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Running example Concrete semantics Abstract semantics Evaluation References Analysis of Program Differences with Numerical Abstract Interpretation Airbus LIP6 day David Delmas 1, 2 e 2 Antoine Min 1 Airbus EYYW 2 LIP6 APR 20 february

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Running example Concrete semantics Abstract semantics Evaluation References

Analysis of Program Differences with Numerical Abstract Interpretation

Airbus – LIP6 day David Delmas 1, 2 Antoine Min´ e 2

1Airbus EYYW 2LIP6 APR

20 february 2019

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Running example Concrete semantics Abstract semantics Evaluation References

Introduction

PhD on“portability analysis” Portability analysis is about comparing the semantics of two syntactically close versions of a program in two different environments

(machine, OS, library, etc.)

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Introduction

PhD on“portability analysis” Portability analysis is about comparing the semantics of two syntactically close versions of a program in two different environments

(machine, OS, library, etc.)

To start with Let us consider two syntactically close versions of a program in the same environment i.e. patch analysis

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Running example Concrete semantics Abstract semantics Evaluation References

Agenda

1

Running example

2

Concrete semantics

3

Abstract semantics

4

Evaluation

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Running example Concrete semantics Abstract semantics Evaluation References

Running example

Unchloop from Trostanetski et al. [2017] Original program 1 { 2 int a; int b; int c; int i ; int r; 3 4 a = input (−1000,1000); 5 b = input (−1000,1000); 6 c = 1; 7 8 i=0; 9 while ( i<a ) { 10 c=c+b; 11 i=i+1; 12 } 13 14 r = c; 15 16 }

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Running example

Unchloop from Trostanetski et al. [2017] Change request 1 { 2 int a; int b; int c; int i ; int r; 3 4 a = input (−1000,1000); 5 b = input (−1000,1000); 6 c = 1; // to change 7 8 i=0; 9 while ( i<a ) { 10 c=c+b; 11 i=i+1; 12 } 13 14 r = c; // to change 15 16 }

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Running example

Unchloop from Trostanetski et al. [2017] Patched program 1 { 2 int a; int b; int c; int i ; int r; 3 4 a = input (−1000,1000); 5 b = input (−1000,1000); 6 c = 0; // 1 7 8 i=0; 9 while ( i<a ) { 10 c=c+b; 11 i=i+1; 12 } 13 14 r = c + 1; // c 15 16 }

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Running example

Unchloop from Trostanetski et al. [2017] Joint representation: double program P 1 { 2 int a; int b; int c; int i ; int r; 3 4 a = input (−1000,1000); 5 b = input (−1000,1000); 6 c = 1 0; 7 8 i=0; 9 while ( i<a ) { 10 c=c+b; 11 i=i+1; 12 } 13 14 r = c c + 1; 15 assert sync(r); 16 }

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Running example

Unchloop from Trostanetski et al. [2017] Left projection: simple program P1 = π1(P) 1 { 2 int a1; int b1; int c1; int i 1; int r1; 3 4 a1 = input (−1000,1000); 5 b1 = input (−1000,1000); 6 c1 = 1; 7 8 i 1=0; 9 while ( i 1<a1 ) { 10 c1=c1+b1; 11 i 1=i1+1; 12 } 13 14 r1 = c1; 15 16 }

π1(x ∈ V)

def

= x1 ∈ V1 π1(s1 s2)

def

= s1

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Running example

Unchloop from Trostanetski et al. [2017] Right projection: simple program P2 = π2(P) 1 { 2 int a2; int b2; int c2; int i 2; int r2; 3 4 a2 = input (−1000,1000); 5 b2 = input (−1000,1000); 6 c2 = 0; 7 8 i 2=0; 9 while ( i 2<a2 ) { 10 c2=c2+b2; 11 i 2=i2+1; 12 } 13 14 r2 = c2 + 1; 15 16 }

π2(x ∈ V)

def

= x2 ∈ V2 π2(s1 s2)

def

= s2

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Running example Concrete semantics Abstract semantics Evaluation References

Agenda

1

Running example

2

Concrete semantics

3

Abstract semantics

4

Evaluation

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Running example Concrete semantics Abstract semantics Evaluation References

Lifting simple program semantics to double programs

Simple programs Pk = πk(P) k ∈ {1; 2} variables in Vk = { xk | x ∈ V } memory states in Ek

def

= Vk → ❘ expression semantics ❊k e ∈ Ek → P(❘) condition semantics ❈k c ∈ Ek → P({true, false}) statement semantics ❙k s ∈ P(Ek × Ek): input-output relation ❉

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Lifting simple program semantics to double programs

Simple programs Pk = πk(P) k ∈ {1; 2} variables in Vk = { xk | x ∈ V } memory states in Ek

def

= Vk → ❘ expression semantics ❊k e ∈ Ek → P(❘) condition semantics ❈k c ∈ Ek → P({true, false}) statement semantics ❙k s ∈ P(Ek × Ek): input-output relation Double program P variables in V memory states in D

def

= E1 × E2 statement semantics ❉ s ∈ P(D × D): input-output relation

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Assignment and blocs

❉ dstat ∈ P(D × D) ❉ skip

def

= ∆D ❉ s1 s2

def

= { ((i1, i2), (o1, o2)) | (i1, o1) ∈ ❙1 s1 ∧ (i2, o2) ∈ ❙2 s2 } ❉ V ← e1 e2

def

= ❉ V ← e1 V ← e2 ❉ s1; s2

def

= ❉ s1 ❉ s2 where R1 R2

def

= { (x, z) | ∃y : (x, y) ∈ R1 ∧ (y, z) ∈ R2 }

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if statement

❉ if c then s1 else s2

def

= ❋ c ❉ s1 ∪ ❋ ¬c ❉ s2 ∪ ❋ c ¬c ❉1 s1 ❉2 s2 ∪ ❋ ¬c c ❉1 s2 ❉2 s1 where ❋ c1 c2

def

= { ((ρ1, ρ2) , (ρ1, ρ2)) | true ∈ ❈1 c1 ρ1 ∩ ❈2 c2 ρ2 } ❋ c

def

= ❋ c c ❉1 s

def

= ❉ π1(s) skip ❉2 s

def

= ❉ skip π2(s)

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while statement

❉ while c do s

def

= (lfp H) ❋ ¬c where H(R)

def

= ∆D ∪ R

  • ❋ c

❉ s ∪ ❋ c ¬c ❉1 s ∪ ❋ ¬c c ❉2 s

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Semantics of the example

1 c = 1 0; 2 3 i=0; 4 while ( i<a ) { 5 c=c+b; 6 i=i+1; 7 } 8 9 r = c c + 1;

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Semantics of the example

1 c = 1 0; 2 3 i=0; 4 while ( i<a ) { 5 c=c+b; 6 i=i+1; 7 } 8 9 r = c c + 1;

❉ = { s0, ((a1, b1, 1, 0, 1), (a2, b2, 0, 0, 1)) | a1 ≤ 0 ∧ a2 ≤ 0 ∧ H0 } ∪ { s0, ((a1, b1, 1 + a1 × b1, a1, 1 + a1 × b1), (a2, b2, 0, 0, 1)) | a1 > 0 ∧ a2 ≤ 0 ∧ H0 } ∪ { s0, ((a1, b1, 1, 0, 1), (a2, b2, a2 × b2, a2, 1 + a2 × b2)) | a1 ≤ 0 ∧ a2 > 0 ∧ H0 } ∪ { s0, ((a1, b1, 1 + a1 × b1, a1, 1 + a1 × b1), (a2, b2, a2 × b2, a2, 1 + a2 × b2)) | a1 > 0 ∧ a2 > 0 ∧ H0 } where s0

def

= ((a1, b1, c1, i1, r1), (a2, b2, c2, i2, r2)) H0

def

= ∀k ∈ {1, 2} : (bk, ck, ik, rk) ∈ ❘4

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Semantics of the example

1 c = 1 0; 2 3 i=0; 4 while ( i<a ) { 5 c=c+b; 6 i=i+1; 7 } 8 9 r = c c + 1;

assuming equal initial states a def = a1 = a2 and b

def

= b1 = b2 ❉ = { s0, ((a, b, 1, 0, 1), (a, b, 0, 0, 1)) | a ≤ 0 ∧ H0 } ∪ ∅ ∪ ∅ ∪ { s0, ((a, b, 1 + a × b, a, 1 + a × b), (a, b, a × b, a, 1 + a × b)) | a > 0 ∧ H0 } where s0

def

= ((a, b, c1, i1, r1), (a, b, c2, i2, r2)) H0

def

= b ∈ ❘ ∧ ∀k ∈ {1, 2} : (ck, ik, rk) ∈ ❘3

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Agenda

1

Running example

2

Concrete semantics

3

Abstract semantics

4

Evaluation

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❉ is not computable in general

Numerical abstraction memory states in D ≈ ❘|V1∪V2| ⇒ any numeric abstract domain on pairs of environments ❉ ❉ ❙ ❙ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉

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❉ is not computable in general

Numerical abstraction memory states in D ≈ ❘|V1∪V2| ⇒ any numeric abstract domain on pairs of environments Abstract semantics ❉ defined by induction the syntax but for ❉ s1 s2

def

= { ... | ...❙1 s1 ∧ ...❙2 s2 } notice ❉ s1 s2 = ❉1 s1 ❉2 s2 e.g. ❉ c ← 1 0 = ❉ c1 ← 1 ❉ c2 ← 0 ❉♯ is standard e.g. ❉♯ s1 s2

def

= ❉♯

1 s1 ♯ ❉♯ 2 s2

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Which numerical abstract domains?

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Running example Concrete semantics Abstract semantics Evaluation References

Which numerical abstract domains?

x y Unable to express non linear invariant Polyhedra r1 = 1 + a × b = r2.

i aixi ≤ b

But able to infer linear invariant −c1 + c2 + 1 = 0, hence r1 = r2.

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Which numerical abstract domains?

x y Unable to express non linear invariant Polyhedra r1 = 1 + a × b = r2.

i aixi ≤ b

But able to infer linear invariant −c1 + c2 + 1 = 0, hence r1 = r2. x y Octagons Able to express necessary linear invariants. ±x ± y ≤ c But unable to infer them using currently proposed transfer functions.

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Which numerical abstract domains?

x y Unable to express non linear invariant Polyhedra r1 = 1 + a × b = r2.

i aixi ≤ b

But able to infer linear invariant −c1 + c2 + 1 = 0, hence r1 = r2. x y Octagons Able to express necessary linear invariants. ±x ± y ≤ c But unable to infer them using currently proposed transfer functions. x y Intervals x, y ∈ [a, b] Unable to express necessary invariants.

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A specialized abstract domain

Remark to express the necessary invariants, it is sufficient to bound the differences x2 − x1 for any variable x. ❉ ❸ ❉ ❸ ❉ ❸ ❉

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A specialized abstract domain

Remark to express the necessary invariants, it is sufficient to bound the differences x2 − x1 for any variable x. State (x1, x2) δx

def

= x2 − x1 (x1, δx) ❉ ❸ ❉ ❸ ❉ ❸ ❉

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A specialized abstract domain

Remark to express the necessary invariants, it is sufficient to bound the differences x2 − x1 for any variable x. State (x1, x2) δx

def

= x2 − x1 (x1, δx) Lattice (P(D × D), ⊆) − − −→ − → ← ← − − − −

α γ

(P(D × D), ⊆) Semantics ❉ ❸

def

= α ◦ ❉ ❸ ❉ ❸ ❉

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A specialized abstract domain

Remark to express the necessary invariants, it is sufficient to bound the differences x2 − x1 for any variable x. State (x1, x2) δx

def

= x2 − x1 (x1, δx) Lattice (P(D × D), ⊆) − − −→ − → ← ← − − − −

α γ

(P(D × D), ⊆) Semantics ❉ ❸

def

= α ◦ ❉ Numerical abstraction ❸♯ Like for ❉, any numeric domain can be used to abstract ❸. Unlike with ❉, intervals are enough to analyse the running example.

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise
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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e e′+e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← π2(e′)+0

if (e = input(a, b) ∧ . . .)

  • r (. . . ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ← π2(e′)+

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e′+e

  • therwise
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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise
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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

δa ∈ [0, 0]

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

δa ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1]

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

δa ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0]

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

δa ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0]

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e e′+e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← π2(e′)+0

if (e = input(a, b) ∧ . . .)

  • r (. . . ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ← π2(e′)+

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e′+e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

δa ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] ∧ δr ∈ [0, 0]

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Symbolic simplifications in ❸♯

Special cases for the assignment transfer function

❸♯ V ← e

def

= ❸♯

1 V ← e ♯

  • ❸♯ δV ← 0

if (e = input(a, b) ∧ input is sync())

  • r (is deterministic(e) ∧ ∀x ∈ Vars(e) : δx = 0)

❸♯ δV ←

x∈V λxδx

if e = µ +

x∈V λxx

❸♯

2 V ← e

  • therwise

1 a = input (−1000,1000); 2 b = input (−1000,1000); 3 c = 1 0; 4 5 i=0; 6 while ( i<a ) { 7 c=c+b; 8 i=i+1; 9 } 10 11 r = c c + 1;

δa ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] δa ∈ [0, 0] ∧ δb ∈ [0, 0] ∧ δc ∈ [−1, −1] ∧ δi ∈ [0, 0] ∧ δr ∈ [0, 0] q.e.d.

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Agenda

1

Running example

2

Concrete semantics

3

Abstract semantics

4

Evaluation

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Evaluation

Prototype abstract interpreter semantics ❉♯ and ❸♯ for toy language 2,000 lines of OCaml source code Apron library for polyhedra and octagons ❉

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Evaluation

Prototype abstract interpreter semantics ❉♯ and ❸♯ for toy language 2,000 lines of OCaml source code Apron library for polyhedra and octagons Limitations collecting semantics ❉ ⇒ compares pairs of terminating executions only assumes the joint representation given

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Evaluation

Prototype abstract interpreter semantics ❉♯ and ❸♯ for toy language 2,000 lines of OCaml source code Apron library for polyhedra and octagons Limitations collecting semantics ❉ ⇒ compares pairs of terminating executions only assumes the joint representation given Benchmarks selected from Trostanetski et al. [2017]a, Partush and Yahav [2013, 2014] focussing on equivalence for now speedups ≥ 1 order of magnitude

aspeedups ≥ 1 order of magnitude wrt. well-established tools Symdiff and RVT

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Benchmarks

Related Benchmark LOC Related ❉♯(polyhedra) ❉♯(octagon) ❸♯(interval)

  • rigin

time (ms) time (ms) time (ms) time (ms) 2017 Comp 13 539 14 ✓ 18 ✗ 2 ✗ Const 9 541 7 ✓ 17 ✓ 1 ✓

  • Fig. 2

14 – 4 ✓ 5 ✓ 1 ✓ LoopMult 14 49,0312 20 ✓ 56 ✗ 1 ✗ LoopSub 15 1,179 19 ✓ 53 ✗ 2 ✗ UnchLoop 13 2,8383 15 ✓ 36 ✗ 2 ✓ 2013 sign 12 – 6 ✓ 8 ✗ 2 ✗ sum 14 4,000 14 ✓ 30 ✓ 6 ✗ copy1 37 7,000 102 ✓ 60 ✓ 2 ✗ remove1 19 1,000 6 ✓ 18 ✓ 2 ✗ 2013-14 seq1 41 11,000 75 ✓ 500 ✗ 2 ✗ test1 158 – 96 ✓ 521 ✓ 4 ✓

1Coreutils 2only 20 loop iterations 3only 5 loop iterations

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Benchmarks

Related Benchmark LOC Related ❉♯(polyhedra) ❉♯(octagon) ❸♯(interval)

  • rigin

time (ms) time (ms) time (ms) time (ms) 2017 Comp 13 539 14 ✓ 18 ✗ 2 ✗ Const 9 541 7 ✓ 17 ✓ 1 ✓

  • Fig. 2

14 – 4 ✓ 5 ✓ 1 ✓ LoopMult 14 49,0312 20 ✓ 56 ✗ 1 ✗ LoopSub 15 1,179 19 ✓ 53 ✗ 2 ✗ UnchLoop 13 2,8383 15 ✓ 36 ✗ 2 ✓ 2013 sign 12 – 6 ✓ 8 ✗ 2 ✓ sum 14 4,000 14 ✓ 30 ✓ 6 ✓ copy1 37 7,000 102 ✓ 60 ✓ 2 ✓ remove1 19 1,000 6 ✓ 18 ✓ 2 ✗ 2013-14 seq1 41 11,000 75 ✓ 500 ✗ 2 ✗ test1 158 – 96 ✓ 521 ✓ 4 ✓

1Coreutils 2only 20 loop iterations 3only 5 loop iterations

✓partitioning in progress

slide-47
SLIDE 47

Running example Concrete semantics Abstract semantics Evaluation References

The seq example: non regression verification

print extra number == false print numbers (long first , long last ) { long i ,x; char ∗ fmt, ∗ separator , ∗ terminator ; for (i = 0; /∗ empty ∗/; i++) { x = first + i ∗ STEP; if ( last < x) break; if (i) printf ( separator ); printf (fmt, x); } if (i) printf (terminator); } print numbers (long first , long last , bool print extra numb { char ∗ fmt, ∗ separator , ∗ terminator ; bool out of range = (last < first ); long x,x0, i ; if (! out of range) { x = first ; for (i = 1; /∗ empty ∗/; i++) { x0 = x; printf (fmt, x); if (out of range) break; x = first + i ∗ STEP;

  • ut of range = (last < x);

if (out of range){ if (! print extra number) break; } printf ( separator ); } printf (terminator); } }

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SLIDE 48

Running example Concrete semantics Abstract semantics Evaluation References

References

  • N. Partush and E. Yahav. Abstract semantic differencing for numerical programs. In
  • F. Logozzo and M. F¨

ahndrich, editors, Static Analysis, pages 238–258, Berlin, Heidelberg,

  • 2013. Springer Berlin Heidelberg. ISBN 978-3-642-38856-9.
  • N. Partush and E. Yahav. Abstract semantic differencing via speculative correlation. In

Proceedings of the 2014 ACM International Conference on Object Oriented Programming Systems Languages & Applications, OOPSLA ’14, pages 811–828, New York, NY, USA,

  • 2014. ACM. ISBN 978-1-4503-2585-1. doi: 10.1145/2660193.2660245. URL

http://doi.acm.org/10.1145/2660193.2660245.

  • A. Trostanetski, O. Grumberg, and D. Kroening. Modular demand-driven analysis of semantic

difference for program versions. In F. Ranzato, editor, Static Analysis - 24th International Symposium, SAS 2017, New York, NY, USA, August 30 - September 1, 2017, Proceedings, volume 10422 of Lecture Notes in Computer Science, pages 405–427. Springer, 2017. ISBN 978-3-319-66705-8. doi: 10.1007/978-3-319-66706-5\ 20. URL https://doi.org/10.1007/978-3-319-66706-5_20.

slide-49
SLIDE 49

Running example Concrete semantics Abstract semantics Evaluation References

Conclusion

Patch analysis by abstract interpretation

1 novel concrete collecting semantics 2 analysis is parametric in abstract domain 3 novel numeric domain 4 experiments on examples from the literature

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SLIDE 50

Running example Concrete semantics Abstract semantics Evaluation References

Conclusion

Patch analysis by abstract interpretation

1 novel concrete collecting semantics 2 analysis is parametric in abstract domain 3 novel numeric domain 4 experiments on examples from the literature

Future work non purely numeric programs

  • ther abstract domains (e.g. zonotopes)

extension to portability analysis

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SLIDE 51

Running example Concrete semantics Abstract semantics Evaluation References

Thank you for your attention. Questions?

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SLIDE 52

Simple programs Double programs

Backup slides

slide-53
SLIDE 53

Simple programs Double programs

Non termination

1 { 2 int x; 3 x = input (−100,100); 4 5 if ( x<0 ) { 6 x=−1; 7 } 8 else { 9 if ( x>=2 x>=4 ) {} 10 else { 11 while (x==2) x=2; 12 x=3; 13 } 14 } 15 16 assert sync (x); // x=2 ignored 17 }

x ≥ 2 x > 4 in Trostanetski et al. [2017]

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SLIDE 54

Simple programs Double programs

Variables switch roles

1 { 2 int x; int y; int a; int b; int c; int i ; 3 x = input (−100,100); 4 y = input (−10,1); 5 6 a = x y; 7 b = y x; 8 9 c = a b; 10 i=0; 11 while ( i<3 ) { 12 c = c−b c−a; 13 i=i+1; 14 } 15 16 assert sync (c); 17 }

a1 = b2 ∧ a2 = b1

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SLIDE 55

Simple programs Double programs

Loop rearrangements

1 first = input (0,100); 2 last = input (0,100); 3 4 step = 2; 5 separator = 128; 6 terminator = 1024; 7 print extra number = 0; 8 9 {/∗ skip ∗/} if ( last < first ) out of range = 1; 10 11 if ( 1==1 out of range == 0 ) { 12 {/∗ skip ∗/} x=first; 13 i = 0 1; 14 while ( break == 0 ) {

slide-56
SLIDE 56

Simple programs Double programs

Loop rearrangements

1 { 2 x = first + i ∗ step; 3 if ( last < x) break = 1; 4 if (break == 0) { 5 if (i!=0) out1 = separator; 6

  • ut2 = x;

7 } 8 } { 9

  • ut2 = x;

10 if (out of range == 1) break = 1; 11 else { 12 x = first + i ∗ step; 13 if ( last < x) out of range = 1; 14 if (out of range == 1) if (print extra number == 0) break = 1; 15 if (break == 0) out1 = separator; 16 } 17 }

slide-57
SLIDE 57

Simple programs Double programs

Loop rearrangements

1 i=i+1; 2 } 3 4 } 5 6 assert sync (out1,out2); 7 if (i!=0) out1 = terminator; out1 = terminator; 8 9 assert sync (out1); 10 }

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SLIDE 58

Simple programs Double programs

Looping at different paces

1 { 2 int i ; 3 int max; 4 5 max = input(0,4359); 6 7 i=0; 8 while ( i<max ) { 9 i = i+1 i+2; 10 } 11 12 {/∗ skip ∗/} if (max>0 && i>max) i=i−1; // when max is odd 13 14 assert sync (i); 15 assert (i==max); 16 }

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SLIDE 59

Simple programs Double programs

Syntax of simple programs

expr ::= V V ∈ V | c c ∈ ❘ | −expr | expr ⋄ expr ⋄ ∈ {+, −, ×, /} | rand(a, b) a, b ∈ ❘ cond ::= expr ⊲ ⊳ expr ⊲ ⊳ ∈ {≤, ≥, =, =, <, >} | ¬cond | cond ⋄ cond ⋄ ∈ {∧, ∨} stat ::= V ← expr | if cond then stat else stat | while cond do stat | stat; stat | skip

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SLIDE 60

Simple programs Double programs

Standard relational concrete semantics of simple programs

Statements

❙ stat ∈ P(E × E) ❙ skip

def

= ∆E ❙ V ← e

def

= { (ρ, ρ[V → v]) | ρ ∈ E, v ∈ ❊ e ρ } ❙ if c then s1 else s2

def

= ❙ c? ❙ s1 ∪ ❙ ¬c? ❙ s2 ❙ while c do s

def

= (lfp λR. ∆E ∪ R ❙ c? ❙ s ) ❙ ¬c? ❙ s1; s2

def

= ❙ s1 ❙ s2 where ❙ c?

def

= { (ρ, ρ) | ρ ∈ E, true ∈ ❈ c ρ } R1 R2

def

= { (x, z) | ∃y : (x, y) ∈ R1 ∧ (y, z) ∈ R2 }

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SLIDE 61

Simple programs Double programs

Standard relational concrete semantics of simple programs

Conditions

❈ cond ∈ E → P({true, false}) ⋄ ∈ {∧, ∨} ❈ ¬c ρ

def

= { ¬v | v ∈ ❈ c ρ } ⊲ ⊳ ∈ {≤, ≥, =, =, <, >} ❈ c1 ⋄ c2 ρ

def

= { v1 ⋄ v2 | v1 ∈ ❈ c1 ρ, v2 ∈ ❈ c2 ρ } ❈ e1 ⊲ ⊳ e2 ρ

def

= { true | ∃v1 ∈ ❊ e1 ρ, v2 ∈ ❊ e2 ρ: v1 ⊲ ⊳ v2 } ∪ { false | ∃v1 ∈ ❊ e1 ρ, v2 ∈ ❊ e2 ρ: v1 ⊲ ⊳ v2 }

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SLIDE 62

Simple programs Double programs

Standard relational concrete semantics of simple programs

Expressions

❊ expr ∈ E → P(❘) ⋄ ∈ {+, −, ×, /} ❊ V ρ

def

= {ρ(V )} ❊ c ρ

def

= {c} ❊ rand(a, b) ρ

def

= { x | a ≤ x ≤ b } ❊ −e ρ

def

= { −v | v ∈ ❊ e ρ } ❊ e1 ⋄ e2 ρ

def

= { v1 ⋄ v2 | v1 ∈ ❊ e1 ρ, v2 ∈ ❊ e2 ρ, ⋄ = / ∨ v2 = 0 }

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SLIDE 63

Simple programs Double programs

Syntax of double programs

dexpr ::= expr | expr expr dcond ::= cond | cond cond dstat ::= stat | stat stat | V ← dexpr | if dcond then dstat else dstat | while dcond do dstat | dstat; dstat

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SLIDE 64

Simple programs Double programs

Concrete semantics

Denotational concrete semantics of double programs

❉ dstat ∈ P(D × D) ❉ skip

def

= ∆D ❉ s1 s2

def

= { ((i1, i2) , (o1, o2)) | ∀k ∈ {1, 2} : (ik, ok) ∈ ❙k sk } ❉ V ← e1 e2

def

= ❉ V ← e1 V ← e2 ❉ s1; s2

def

= ❉ s1 ❉ s2 ❉ if c then s1 else s2 def = ❋ c ❉ s1 ∪ ❋ ¬c ❉ s2 ∪ ❋ c ¬c ❉1 s1 ❉2 s2 ∪ ❋ ¬c c ❉1 s2 ❉2 s1 ❉ while c do s

def

= (lfp H) ❋ ¬c ❋ c1 c2 def = { ((ρ1, ρ2) , (ρ1, ρ2)) | ∀ k ∈ {1, 2} : true ∈ ❈k ck ρk } ❋ c

def

= ❋ c c ❉1 s

def

= ❉ π1(s) skip ❉2 s

def

= ❉ skip π2(s) H(R)

def

= ∆D ∪ R

  • (

❋ c c ❉ s ∪ ❋ c ¬c ❉1 s ∪ ❋ ¬c c ❉2 s )