Two Approaches to Inte terprocedural l Data ta Flow w Analys - - PowerPoint PPT Presentation

Two Approaches to Inte terprocedural l Data ta Flow w Analys lysis is

Micha Sharir ir Amir Pnueli li

12.06.2010 Nikolai Knopp

Part Two: The Call String g Approach ch

1

ℎ(𝑜1,𝑜2) 𝑠

𝑛𝑏𝑗𝑜

𝑜1 𝑔

(𝑠𝑛𝑏𝑗𝑜,𝑜1)

𝑔

(𝑜2,𝑓𝑛𝑏𝑗𝑜)

Recap cap

12.06.2010 Nikolai Knopp 2

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Functional approach

𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t if a=0 return

ℎ(𝑜1,𝑜2) 𝑠

𝑛𝑏𝑗𝑜

𝑜1 𝑔

(𝑠𝑛𝑏𝑗𝑜,𝑜1)

𝑔

(𝑜2,𝑓𝑛𝑏𝑗𝑜)

Recap cap

12.06.2010 Nikolai Knopp 3

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Functional approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t if a=0 return

ℎ(𝑜1,𝑜2) 𝑠

𝑛𝑏𝑗𝑜

𝑜1 𝑔

(𝑠𝑛𝑏𝑗𝑜,𝑜1)

𝑔

(𝑜2,𝑓𝑛𝑏𝑗𝑜)

Recap cap

12.06.2010 Nikolai Knopp 4

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Functional approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝜚(𝑠𝑞,𝑓𝑞)

if a=0 return

ℎ(𝑜1,𝑜2) 𝑠

𝑛𝑏𝑗𝑜

𝑜1 𝑔

(𝑠𝑛𝑏𝑗𝑜,𝑜1)

𝑔

(𝑜2,𝑓𝑛𝑏𝑗𝑜)

Recap cap

12.06.2010 Nikolai Knopp 5

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Functional approach

a*b avail.? 𝑜2

𝜚(𝑠𝑛𝑏𝑗𝑜,𝑜2)

𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

⊤ = 1 𝜚(𝑠𝑞,𝑓𝑞)

if a=0 return

Motivat ation for a different approa

• ach

ch

 Avoid expensive functional compositions  Possibility to trade off precision vs.

performance to reduce complexity

12.06.2010 Nikolai Knopp 6

Struct cture

1.

• 1. Defini

nition n of a new DF problem (𝑴∗,𝑮∗)

• 2. Proof: Solution to 𝑀∗,𝐺∗ ≡ MOP solution
• 3. Feasibility and precise variants
• 4. Approximative solution

12.06.2010 Nikolai Knopp 7

Recap cap: Interproced

• cedural

al Grap aphs

Two representations:

1.

𝐻 = 𝑂𝑞

𝑞

, 𝐹𝑞

𝑞

2.

𝐻∗ = 𝑂𝑞

𝑞

, 𝐹∗

12.06.2010 Klaas Boesche 8

p

… … call p

s

𝐹𝑡 𝐹𝑡 𝐹𝑡

1

return …

𝑠

𝑞

𝑓𝑞 𝐹1 𝐹1 𝐹𝑞 𝐹𝑞

𝐹𝑞 = 𝐹𝑞

0 ∪ 𝐹𝑞 1

𝐹∗ = 𝐹0 ∪ 𝐹1 𝐹0 = 𝐹𝑞

𝑞

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 9

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 10

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 11

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

( ,⊤)

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 12

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

( ,1)

𝑑2

if a=0

( 𝑑1 ,1)

return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 13

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 14

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

( 𝑑1 ,1)

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 15

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

( 𝑑1 ,⊤)

𝑑2

if a=0

( 𝑑1𝑑2 , ⊤)

return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 16

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝑑2

if a=0 return

( 𝑑1𝑑2 , ⊤)

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 17

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 18

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

( 𝑑1 ,⊤)

𝑑2

if a=0 return

( 𝑑1 ,1)

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 19

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

𝑑2

if a=0 return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

Basi sic c idea

12.06.2010 Nikolai Knopp 20

call p t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜

Call String (CS) approach

a*b avail.? 𝑜2 𝑠

𝑞

𝑓𝑞

read a, b t := a * b t := a * b print t

( , 1)

𝑑2

Definition:

• n: Call strings

 Let 𝑟 ∈ 𝐽𝑊𝑄 𝑠𝑛𝑏𝑗𝑜,𝑜

decomposed as:



𝑑1𝑑2 … 𝑑𝑘 =: 𝛿 ∈ Γ call string (CS) to 𝑟 in 𝐻∗.

𝜇 ∈ Γ is empty call string (𝑘 = 0 in 𝑛𝑏𝑗𝑜).

 Γ = space of valid call strings in 𝐻∗  𝐷𝑁: 𝐽𝑊𝑄 → Γ with 𝐷𝑁 𝑟 = 𝛿

12.06.2010 Nikolai Knopp 21

𝑛𝑏𝑗𝑜

𝑜

call call

𝑑1 𝑑𝑘

…

𝑟1 𝑟𝑘 𝑟𝑘+1

𝑞𝑘 𝑞𝑘+1

New DF framew mework

• rk



𝑀∗,𝐺∗ ≔ “Extended” version of (L,F)

• Uses interprocedura

ural flow graph 𝐻∗

• 𝑀∗: Values ∈ 𝑀 tagged with call strings ∈ Γ
• 𝐺∗: For data set 𝜊 ∈ L∗ apply edge effect to call

string and value

12.06.2010 Nikolai Knopp

Value dom

• mai

ain 𝑀∗

 𝑀∗ ≔ Γ → 𝑀 semi-lattice

• ⊤∗ = 𝛿 ↦ ⊤
• ⊥∗ = 𝛿 ↦ ⊥
• ⋀ pointwise in 𝑀:

𝜊1 ∧ 𝜊2 𝛿 = 𝜊1 𝛿 ∧ 𝜊2 𝛿

 “𝜊 ∈ 𝑀∗ maps call strings 𝛿

to values in 𝑀 propagated via all paths 𝑟 ∈ 𝐷𝑁−1 𝛿 ”

12.06.2010 Nikolai Knopp 23

if a=0 call p a := a - 1

T F

𝑠

𝑞

𝑑2

𝜊

Value dom

• mai

ain 𝑀∗

 𝑀∗ ≔ Γ → 𝑀 semi-lattice

• ⊤∗ = 𝛿 ↦ ⊤
• ⊥∗ = 𝛿 ↦ ⊥
• ⋀ pointwise in 𝑀:

𝜊1 ∧ 𝜊2 𝛿 = 𝜊1 𝛿 ∧ 𝜊2 𝛿

 “𝜊 ∈ 𝑀∗ maps call strings 𝛿

to values in 𝑀 propagated via all paths 𝑟 ∈ 𝐷𝑁−1 𝛿 ”

12.06.2010 Nikolai Knopp 24

𝜊 = 𝑑1 , 1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 𝜊 𝑑1 = 1 𝜊 𝑑2 = ⊥

if a=0 call p a := a - 1

T F

𝑠

𝑞

𝑑2

𝜊

Edge effect cts s doma main 𝐺∗

 𝐺∗ ≔ 𝑀∗ → 𝑀∗ embeds edge effects from 𝐺

into 𝑀∗ domain

 For 𝑛, 𝑜 ∈ 𝐹∗ and data 𝜊 = . . , 𝛿′, 𝑤′ , . .

∈ 𝑀∗ at 𝑛, data at 𝑜 is 𝑔 𝑛,𝑜

∗

𝜊 = . . , 𝛿, 𝑤 ,. .

• update each 𝛿′ to reflect 𝑛, 𝑜 taken:

𝛿 = 𝛿′ ∘ 𝑛,𝑜

• propagate effect 𝑔 𝑛,𝑜 ∈ 𝑀 applied to each 𝑤′:

𝑤 = 𝑔 𝑛,𝑜 𝑤′

CS updat ate operat ation

• n ∘

12.06.2010 Nikolai Knopp 26

𝑞 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞

𝑑1

call 𝑞1 return

𝑜1 𝑓𝑞 𝑜2

 ∘∶ Γ × 𝐹∗ → Γ updates call strings along edges

CS updat ate operat ation

• n ∘

12.06.2010 Nikolai Knopp 27

𝑞 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞

𝑑1 ∘ 𝑠

𝑞, 𝑑2 = 〈𝑑1〉

𝑑1

call 𝑞1 return

𝑜1 𝑓𝑞 𝑜2 𝑑1 ∘ 𝑜2,𝑓𝑞 = 〈𝑑1〉

 ∘∶ Γ × 𝐹∗ → Γ updates call strings along edges

• 1. Interproc.

CS updat ate operat ation

• n ∘

12.06.2010 Nikolai Knopp 28

𝑞 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞

𝑑1 ∘ 𝑑2, 𝑠

𝑞 = 〈𝑑1, 𝑑2〉

𝑑1 ∘ 𝑠

𝑞, 𝑑2 = 〈𝑑1〉

𝑑1

call 𝑞1

𝜇 ∘ 𝑑1, 𝑠

𝑞 = 〈𝑑1〉

return

𝑜1 𝑓𝑞 𝑜2 𝑑1 ∘ 𝑜2,𝑓𝑞 = 〈𝑑1〉

 ∘∶ Γ × 𝐹∗ → Γ updates call strings along edges

• 1. Interproc.
• 2. Call

CS updat ate operat ation

• n ∘

12.06.2010 Nikolai Knopp 29

𝑞 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞

𝑑1 ∘ 𝑑2, 𝑠

𝑞 = 〈𝑑1, 𝑑2〉

𝑑1 ∘ 𝑠

𝑞, 𝑑2 = 〈𝑑1〉

𝑑1

call 𝑞1

𝜇 ∘ 𝑑1, 𝑠

𝑞 = 〈𝑑1〉

return

𝑜1 𝑓𝑞 𝑑1 ∘ 𝑓𝑞, 𝑜1 = 𝜇 𝑑1𝑑2 ∘ 𝑓𝑞, 𝑜2 = 𝑑1 𝑜2 𝑑1 ∘ 𝑜2,𝑓𝑞 = 〈𝑑1〉

 ∘∶ Γ × 𝐹∗ → Γ updates call strings along edges

• 1. Interproc.
• 2. Call
• 3. Return

CS updat ate operat ation

• n ∘

12.06.2010 Nikolai Knopp 30

𝑞 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞

𝑑1 ∘ 𝑑2, 𝑠

𝑞 = 〈𝑑1, 𝑑2〉

𝑑1 ∘ 𝑠

𝑞, 𝑑2 = 〈𝑑1〉

𝑑1

call 𝑞1

𝜇 ∘ 𝑑1, 𝑠

𝑞 = 〈𝑑1〉

return

𝑜1 𝑓𝑞 𝑑1 ∘ 𝑓𝑞, 𝑜1 = 𝜇 𝑑1𝑑2 ∘ 𝑓𝑞, 𝑜2 = 𝑑1 𝑜2 𝑑1 ∘ 𝑜2,𝑓𝑞 = 〈𝑑1〉

 ∘∶ Γ × 𝐹∗ → Γ updates call strings along edges

𝑑1 ∘ 𝑓𝑞, 𝑜2 undef. 𝑑1𝑑2 ∘ 𝑓𝑞, 𝑜1 undef.

• 1. Interproc.
• 2. Call
• 3. Return
• 4. invalid

CS updat ate operat ation

• n ∘

12.06.2010 Nikolai Knopp 31

𝑞 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞

𝑑1 ∘ 𝑑2, 𝑠

𝑞 = 〈𝑑1, 𝑑2〉

𝑑1 ∘ 𝑠

𝑞, 𝑑2 = 〈𝑑1〉

𝑑1

call 𝑞1

𝜇 ∘ 𝑑1, 𝑠

𝑞 = 〈𝑑1〉

return

𝑜1 𝑓𝑞 𝑑1 ∘ 𝑓𝑞, 𝑜1 = 𝜇 𝑑1𝑑2 ∘ 𝑓𝑞, 𝑜2 = 𝑑1 𝑜2 𝑑1 ∘ 𝑜2,𝑓𝑞 = 〈𝑑1〉

 ∘∶ Γ × 𝐹∗ → Γ updates call strings along edges

𝑑1 ∘ 𝑓𝑞, 𝑜2 undef. 𝑑1𝑑2 ∘ 𝑓𝑞, 𝑜1 undef.

• 1. Interproc.
• 2. Call
• 3. Return
• 4. invalid

𝑑1 ∘ 𝑑1, 𝑠

𝑞 undef.

∘ consi sist stent with 𝐽𝑊𝑄

 𝛿′ ∘ 𝑛, 𝑜 = 𝛿 only defined iff:

• 1. (By definition)

A path 𝑟′ ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑛 exists with 𝐷𝑁 𝑟′ = 𝛿′

• 2. (Lemma)

𝑟 = 𝑟′||(𝑛,𝑜) lies in 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 and 𝐷𝑁 𝑟 = 𝛿

⇒ “∘ only updates and yields call strings for interprocedurally valid paths”

12.06.2010 Nikolai Knopp 32

Edge effect cts s doma main 𝐺∗

 𝐺∗ ≔ 𝑀∗ → 𝑀∗ embeds edge effects from 𝐺

into 𝑀∗ domain

 For 𝑛, 𝑜 ∈ 𝐹∗ and data 𝜊 = . . , 𝛿′, 𝑤′ , . .

∈ 𝑀∗ at 𝑛, data at 𝑜 is 𝑔 𝑛,𝑜

∗

𝜊 = . . , 𝛿, 𝑤 ,. .

• update each 𝛿′ to reflect 𝑛, 𝑜 taken:

𝛿 = 𝛿′ ∘ 𝑛,𝑜

• propagate effect 𝑔 𝑛,𝑜 ∈ 𝑀 applied to each 𝑤′:

𝑤 = 𝑔 𝑛,𝑜 𝑤′

𝑑1 , 1 , 𝑑1𝑑2 ,⊤ 𝜊

call p a := a - 1

𝑑2

𝑔 𝑛,𝑑2

𝑛

Edge effect cts s 𝑔 𝑛,𝑜

∗

Defini nition: n: Let 𝑛, 𝑜 ∈ 𝐹∗, 𝑔 𝑛,𝑜 ∈ 𝐺, 𝜊 ∈ 𝑀∗.

12.06.2010 34 Nikolai Knopp

𝑑1 , 1 , 𝑑1𝑑2 ,⊤ 𝜊 𝑑1 , ⊤ , 𝑑1𝑑2 ,⊤

call p a := a - 1

𝑑2

𝑔 𝑛,𝑑2 𝑔 𝑛,𝑑2

∗

𝜊

𝑛

Edge effect cts s 𝑔 𝑛,𝑜

∗

Defini nition: n: Let 𝑛, 𝑜 ∈ 𝐹∗, 𝑔 𝑛,𝑜 ∈ 𝐺, 𝜊 ∈ 𝑀∗.

12.06.2010 35 Nikolai Knopp

𝑑1 , 1 , 𝑑1𝑑2 ,⊤ 𝜊 𝑑1 , ⊤ , 𝑑1𝑑2 ,⊤

call p a := a - 1

𝑑2

𝑔 𝑛,𝑑2 𝜊 𝑑1 𝑔 𝑛,𝑑2

∗

𝜊 𝑔 𝑛,𝑑2

∗

𝜊 𝑑1

𝑛

Edge effect cts s 𝑔 𝑛,𝑜

∗

Defini nition: n: Let 𝑛, 𝑜 ∈ 𝐹∗, 𝑔 𝑛,𝑜 ∈ 𝐺, 𝜊 ∈ 𝑀∗.

12.06.2010 36 Nikolai Knopp

𝑔 𝑛,𝑜 if ∃𝛿′ with 𝛿 = 𝛿′ ∘ 𝑛, 𝑜 else ⊥ 𝑔 𝑛,𝑜

∗

≔ 𝜊 𝛿 𝜊 𝛿′ ∀ 𝛿 ∈ Γ

Edge effect cts s doma main 𝐺∗

Define 𝐺∗ ⊆ 𝑀∗ → 𝑀∗ such that

1.

𝐺∗ smallest set containing 𝑔 𝑛,𝑜

∗

| 𝑛, 𝑜 ∈ 𝐹∗ ∪ 𝑗𝑒𝑀∗

2.

𝐺∗ closed under functional composition and ⋀

Lemm mma: a: Prop

• perties

s of 𝐺∗

1.

𝐺 monotone distributive in 𝑀 ⇒ 𝐺∗ monotone distributive in 𝑀∗

2.

𝐺 distributive in 𝑀 ⇒ 𝐺∗ continuo nuous us in 𝑀∗: For an infinite collection 𝜊𝑙 𝑙≥1 ⊆ 𝑀∗ and 𝑛, 𝑜 ∈ 𝐹∗: 𝑔 𝑛,𝑜

∗

𝜊𝑙

𝑙

= 𝑔 𝑛,𝑜

∗

𝜊𝑙

𝑙

12.06.2010 Nikolai Knopp 38

Dataf aflow equat ations

• ns 𝑦𝑜

∗

 The dataflow problem for 𝐺∗,𝑀∗ :

• 𝑦𝑠𝑛𝑏𝑗𝑜

∗

= 𝜇, ⊤

• 𝑦𝑜

∗ = ⋀

𝑔 𝑛,𝑜

∗

𝑦𝑛

∗ 𝑛,𝑜 ∈𝐹∗

𝑜 ∈ 𝑂∗ − 𝑠

𝑛𝑏𝑗𝑜

 Claim:

∃ maximum fixed-point solution (MFP MFPCS

CS) for

𝑦𝑜

∗ 𝑜∈𝑂∗

12.06.2010 Nikolai Knopp 40

Proo

• of: Exist

stence ce of max. FP solution

• n

 Iteratively solving the equations yields decreasing

(𝐺∗ monotone) chains 𝑦𝑜

∗ 𝑗 ≥ 𝑦𝑜 ∗ 𝑗+1 ∀𝑜 ∈ 𝑂∗.

Then ∀𝛿 ∈ Γ 𝑦𝑜

∗ 𝑗 𝛿 ≥ 𝑦𝑜 ∗ 𝑗+1 𝛿

a decreasing chain in 𝑀 with limit lim𝑦𝑜

∗ 𝛿 (exists as

𝑀 bounded).

 𝑦𝑜

∗ ≔ lim 𝑗→∞ 𝑦𝑜 ∗ 𝑗 possibly infinite but well-defined

⇒ 𝑦𝑜

∗ 𝑜∈𝑂∗ is maximal FP solution to DF problem.

12.06.2010 Nikolai Knopp 41

Maximum mum FP solution

• n of (𝑀∗, 𝐺∗)

12.06.2010 Nikolai Knopp 42

 Initialise 𝑦𝑜

∗ 𝑜∈𝑂∗ with

• 𝑦𝑠𝑛𝑏𝑗𝑜

∗ 0

= 𝜇, ⊤

• 𝑦𝑜

∗ 0 = ⊥∗

 Result if Γ

• finite

te: Any iterative algorithm terminates and reaches MFP.

• infini

nite te: Iteration may diverge, intermediate steps unsafe approx. of MFP 𝑜 ∈ 𝑂∗ − 𝑠

𝑛𝑏𝑗𝑜

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 43

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 44

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 45

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 46

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 〈𝑑1〉,1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 47

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 〈𝑑1〉,1 〈𝑑1〉,1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 48

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 〈𝑑1〉,1 〈𝑑1〉,1 𝑑1 ,⊤ 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Exam ample

12.06.2010 Nikolai Knopp 49

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 𝜇, 1 〈𝑑1〉,1 𝑑1 ,⊤ 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

Exam ample

12.06.2010 Nikolai Knopp 50

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 𝜇, 1 𝑑1 ,⊤ 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

Exam ample

12.06.2010 Nikolai Knopp 51

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

Exam ample

12.06.2010 Nikolai Knopp 52

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

Exam ample

...

12.06.2010 Nikolai Knopp 53

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

Chap apter summ mmar ary

Call string ng approach: ch:

 New dataflow problem on top of (𝑀, 𝐺)

• Data “tagged” with propagation history
• Interprocedural flow explicit

 Existence of MFPCS

• No functional compositions
• Only iteratively computable if Γ finite

12.06.2010 Nikolai Knopp 54

Struct cture

• 1. Definition of a new DF problem (𝑀∗, 𝐺∗)

2.

• 2. Proof: Solut

ution n to 𝑴∗, 𝑮∗ ≡ MOP solution

• 3. Feasibility and precise variants
• 4. Approximative solution

12.06.2010 Nikolai Knopp 55

MOP P solution

• n and MFPCS

CS

 This chapter: Show that MOP solution

𝑧𝑜 = 𝑔

𝑟: 𝑟 ∈ 𝐽𝑊𝑄 𝑠𝑛𝑏𝑗𝑜,𝑜

⊤ can be acquired from MFPCS.

 Chapter outline:

• 1. Define MOPCS
• 2. Show MOPCS = MFPCS
• 3. Extract results 𝑦𝑜

′ ∈ 𝑀 from MFPCS

• 4. Using MOPCS and MFPCS, show 𝑦𝑜

′ = MOP

12.06.2010 Nikolai Knopp 56

MOP P solution

• n for call strings

s (MOPCS

CS)

 Propagation of data along paths in 𝐻∗

For 𝑟 = 𝑠

𝑛𝑏𝑗𝑜, 𝑡1, … ,𝑡𝑙,𝑜 ∈ 𝑞𝑏𝑢ℎ𝐻∗:

𝑔

𝑟 ∗ ≔ 𝑔 𝑡𝑙,𝑜 ∗

∘ 𝑔 𝑡𝑙−1,𝑡𝑙

∗

∘ ⋯∘ 𝑔 𝑡1,𝑡2

∗

∘ 𝑔

(𝑠𝑛𝑏𝑗𝑜,𝑡1) ∗

 Call string MOP solution (MOPCS

CS)

∀𝑜 ∈ 𝑂∗: 𝑧𝑜

∗ ≔ ⋀ 𝑔 𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

∶ 𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗ 𝑠

𝑛𝑏𝑗𝑜, 𝑜

12.06.2010 Nikolai Knopp 57

𝑛𝑏𝑗𝑜

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑔

𝑟1 ∗

𝑔

𝑟2 ∗ …

MOP P solution

• n for call strings

s (MOPCS

CS)

 Propagation of data along paths in 𝐻∗

For 𝑟 = 𝑠

𝑛𝑏𝑗𝑜, 𝑡1, … ,𝑡𝑙,𝑜 ∈ 𝑞𝑏𝑢ℎ𝐻∗:

𝑔

𝑟 ∗ ≔ 𝑔 𝑡𝑙,𝑜 ∗

∘ 𝑔 𝑡𝑙−1,𝑡𝑙

∗

∘ ⋯∘ 𝑔 𝑡1,𝑡2

∗

∘ 𝑔

(𝑠𝑛𝑏𝑗𝑜,𝑡1) ∗

 Call string MOP solution (MOPCS

CS)

∀𝑜 ∈ 𝑂∗: 𝑧𝑜

∗ ≔ ⋀ 𝑔 𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

∶ 𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗ 𝑠

𝑛𝑏𝑗𝑜, 𝑜

12.06.2010 Nikolai Knopp 58

𝑞𝑏𝑢ℎ𝐻∗ ⊇ 𝐽𝑊𝑄 𝑛𝑏𝑗𝑜

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑔

𝑟1 ∗

𝑔

𝑟2 ∗ …

?

Valid–pat ath-only prop

• pag

agat ation

• n

Lemma: : Let 𝑜 ∈ 𝑂∗,𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗ 𝑠𝑛𝑏𝑗𝑜,𝑜 , 𝛿 ∈ Γ. Then

1.

𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 and 𝐷𝑁 𝑟 = γ

⇔ 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 defined

2.

𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 = 𝑔

𝑟 ⊤

⇒ “MOPCS only contains paths in 𝐽𝑊𝑄 and for each such 𝑞 yields the same result as 𝑀, 𝐺 ”

12.06.2010 Nikolai Knopp 59

Valid–pat ath-only prop

• pag

agat ation

• n (Proo
• of)

By induc uction on 𝑚 𝑟 for 𝑟 = (𝑠𝑛𝑏𝑗𝑜, 𝑡1,… ,𝑡𝑙, 𝑜):

 𝒎 𝒓 = 𝟏: Let 𝑜 ∈ 𝑂∗.

𝑟 = 𝑠

𝑛𝑏𝑗𝑜 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜,𝑠 𝑛𝑏𝑗𝑜 ⇒ 𝐷𝑁 𝑟 = 𝜇.

𝑔

𝑟 ∗ 𝑠 𝑛𝑏𝑗𝑜 =

𝜇, ⊤

• nly def. for 𝜇 with ⊤ = 𝑔

𝑟 ⊤ .

12.06.2010 Nikolai Knopp 60

 IH:

Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟

then 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝜇 = 𝑔

𝑞 ⊤ .

Valid–pat ath-only prop

• pag

agat ation

• n (Proo
• of)

 IH:

Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟

then 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝜇 = 𝑔

𝑞 ⊤

 IS: 𝑚 𝑟 = 𝑙.

12.06.2010 Nikolai Knopp 61

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑡𝑙−1 𝑡𝑙 ...

𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 𝑟 ≝ 𝑔 𝑡𝑙,𝑜 𝑦𝑡𝑙

∗

Valid–pat ath-only prop

• pag

agat ation

• n (Proo
• of)

 IH:

Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟

then 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝜇 = 𝑔

𝑞 ⊤

 IS: 𝑚 𝑟 = 𝑙.

12.06.2010 Nikolai Knopp 62

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑡𝑙−1 𝑡𝑙 ...

𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 𝑔

𝑟′ ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿′ 𝑟′ 𝑟 𝛿 = 𝛿′ ∘ 𝑛, 𝑜 , 𝛿′ = 𝐷𝑁 𝑟′ ≝ 𝑔 𝑡𝑙,𝑜 𝑦𝑡𝑙

∗

Valid–pat ath-only prop

• pag

agat ation

• n (Proo
• of)

 IH:

Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟

then 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝜇 = 𝑔

𝑞 ⊤

 IS: 𝑚 𝑟 = 𝑙.

12.06.2010 Nikolai Knopp 63

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑡𝑙−1 𝑡𝑙 ...

𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 𝑔

𝑟′ ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿′ 𝑟′ 𝑟 =𝑱𝑰 𝑔𝑟′ ⊤ 𝛿 = 𝛿′ ∘ 𝑛, 𝑜 , 𝛿′ = 𝐷𝑁 𝑟′ ≝ 𝑔 𝑡𝑙,𝑜 𝑦𝑡𝑙

∗

Valid–pat ath-only prop

• pag

agat ation

• n (Proo
• of)

 IH:

Iff 𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 , 𝑚 𝑟 < 𝑙 and 𝛿 = 𝐷𝑁 𝑟

then 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝜇 = 𝑔

𝑞 ⊤

 IS: 𝑚 𝑟 = 𝑙.

12.06.2010 Nikolai Knopp 64

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑡𝑙−1 𝑡𝑙 ...

𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 𝑔

𝑟′ ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿′ = 𝑔 𝑡𝑙,𝑜 𝑔𝑟′ ⊤ ≝ 𝑔

𝑟 ⊤

𝑟′ 𝑟 =𝑱𝑰 𝑔𝑟′ ⊤

∎

𝛿 = 𝛿′ ∘ 𝑛, 𝑜 , 𝛿′ = 𝐷𝑁 𝑟′

MFPCS = MOPCS

CS:

: 𝑦𝑜

∗ = 𝑧𝑜 ∗

Theorem: (𝑀, 𝐺) distributive ⇒ ∀𝑜 ∈ 𝑂∗: 𝑦𝑜

∗ = 𝑧𝑜 ∗

Proof:

 𝒚𝒐

∗ ≤ 𝒛𝒐 ∗ : Let 𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗ = 𝑠𝑛𝑏𝑗𝑜,𝑡1, … , 𝑡𝑙, 𝑜 .

12.06.2010 Nikolai Knopp 65

:𝑟

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑡1 𝑡2 𝑢

𝑦𝑡1

∗ ≤ 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑡1 ∗

𝑦𝑠𝑛𝑏𝑗𝑜

∗

𝑦𝑡2

∗ ≤ 𝑔 𝑡1,𝑡2 ∗

𝑦𝑡1

∗

𝑦𝑜

∗ ≤ 𝑔 𝑡𝑙,𝑜 ∗

𝑦𝑡𝑙

∗

...

MFPCS = MOPCS

CS:

: 𝑦𝑜

∗ = 𝑧𝑜 ∗

Theorem: (𝑀, 𝐺) distributive ⇒ ∀𝑜 ∈ 𝑂∗: 𝑦𝑜

∗ = 𝑧𝑜 ∗

Proof:

 𝒚𝒐

∗ ≤ 𝒛𝒐 ∗ : Let 𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗ = 𝑠𝑛𝑏𝑗𝑜,𝑡1, … , 𝑡𝑙, 𝑜 .

12.06.2010 Nikolai Knopp 66

:𝑟

𝑠

𝑛𝑏𝑗𝑜

𝑜 𝑡1 𝑡2 𝑢

𝑦𝑡1

∗ ≤ 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑡1 ∗

𝑦𝑠𝑛𝑏𝑗𝑜

∗

𝑦𝑡2

∗ ≤ 𝑔 𝑡1,𝑡2 ∗

𝑦𝑡1

∗

𝑦𝑜

∗ ≤ 𝑔 𝑡𝑙,𝑜 ∗

𝑦𝑡𝑙

∗

...

By monotonicity ⇒ 𝑦𝑜

∗ ≤ 𝑔 𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝑁𝑃𝑄𝐷𝑇 = ⋀ 𝑔

𝑟 ∗ 𝑟

⇒ 𝑦𝑜

∗ ≤ ⋀ 𝑔 𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗ 𝑟

MFPCS = MOPCS

CS:

: 𝑦𝑜

∗ = 𝑧𝑜 ∗

 𝒚𝒐

∗ ≥ 𝒛𝒐 ∗:

By induction on 𝑗: ∀𝑗 ≥ 0, 𝑜 ∈ 𝑂∗: 𝑦𝑜

∗ 𝑗 ≥ 𝑧𝑜 ∗

• 𝒋 = 𝟏:

𝑜 = 𝑠𝑛𝑏𝑗𝑜 ∶ 𝑦𝑠𝑛𝑏𝑗𝑜

∗(0)

≝ 𝜇, ⊤ = 𝑔

𝑟0 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

≥ 𝑧𝑠𝑛𝑏𝑗𝑜

∗

𝑜 ≠ 𝑠𝑛𝑏𝑗𝑜 ∶ 𝑦𝑜

∗ 0 ≝ ⊥∗ ≥ 𝑧𝑜 ∗

• IH:

∀𝑗 ≤ 𝑙, 𝑜 ∈ 𝑂∗: 𝑦𝑜

∗ 𝑗 ≥ 𝑧 𝑜 ∗

• IS:

𝑦𝑜

∗ 𝑙+1 ≝

𝑔 𝑛,𝑜

∗

𝑦𝑛

∗ 𝑙 𝑛,𝑜 ∈𝐹∗

IH ≥ 𝑔 𝑛,𝑜

∗

𝑧𝑛

∗ 𝑛,𝑜 ∈𝐹∗

≥ 𝑧𝑜

∗

12.06.2010 Nikolai Knopp 67

∎ Cont.+Dist.

• f 𝑔 𝑛,𝑜

∗

Resu sult for 𝑀, 𝐺 from

• m MFPCS

CS

Define 𝑦𝑜

′ ∈ 𝑀 as follows:

𝑦𝑜

′ ≔ 𝑦𝑜 ∗ 𝛿 𝛿∈Γ

Claim: 𝑦𝑜

′ is the MOP solution in (𝑀, 𝐺)

12.06.2010 Nikolai Knopp 68

𝑑1 ,1 , 𝑑1𝑑2 , ⊤ , 𝑑1𝑑2𝑑2 ,⊤ , …

a := a - 1

= 𝑦𝑜

∗ 𝑦𝑜

′ = 𝑦𝑜 ∗ 𝜇 ∧ 𝑦𝑜 ∗

𝑑1 ∧ 𝑦𝑜

∗

𝑑1𝑑2 ∧ ⋯ = ⊥ ∧ 1 ∧ ⊤ ∧ ⋯ = 1 ∧ ⊤ ∧ ⋯ = ⊤

𝑜

MFPCS

CS ≡ MOP

Theorem: For each 𝑜 ∈ 𝑂∗: 𝑦𝑜

′ = 𝑧𝑜

Proof:

𝑦𝑜

′

12.06.2010 Nikolai Knopp 69

= 𝑦𝑜

∗(𝛿) 𝛿∈Γ

= 𝑔

𝑟 ∗ 𝑦𝑠𝑛𝑏𝑗𝑜 ∗

𝛿 |𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗(𝑠

𝑛𝑏𝑗𝑜,𝑜) 𝛿∈Γ

= 𝑔

𝑟 ⊤ |𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜, 𝑜 sth. 𝐷𝑁 𝑟 = 𝛿 𝛿∈Γ

= 𝑔

𝑟 ⊤ |𝑟 ∈ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜, 𝑜

= 𝑧𝑜

∎

Def 𝑦𝑜

′

MFPCS= MOPCS

Only-valid-path

• propag. Lemma

𝐷𝑁−1 Γ ∩ 𝐽𝑊𝑄 = 𝐽𝑊𝑄

Chap apter summ mmar ary

 Shown:

• Get MOP solution from MFPCS
• Conversion 𝑦𝑜

∗ → 𝑦𝑜 ′ straightforward and without

functional composition

⇒ Call string approach is actually useful

 But: Computing MFPCS still problematic

12.06.2010 Nikolai Knopp 70

Struct cture

• 1. Definition of a new DF problem (𝑀∗, 𝐺∗)
• 2. Proof: Solution to 𝑀∗,𝐺∗ ≡ MOP solution

3.

• 3. Feasibility and preci

cise variant nts

• 4. Approximative solution

12.06.2010 Nikolai Knopp 71

Motivat ation

 Observation: 𝑀 finite ⇒ functional approach

converges, CS approach not necessarily

 Idea: Ensure termination of CS approach by

limiting to finite CS subset Γ0 ⊆ Γ

 Show: Can be done for all (𝑀, 𝐺) with finite 𝑀

without losing precision

12.06.2010 Nikolai Knopp 72

Redefinitions

• ns using Γ

𝑝

 Γ0 is finite subset of Γ fulfilling:

If 𝛿 ∈ Γ0 and 𝛿′ initial subtuple of 𝛿 ⇒ 𝛿′ ∈ Γ0

 𝐽𝑊𝑄′ ≔ 𝑟 ∈ 𝐽𝑊𝑄 | ∀ prefix 𝑟′of 𝑟: 𝐷𝑁 𝑟′ ∈ Γ0  Let ∘0 only act in Γ0

• discards 𝛿′ entirely iff 𝛿′ ∘0 𝑛,𝑜 ∉ Γ0
• ∘0 is consistent with 𝐽𝑊𝑄′

12.06.2010 Nikolai Knopp 73

Definitions:

• ns: DF anal

alysis s with Γ

Now consider 𝑀0

∗ ,𝐺0 ∗ with 𝑀0 ∗ ≔ Γ0 → 𝑀 finite

• Dataflow equations now iteratively solvable

𝑦𝑜

′ 0 ≔ 𝑦𝑜 ∗ 𝛿 𝛿∈Γ0

• MOP solution using only paths in 𝐽𝑊𝑄′

𝑧𝑜

′ 0 ≔ 𝑔 𝑟 0 | 𝑟 ∈ 𝐽𝑊𝑄′ 𝑠 𝑛𝑏𝑗𝑜, 𝑜

12.06.2010 Nikolai Knopp 74

(MFPCS0) (MOPCS0)

MOPCS0 = MFPCS0

Theorem: ∀𝑜 ∈ 𝑂∗: 𝑦𝑜

′ 0 = 𝑧𝑜 ′

Proof:

• Completely analogous to MOPCS=MFPCS proof by

replacing Γ, 𝐽𝑊𝑄,∘ by Γ0, 𝐽𝑊𝑄′,∘0

• No reasoning about infinite meets or continuity
• f 𝐺0

∗ required

12.06.2010 Nikolai Knopp 75

∎

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

12.06.2010 Nikolai Knopp 76

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

12.06.2010 Nikolai Knopp 77

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

12.06.2010 Nikolai Knopp 78

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

12.06.2010 Nikolai Knopp 79

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

12.06.2010 Nikolai Knopp 80

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ , 𝑑1𝑑2𝑑2𝑑2 ,⊤

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

 No information gain on further iteration  Stop if data tagged by longer CS „irrelevant“

12.06.2010 Nikolai Knopp 81

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉,1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 ,⊤ , 𝑑1𝑑2𝑑2 , ⊤ , 𝑑1𝑑2𝑑2𝑑2 ,⊤

𝑔 𝑑2,𝑠𝑞

∗

𝑔 𝑑1,𝑠𝑞

∗

Motivat ation: CS of limi mited length

 No information gain on further iteration  Stop if data tagged by longer CS „irrelevant“

12.06.2010 Nikolai Knopp 82

if a=0

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p a := a - 1

T F

p

call p

main

𝑠

𝑞

read a, b t := a * b

𝑑2

𝜇, ⊤ 𝜇, 1 𝑔 𝑠𝑛𝑏𝑗𝑜,𝑑1

∗

𝑔 𝑠𝑞,𝑜2

∗

𝑜2

𝑔 𝑜2,𝑑2

∗

〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤ 〈𝑑1〉, 1 , 𝑑1𝑑2 , ⊤

Redundan ant inform

• rmat

ation

• n in anal

alysis

 Why paths of arbitrary length?

• 𝐿 ≔ number of call sites in program ∈ ℕ

⇒ only possible by recurring on call site(s)

• At each call, analysis arrives with value 𝑦 ∈ 𝑀 and

returns with value 𝑧 ∈ 𝑀. 𝑀 is finite ⇒ 𝑀 2 many distinct begin/end value combinations per call

 Idea: Recurring on any of the 𝐿 calls more

than 𝑀 2 times is redundant.

12.06.2010 Nikolai Knopp 83

Lemm mma: a: Path short

• rtening

Let

 (𝑀, 𝐺) with 𝑀 finite  𝑁 ≔ 𝐿 ∗ 𝑀 2

• 𝐿 ≔ # call blocks in 𝐻∗

 Γ0 = Γ𝑁.

Then ∀ 𝑜 ∈ 𝑂∗: If 𝑟 ∈ 𝐽𝑊𝑄 𝑠𝑛𝑏𝑗𝑜, 𝑜 then exists 𝑟′ ∈ 𝐽𝑊𝑄′(𝑠𝑛𝑏𝑗𝑜, 𝑜) with 𝑔

𝑟 0 = 𝑔𝑟′ 0 .

12.06.2010 Nikolai Knopp

𝑞1 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞2

𝑑1 call 𝑞1

return

𝑜1 𝑓𝑞2 𝑜2

84

Lemm mma: a: Path short

• rtening

Let

 (𝑀, 𝐺) with 𝑀 finite  𝑁 ≔ 𝐿 ∗ 𝑀 2

• 𝐿 ≔ # call blocks in 𝐻∗

 Γ0 = Γ𝑁.

Then ∀ 𝑜 ∈ 𝑂∗: If 𝑟 ∈ 𝐽𝑊𝑄 𝑠𝑛𝑏𝑗𝑜, 𝑜 then exists 𝑟′ ∈ 𝐽𝑊𝑄′(𝑠𝑛𝑏𝑗𝑜, 𝑜) with 𝑔

𝑟 0 = 𝑔𝑟′ 0 .

12.06.2010 Nikolai Knopp

𝑞1 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞2

𝑑1 call 𝑞1

return

𝑜1 𝑓𝑞2 𝑜2

85

𝑟

Lemm mma: a: Path short

• rtening

Let

 (𝑀, 𝐺) with 𝑀 finite  𝑁 ≔ 𝐿 ∗ 𝑀 2

• 𝐿 ≔ # call blocks in 𝐻∗

 Γ0 = Γ𝑁.

Then ∀ 𝑜 ∈ 𝑂∗: If 𝑟 ∈ 𝐽𝑊𝑄 𝑠𝑛𝑏𝑗𝑜, 𝑜 then exists 𝑟′ ∈ 𝐽𝑊𝑄′(𝑠𝑛𝑏𝑗𝑜, 𝑜) with 𝑔

𝑟 0 = 𝑔𝑟′ 0 .

12.06.2010 Nikolai Knopp

𝑞1 𝑛𝑏𝑗𝑜

𝑑2

call 𝑞1

𝑠

𝑞2

𝑑1 call 𝑞1

return

𝑜1 𝑓𝑞2 𝑜2

86

𝑟′

Lemm mma: a: Path short

• rtening (proo
• of)

 Proof: By induction on 𝑚(𝑟) ≔ length of 𝑟.

• 𝒎 𝒓 = 𝟏: 𝜇 ∈ Γ and 𝜇 ∈ Γ0.
• IH: Lemma holds for all paths with 𝑚 𝑟 < 𝑙, 𝑙 ∈ ℕ+.
• IS: Let 𝑜 ∈ 𝑂∗, 𝑟 ∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 , 𝑚 𝑟 = 𝑙.

Assume 𝑟 ∉ 𝐽𝑊𝑄′ 𝑠

𝑛𝑏𝑗𝑜, 𝑜 .

Let 𝑟′ the shortest prefix with 𝑟′ ∉ 𝐽𝑊𝑄′ 𝑠

𝑛𝑏𝑗𝑜, 𝑜 .

Then 𝑟′ contains 𝑁 + 1 unreturned calls.

By decomposition Lemma:

12.06.2010 Nikolai Knopp 87

𝑟′ = 𝑟0|| 𝑑1, 𝑠

𝑞1 ||𝑟1|| 𝑑2, 𝑠 𝑞2 || … || 𝑑𝑁+1, 𝑠 𝑞𝑁+1 ||𝑟𝑁+1

Lemm mma: a: Path short

• rtening (proo
• of)

 𝑁 + 1 calls, but max. 𝑁 distinct elements. Let 𝑑 , 𝑠𝑞

a duplicate call. If 𝑑 returns, let 𝑓𝑞

, 𝑜 the return

edge.

 Rewrite 𝑟 as

𝑟0

′ || 𝑑 , 𝑠𝑞 || 𝑟1 ′ || 𝑑 , 𝑠𝑞 || 𝑟2 ′ || 𝑓𝑞 , 𝑜 || 𝑟3 ′ || 𝑓𝑞 ,𝑜 || 𝑟4 ′

 Shorter 𝑟

∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 with 𝑔 𝑟 = 𝑔 𝑟 by dropping

redundant parts

 By IH

IH, ∃𝑟′ ∈ 𝐽𝑊𝑄′ 𝑠

𝑛𝑏𝑗𝑜,𝑜 for 𝑟

with 𝑔𝑟′ = 𝑔

𝑟 and thus

𝑔𝑟′ = 𝑔

𝑟.

12.06.2010 Nikolai Knopp 88

𝑟′ = 𝑟0|| 𝑑1, 𝑠

𝑞1 ||𝑟1|| 𝑑2, 𝑠 𝑞2 || … || 𝑑𝑁+1, 𝑠 𝑞𝑁+1 ||𝑟𝑁+1

∎

Lemm mma: a: Path short

• rtening (proo
• of)

 𝑁 + 1 calls, but max. 𝑁 distinct elements. Let 𝑑 , 𝑠𝑞

a duplicate call. If 𝑑 returns, let 𝑓𝑞

, 𝑜 the return

edge.

 Rewrite 𝑟 as

𝑟0

′ || 𝑑 , 𝑠𝑞 || 𝑟1 ′ || 𝑑 , 𝑠𝑞 || 𝑟2 ′ || 𝑓𝑞 , 𝑜 || 𝑟3 ′ || 𝑓𝑞 ,𝑜 || 𝑟4 ′

 Shorter 𝑟

∈ 𝐽𝑊𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 with 𝑔 𝑟 = 𝑔 𝑟 by dropping

redundant parts

 By IH

IH, ∃𝑟′ ∈ 𝐽𝑊𝑄′ 𝑠

𝑛𝑏𝑗𝑜,𝑜 for 𝑟

with 𝑔𝑟′ = 𝑔

𝑟 and thus

𝑔𝑟′ = 𝑔

𝑟.

12.06.2010 Nikolai Knopp 89

𝑟′ = 𝑟0|| 𝑑1, 𝑠

𝑞1 ||𝑟1|| 𝑑2, 𝑠 𝑞2 || … || 𝑑𝑁+1, 𝑠 𝑞𝑁+1 ||𝑟𝑁+1

∎

MOPCS0=MOP

 For 𝑁 ∈ ℕ ≥ 0, define Γ𝑁 as set of all 𝛿 ∈ Γ with

length ≤ 𝑁.

 Theorem:

Let 𝑀, 𝐺 a distributive DF framework with 𝑀 finite, and Γ0 = Γ𝑁 with 𝑁 = 𝐿 ∗ 𝑀 2. Then ∀𝑜 ∈ 𝑂∗: 𝑦𝑜

′ 0 = 𝑧𝑜

⇒ “If 𝛿 is too long and gets discarded, a shorter one in the set represents the same data.”

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Better bounds s for 𝑁

 𝑁 impractically large for most problems  Lower bounds exists for certain problem

classes. Example: Decomposable frameworks

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Decom

• mposab

sable fram amework

• rks

𝑀 = , 𝑢 , 𝑣 , 𝑢, 𝑣 , ⊥ 𝑁 = 50

12.06.2010 Nikolai Knopp 92

if … return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p b := b + 1 t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜 𝑜2 𝑠

𝑞

𝑓𝑞

read a,b t := a * b u := 2 * b t := a * b u := 2 * b print t,u

𝑑2 a*b,2*b avail.?

Decom

• mposab

sable fram amework

• rks

𝑀 = , 𝑢 , 𝑣 , 𝑢, 𝑣 , ⊥ 𝑁 = 50

12.06.2010 Nikolai Knopp 93

if … return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p b := b + 1 t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜 𝑜2 𝑠

𝑞

𝑓𝑞

read a,b t := a * b u := 2 * b t := a * b u := 2 * b print t,u

𝑑2

+𝑢 + 𝑣 𝑗𝑒 −𝑢 𝑗𝑒 +𝑢 − 𝑣

a*b,2*b avail.?

Decom

• mposab

sable fram amework

• rks

𝑀 = , 𝑢 , 𝑣 , 𝑢, 𝑣 , ⊥ 𝑁 = 50

12.06.2010 Nikolai Knopp 94

if … return

𝑠

𝑛𝑏𝑗𝑜

𝑑1

call p b := b + 1 t := a * b a := a - 1

T F

p

call p stop

main

𝑓𝑛𝑏𝑗𝑜 𝑜2 𝑠

𝑞

𝑓𝑞

read a,b t := a * b u := 2 * b t := a * b u := 2 * b print t,u

𝑑2

+𝑢 + 𝑣 𝑗𝑒 −𝑢 𝑗𝑒 +𝑢 − 𝑣 𝑣 𝑣 𝑢, 𝑣 𝑣 𝑢

a*b,2*b avail.?

Decom

• mposab

sable fram amework

• rks

𝑀𝑢 = , 𝑢 ,⊥ , 𝑀𝑣 = , 𝑣 , ⊥ , ⇒ 𝑀 = 𝑀𝑢 × 𝑀𝑣 𝑁 = 18

12.06.2010 Nikolai Knopp 95

if … return call p b := b + 1 t := a * b a := a - 1

T F

call p stop read a,b t := a * b u := 2 * b t := a * b u := 2 * b print t,u

𝑠

𝑛𝑏𝑗𝑜

𝑑1

p main

𝑓𝑛𝑏𝑗𝑜 𝑜2 𝑠

𝑞

𝑓𝑞 𝑑2

+𝑢 +𝑣 𝑢 𝑣 𝑣 id 𝑗𝑒 𝑣 −𝑢 𝑗𝑒 𝑣 𝑗𝑒 𝑗𝑒 +𝑢 −𝑣 𝑢

a*b,2*b avail.?

Bound for decom compos

• sab

able framew mework

• rks

 Theorem: Let 𝑀, 𝐺 decomposable into 𝑙

frameworks 𝑀𝑗,𝐺𝑗 𝑗=1

𝑙

. Setting the maximum callstring length to 𝑁 = 𝐿 ∗ max

𝑗∈ 1..𝑙

𝑀𝑗

2

yields 𝑧𝑜

′ 0 = 𝑧𝑜 ∀𝑜 ∈ 𝑂∗.

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1-relat ated fram amework

• rks

 Defini

nition: n: A decomposable DF framework (L,F) is 1-related if each 𝐺𝑗 only consists of constant and identity functions.

 Theorem: In this case, using Γ0 = Γ3𝐿 yields

𝑧𝑜

′ 0 = 𝑧𝑜 ∀𝑜 ∈ 𝑂∗.

 Example: Available expressions is 1-related.

• Decomp. into subproblems 𝑀𝑓,𝐺

𝑓 for each

expression 𝑓 with 𝑀𝑓 = ⊤, 1,⊥ , 𝐺

𝑓 = 𝑗𝑒, 𝑔 ⊤, 𝑔 1, 𝑔 ⊥

12.06.2010 Nikolai Knopp 97

Chap apter summ mmar ary

 If 𝑀 finite, MFPCS iteratively computable on Γ0

• Precision/safeness depending on choice for Γ0

 Enforce termination by limiting max. CS

length

• Precision preserving bounds exist, but only of

theoretical value

• Better bounds exist for special problem classes

12.06.2010 Nikolai Knopp 98

Struct cture

• 1. Definition of a new DF problem (𝑀∗, 𝐺∗)
• 2. Proof: Solution to 𝑀∗,𝐺∗ ≡ MOP solution
• 3. Feasibility and precise variants

4.

• 4. Approximative solut

utions ns

12.06.2010 Nikolai Knopp 99

Motivat ation

 Prefer a safe approximative solution 𝑦

𝑜 ≤ 𝑧𝑜

• ver MFPCS if
• MFPCS not (iteratively) computable
• computation not feasible MFPCS by time/space

constraints

and computation of 𝑦 𝑜 ≤ 𝑧𝑜 feasible. ⇒ Accept precision loss for less complexity.

12.06.2010 Nikolai Knopp 100

Reduci cing comp mplexity of CS approa

• ach

ch

12.06.2010 Nikolai Knopp 101

Γ Γ

𝜏

Embedding Γ into

• Γ

 Let ∗ an operation in Γ

with left identity 𝑥 ∈ Γ .

 Encoding function 𝜏: Γ → Γ

• Define 𝜏 𝑜 for each call node 𝑜 in 𝐻∗.
• For 𝛿 = 𝑑1𝑑2 … 𝑑𝑜 ∈ Γ:

𝜏 𝛿 = 𝜏 𝑑1 ∗ ⋯∗ 𝜏 𝑑𝑜 and 𝜏 𝛿 = 𝑥.

12.06.2010 Nikolai Knopp 102

Tran ansf sfer relat ation

• n 𝑆 𝑛,𝑜

 For each 𝑛, 𝑜 ∈ 𝐹∗, define transfer relation

𝑆 𝑛,𝑜 ∈ Γ × Γ :

• If 𝑛,𝑜 intraproc.:

∀𝑏 ∈ Γ : 𝑏 𝑆 𝑛,𝑜 𝑏

• If 𝑛,𝑜 interproc.:

𝜏 𝛿′ 𝑆 𝑛,𝑜 𝜏 𝛿 ⇒ 𝑆𝑑𝑏𝑚𝑚 = 𝑆𝑠𝑓𝑢𝑣𝑠𝑜

−1

12.06.2010 Nikolai Knopp 103

𝑛 𝑜 𝑜′ 𝑛′ 𝛿′ 𝛿 𝛿 𝛿′ 𝑛 𝑜 𝛿′ 𝛿′ 𝜏 𝛿′ = 𝑏

Acceptab able paths

Let 𝑟 = 𝑜1, 𝑜2, … , 𝑜𝑙 ∈ 𝑞𝑏𝑢ℎ𝐻∗.

 Define 𝑆𝑟 ≔ 𝑆 𝑜1,𝑜2 ∘ 𝑆 𝑜2,𝑜3 ∘ ⋯ ∘ 𝑆 𝑜𝑙−1,𝑜𝑙  𝑟 acceptable ∶⇔ 𝑆𝑟 ≠ ∅  Intraprocedurally acceptable paths:

𝐽𝐵𝑄 𝑠𝑛𝑏𝑗𝑜, 𝑜 ≔ 𝑟 ∈ 𝑞𝑏𝑢ℎ𝐻∗ 𝑠𝑛𝑏𝑗𝑜,𝑜 |𝑆𝑟 𝑥 ≠ ∅

12.06.2010 Nikolai Knopp 104

Inval alid pat aths s in 𝐽𝐵𝑄

12.06.2010 Nikolai Knopp 105

𝑏 𝑆 𝑛,𝑜 𝑐

𝜏

𝐷𝑁 𝑟′ = 𝛿′ 𝐷𝑁 𝑟′|| 𝑛, 𝑜 = 𝛿

𝑏 𝑐

𝜏 𝑛 𝑜 𝑠

𝑛𝑏𝑗𝑜

𝑟′

𝑥 𝑆𝑟′ 𝜏 𝛿

Inval alid pat aths s in 𝐽𝐵𝑄

12.06.2010 Nikolai Knopp 106

𝑏 𝑆 𝑛,𝑜 𝑐

𝜏

𝐷𝑁 𝑟′ = 𝛿′ 𝐷𝑁 𝑟′|| 𝑛, 𝑜 = 𝛿

𝑏 𝑐

𝜏 𝑦

𝑟

𝑛 𝑜 𝑠

𝑛𝑏𝑗𝑜

𝑟′

𝑥 𝑆𝑟′ 𝜏 𝛿

Inval alid pat aths s in 𝐽𝐵𝑄

12.06.2010 Nikolai Knopp 107

𝑏 𝑆 𝑛,𝑜 𝑐

𝜏

𝐷𝑁 𝑟′ = 𝛿′ 𝐷𝑁 𝑟′|| 𝑛, 𝑜 = 𝛿

𝑏 𝑐

𝜏

𝐷𝑁 𝑟 = 𝛿

𝜏 𝑦

𝑟

𝑛 𝑜 𝑠

𝑛𝑏𝑗𝑜

𝑟′

𝑥 𝑆𝑟′ 𝜏 𝛿

Inval alid pat aths s in 𝐽𝐵𝑄

12.06.2010 Nikolai Knopp 108

𝑏 𝑆 𝑛,𝑜 𝑐

𝜏

𝐷𝑁 𝑟′ = 𝛿′ 𝐷𝑁 𝑟′|| 𝑛, 𝑜 = 𝛿

𝑏 𝑐

𝜏

𝐷𝑁 𝑟 = 𝛿

𝜏 𝑦

𝑟

𝑛 𝑜 𝑠

𝑛𝑏𝑗𝑜

𝑟′

𝑥 𝑆𝑟′ 𝜏 𝛿 𝑥 𝑆𝑟

𝜏 𝛿

Inval alid pat aths s in 𝐽𝐵𝑄

12.06.2010 Nikolai Knopp 109

𝑏 𝑆 𝑛,𝑜 𝑐

𝜏

𝐷𝑁 𝑟′ = 𝛿′ 𝐷𝑁 𝑟′|| 𝑛, 𝑜 = 𝛿

𝑏 𝑐

𝜏

𝐷𝑁 𝑟 = 𝛿

𝜏 𝑦

𝑟

𝑛 𝑜 𝑠

𝑛𝑏𝑗𝑜

𝑟′

𝑥 𝑆𝑟′ 𝜏 𝛿 𝑥 𝑆𝑟

𝜏 𝛿

⇒ 𝐽𝐵𝑄 𝑠

𝑛𝑏𝑗𝑜, 𝑜 ⊇ 𝐽𝑊𝑄 𝑠 𝑛𝑏𝑗𝑜, 𝑜

Dataf aflow anal alysi sis s using Γ

Now define framework 𝑀 , 𝐺

• 𝑀

≔ Γ → 𝑀

• For edge 𝑛,𝑜 , data in 𝜊𝑛 is propagated to 𝜊𝑜

from all 𝑏 ∈ Γ to 𝑐 ∈ Γ for which 𝑏 𝑆 𝑛,𝑜 𝑐: 𝑔 𝑛,𝑜

∗

𝜊𝑛 𝑐 ≔ ⋀ 𝑔 𝑛,𝑜 𝜊𝑛 𝑏 |𝑏 𝑆 𝑛,𝑜 𝑐

• As before, define DF equations, the MFP solution

𝑦 𝑜

∗ 𝑜∈𝑂∗ (which exists as Γ

finite), and the MOP solution 𝑧 𝑜 using 𝐽𝐵𝑄. ⇒ Obviously 𝑧 𝑜 ≤ 𝑧𝑜, so a safe approximation.

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Dataf aflow anal alysi sis s using Γ (cont

• ntd.)

 Without proof:

• 𝑀 distributive:

𝑦 𝑜 = 𝑧 𝑜 ≤ 𝑧𝑜

• 𝑀 only monotone:

𝑦 𝑜 ≤ 𝑧 𝑜 ≤ 𝑧𝑜

 Examples:

12.06.2010 Nikolai Knopp 111

 Γ

= 𝑥 𝜏: 𝛿 ↦ 𝑥 ∗: 𝑥, 𝑥 ↦ 𝑥 ⇒ intraproc. analysis

 Γ

= ℤ/𝑙ℤ, 𝑙 ∈ ℕ ∀ call 𝑑𝑗 ∈ 𝑂∗: 𝜏 𝑑𝑗 = 𝑗 𝑏 ∗ 𝑐 ≔ 𝑏 + 𝑐 𝑛𝑝𝑒 𝑙 ⇒ only 𝑙 call strings

Summary

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Summ mmar ary: CS Approach ach struct cture

 CS approach makes interprocedural flow

explicit

• Avoids functional composition of propagation

functions

• Keeps track of origin of data by tagging with CS
• Complexity of approach “controlled” by Γ

12.06.2010 Nikolai Knopp 113

Summ mmar ary: Results s of CS approa

• ach

ch (1/2)

Assume 𝑀, 𝐺 distributive.

 𝑀 and Γ finite:

MFPCS iteratively computable and yields MOP solution.

 Only 𝑀 finite:

Iterative calculation of MFPCS will converge

• n MOP if Γ replaced by Γ𝑁, else likely to

diverge

• Depending on problem, better bounds exist.

12.06.2010 Nikolai Knopp 114

Summ mmar ary: Results s of CS approa

• ach

ch (2/2)

All other cases combinations of:

• 𝑀, 𝐺 not distributive
• 𝑀 infinite
• 𝐺 not bounded

 Convergence then only guaranteed if Γ

embedded into finite subset Γ

• Also likely to be the only feasible approach to the
• ther situations

 Solution then is safe approximation of MOP

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12.06.2010 Nikolai Knopp 116

Thanks!