y 0 x=0 res=0 } Inv = { y 0 res=Sum(0, x) } = { y 0 res=m x=n m=Sum(0, n) } while (x y) do { y 0 res=m x=n m=Sum(0, n) x y n y } x := x+1 { y 0 res=m x=n+1 m=Sum(0, n) n y} res := res+x { y 0 res=m+x x=n+1 m=Sum(0, n) n y} { y 0 res-x=Sum(0, x-1) n y} { y 0 res=Sum(0, x) } { y 0 res=Sum(0, x) x>y } {res = Sum(0, y) }
43
Handling data structures 43
44
Problems with Hoare logic and heaps { P[a/ x ] } x := a { P } Consider the annotated program { y=5 } y := 5; { y=5 } x := &y; { y=5 } *x := 7; { y=5 } Is it correct? 44 The rule works on a syntactic level unaware of possible aliasing between different terms (y and *x in our case)
45
Problems with Hoare logic and heaps { P[a/ x ] } x := a { P } {(x=&y z=5) (x &y y=5)} *x = z; { y=5 } What should the precondition be? 45
46
Problems with Hoare logic and heaps { P[a/ x ] } x := a { P } {(x=&y z=5) (x &y y=5)} *x = z; { y=5 } We split into cases depending on possible aliasing 46
47
Problems with Hoare logic and heaps { P[a/ x ] } x := a { P } {(x=&y z=5) (x &y y=5)} *x = z; { y=5 } What should the precondition be? Joseph M. Morris: A General Axiom of AssignmentA General Axiom of Assignment A different approach: heaps as arrays Really successful approach for heaps is based on Separation Logic Separation Logic 47
48
Axiomatizing data types 48 We added a new type of variables – array variables – Model array variable as a function y : Z Z Re-define program states State = Define operational semantics x := y [ a ], y [ a ] := x, S ::= x := a | x := y [ a ] | y [ a ] := x | skip | S 1 ; S 2 | if b then S 1 else S 2 | while b do S
49
Axiomatizing data types 49 We added a new type of variables – array variables – Model array variable as a function y : Z Z We need the two following axioms: S ::= x := a | x := y [ a ] | y [ a ] := x | skip | S 1 ; S 2 | if b then S 1 else S 2 | while b do S { y[x a](x) = a } { z x y[x a](z) = y(z) }
50
Array update rules (wlp) Treat an array assignment y[a] := x as an update to the array function y – y := y[a x] meaning y’= v. v=a ? x : y(v) 50 S ::= x := a | x := y [ a ] | y [ a ] := x | skip | S 1 ; S 2 | if b then S 1 else S 2 | while b do S [array-update] { P[y[a x]/y] } y[a] := x { P } [array-load] { P[y(a)/x] } x := y[a] { P } A very general approach – allows handling many data types
51
Array update rules (wlp) example Treat an array assignment y[a] := x as an update to the array function y – y := y[a x] meaning y’= v. v=a ? x : y(v) 51 [array-update] { P[y[a x]/y] } y[a] := x { P } {x=y[i 7](i)} y[i]:=7 {x=y(i)} {x=7} y[i]:=7 {x=y(i)} [array-load] { P[y(a)/x] } x := y[a] { P } {y(a)=7} x:=y[a] {x=7}
52
Array update rules (sp) 52 [array-update F ] { x=v y=g a=b } y[a] := x { y=g[b v] } [array-load F ] { y=g a=b } x := y[a] { x=g(b) } In both rules v, g, and b are fresh
53
practice proving programs with arrays 53
54
Array-max program – specify 54 nums : array N : int // N stands for num’s length { N 0 nums=orig_nums } x := 0 res := nums[0] while x res then res := nums[x] x := x + 1 1.{ x=N } 2.{ m. (m 0 m<N) nums(m) res } 3.{ m. m 0 m<N nums(m)=res } 4.{ nums=orig_nums }
55
Array-max program 55 nums : array N : int // N stands for num’s length { N 0 nums=orig_nums } x := 0 res := nums[0] while x res then res := nums[x] x := x + 1 Post 1 : { x=N } Post 2 : { nums=orig_nums } Post 3 : { m. 0 m<N nums(m) res } Post 4 : { m. 0 m<N nums(m)=res }
56
Proof strategy Prove each goal 1, 2, 3, 4 separately and use conjunction rule to prove them all After proving – {N 0} C {x=N} – {nums=orig_nums} C {nums=orig_nums} We have proved – {N 0 nums=orig_nums} C {x=N nums=orig_nums} We can refer to assertions from earlier proofs in writing new proofs 56
57
Array-max example: Post 1 57 nums : array N : int // N stands for num’s length { N 0 } x := 0 { N 0 x=0 } res := nums[0] { x=0 } Inv = { x N } while x res then { x=k k<N } res := nums[x] { x=k k<N } { x=k k<N } x := x + 1 { x=k+1 k<N } { x N x N } { x=N }
58
Array-max example: Post 2 58 nums : array N : int // N stands for num’s length { nums=orig_nums } x := 0 { nums=orig_nums } res := nums[0] { nums=orig_nums } Inv = { nums=orig_nums } while x res then { nums=orig_nums } res := nums[x] { nums=orig_nums } { nums=orig_nums } x := x + 1 { nums=orig_nums } { nums=orig_nums x N } { nums=orig_nums }
59
Array-max example: Post 3 59 nums : array { N 0 0 m res then { nums(x)>oRes res=oRes x=k 0 m oRes x=k 0 m<k nums(m) oRes } { x=k 0 m k nums(m) res } { (x=k 0 m k nums(m) res) (oRes nums(x) res=oRes x=k res=oRes 0 m<k nums(m) oRes)} { x=k 0 m k nums(m) res } x := x + 1 { x=k+1 0 m k nums(m) res } { 0 m<x nums(m) res } { x=N 0 m<x nums(m) res} [univ p ] { m. 0 m<N nums(m) res }
60
Proving termination 60 By Noble0 (Own work) [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0)], via Wikimedia Commons
61
Total correctness semantics for While 61 [ P[a/ x ] ] x := a [ P ] [ass p ] [ P ] skip [ P ] [skip p ] [ P ] S 1 [ Q ],[ Q ] S 2 [ R ] [ P ] S 1 ; S 2 [ R ] [comp p ] [ b P ] S 1 [ Q ], [ b P ] S 2 [ Q ] [ P ] if b then S 1 else S 2 [ Q ] [if p ] [ P’ ] S [ Q’ ] [ P ] S [ Q ] [cons p ] if P P’ and Q’ Q [while p ] [ P(z+1) ] S [ P(z) ] [ z. P(z) ] while b do S [ P(0) ] P(z+1) b P(0) b
62
Total correctness semantics for While 62 [ P[a/ x ] ] x := a [ P ] [ass p ] [ P ] skip [ P ] [skip p ] [ P ] S 1 [ Q ],[ Q ] S 2 [ R ] [ P ] S 1 ; S 2 [ R ] [comp p ] [ b P ] S 1 [ Q ], [ b P ] S 2 [ Q ] [ P ] if b then S 1 else S 2 [ Q ] [if p ] [ P’ ] S [ Q’ ] [ P ] S [ Q ] [cons p ] if P P’ and Q’ Q [while p ] [ b P t=k ] S [ P t<k ] [ P ] while b do S [ b P ] P t0P t0 Rank, or Loop variant
63
Proving termination There is a more general rule based on well- founded relations – Partial orders with no infinite strictly decreasing chains Exercise: write a rule that proves only that a program S, started with precondition P terminates [] S [] 63
64
Proving termination There is a more general rule based on well- founded relations – Partial orders with no infinite strictly decreasing chains Exercise: write a rule that proves only that a program S, started with precondition P terminates [ P ] S [ true ] 64
65
Array-max – specify termination 65 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [?] while x res then res := nums[x] x := x + 1 [?]
66
Array-max – specify termination 66 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [ N-x ] while x < N [ ? ] if nums[x] > res then res := nums[x] x := x + 1 [ ? ] [ true ]
67
Array-max – prove loop variant 67 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [ t=N-x ] while x < N [ x<N N-x=k N-x 0 ] if nums[x] > res then res := nums[x] x := x + 1 // [ N-x<k N-x 0 ] [ true ]
68
Array-max – prove loop variant 68 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [ t=N-x ] while x < N [ x=x0 x0<N N-x0=k N-x0 0 ] if nums[x] > res then res := nums[x] x := x + 1 // [ N-x<k N-x 0 ] [ true ] Capture initial value of x, since it changes in the loop
69
Array-max – prove loop variant 69 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [ t=N-x ] while x < N [ x=x0 x0<N N-x0=k N-x0 0 ] if nums[x] > res then res := nums[x] [ x=x0 x0<N N-x0=k N-x0 0 ] // Frame x := x + 1 // [ N-x<k N-x 0 ] [ true ]
70
Array-max – prove loop variant 70 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [ t=N-x ] while x < N [ x=x0 x0<N N-x0=k N-x0 0 ] if nums[x] > res then res := nums[x] [ x=x0 x0<N N-x0=k N-x0 0 ] // Frame x := x + 1 [ x=x0+1 x0<N N-x0=k N-x0 0 ] // [ N-x<k N-x 0 ] [ true ]
71
Array-max – prove loop variant 71 nums : array N : int // N stands for num’s length x := 0 res := nums[0] Variant = [ t=N-x ] while x < N [ x=x0 x0<N N-x0=k N-x0 0 ] if nums[x] > res then res := nums[x] [ x=x0 x0<N N-x0=k N-x0 0 ] // Frame x := x + 1 [ x=x0+1 x0<N N-x0=k N-x0 0 ] [ N-x<k N-x 0 ] // cons [ true ]
72
Zune calendar bugbug while (days > 365) { if (IsLeapYear(year)) { if (days > 366) { days -= 366; year += 1; } } else { days -= 365; year += 1; } 72
73
Fixed code while (days > 365) { if (IsLeapYear(year)) { if (days > 366) { days -= 366; year += 1; } else { break; } } else { days -= 365; year += 1; } 73
74
Fixed code – specify termination [ ? ] while (days > 365) { if (IsLeapYear(year)) { if (days > 366) { days -= 366; year += 1; } else { break; } } else { days -= 365; year += 1; } [ ? ] 74
75
Fixed code – specify variant [ true ] Variant = [ ? ] while (days > 365) { if (IsLeapYear(year)) { if (days > 366) { days -= 366; year += 1; } else { break; } } else { days -= 365; year += 1; } [?] } [ true ] 75
76
Fixed code – proving termination [ true ] Variant = [ t=days ] while (days > 365) { [ days>365 days=k days 0 ] if (IsLeapYear(year)) { if (days > 366) { days -= 366; year += 1; } else { break; [ false ] } } else { days -= 365; year += 1; } // [ days 0 days<k ] } [ true ] 76 Let’s model break by a small cheat – assume execution never gets past it
77
Fixed code – proving termination [ true ] Variant = [ t=days ] while (days > 365) { [ days 0 k=days days>365 ] -> [ days 0 k=days days>365 ] if (IsLeapYear(year)) { [ k=days days>365 ] if (days > 366) { [ k=days days>365 days>366 ] -> [ k=days days>366 ] days -= 366; [ days=k-366 days>0 ] year += 1; [ days=k-366 days>0 ] } else { [ k=days days>365 days 366 ] break; [ false ] } [ (days=k-366 days>0) false ] -> [ days 0 ] } else { [ k=days days>365 ] days -= 365; [ k-365=days days-365>365 ] -> [ k-365=days days 0 ] -> [ days<k days 0 ] year += 1; [ days<k days 0 ] } [ days<k days 0 ] } [ true ] 77
78
Challenge: proving non-termination Write a rule for proving that a program does not terminate when started with a precondition P Prove that the buggy Zune calendar program does not terminate 78 { b P } S { b } { P } while b do S { false } [while-nt p ]
79
conclusion 79
80
Extensions to axiomatic semantics Assertions for execution time – Exact time – Order of magnitude time Assertions for dynamic memory – Separation Logic Separation Logic Assertions for parallelism – Owicki-Gries Owicki-Gries – Concurrent Separation Logic – Rely-guarantee 80
81
Axiomatic verification conclusion Very powerful technique for reasoning about programs – Sound and complete Extendible Static analysis can be used to automatically synthesize assertions and loop invariants 81
82
Next lecture: static analysis
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.