Theorem Knaster – Tarski

(1)

Theorem Knaster – Tarski

Assume D is a complete lattice. Then every monotonic function f : D

→

D _{has a} least fixpoint d₀

∈

D_.

Let P

= {

d

∈

D

|

f d

⊑

d

}

. Then d₀

=

_F P .

(2)

(3)

Theorem Knaster – Tarski

→

∈

D_.

Let P

= {

d

∈

D

|

f d

⊑

d

}

. Then d₀

=

_F P .

Proof:

(1) d₀

∈

P :

(4)

Theorem Knaster – Tarski

→

∈

D_.

Let P

= {

d

∈

D

|

f d

⊑

d

}

. Then d₀

=

_F P .

Proof:

(1) d₀

∈

P :

f d₀

⊑

f d

⊑

d for all d

∈

P

==⇒ f d₀ is a lower bound of P

==⇒ f d₀

⊑

d₀ since d₀

=

_F P

==⇒ d₀

∈

P :-)

(5)

(2) f d₀

=

d₀ :

(6)

(2) f d₀

=

d₀ :

f d₀

⊑

d₀ by (1)

==⇒ f

(

f d₀

) ⊑

f d₀ by monotonicity of f

==⇒ f d₀

∈

P

==⇒ d₀

⊑

f d₀ and the claim follows :-)

(7)

(2) f d₀

=

d₀ :

f d₀

⊑

d₀ by (1)

==⇒ f

(

f d₀

) ⊑

==⇒ f d₀

∈

P

==⇒ d₀

⊑

(3) d₀ is least fixpoint:

(8)

(2) f d₀

=

d₀ :

f d₀

⊑

d₀ by (1)

==⇒ f

(

f d₀

) ⊑

==⇒ f d₀

∈

P

==⇒ d₀

⊑

(3) d₀ is least fixpoint:

f d₁

=

d₁

⊑

d₁ an other fixpoint

==⇒ d₁

∈

P

==⇒ d₀

⊑

d₁ :-))

(9)

Remark:

The least fixpoint d₀ is in P and a lower bound :-)

==⇒ d₀ is the least value x with x

⊒

f x

(10)

Remark:

⊒

f x

Application:

Assume x_i

⊒

f_i

(

x₁, . . . , x_n

)

, i

=

1, . . . ,n

(∗)

is a system of constraints where all f_i : Dⁿ

→

D are monotonic.

(11)

Remark:

⊒

f x

Application:

Assume x_i

⊒

f_i

(

x₁, . . . , x_n

)

, i

=

1, . . . ,n

(∗)

is a system of constraints where all f_i : Dⁿ

→

D are monotonic.

==⇒ least solution of

(∗) ==

least fixpoint of F :-)

(12)

Example 1:

^D

⁼

²^U^, ^{f x}

⁼

^x

^∩

^a

^∪

^b

(13)

Example 1:

^D

⁼

²^U^, ^{f x}

⁼

^x

^∩

^a

^∪

^b

f f^k

⊥

f^k

⊤

0

∅

U

(14)

Example 1:

^D

⁼

²^U^, ^{f x}

⁼

^x

^∩

^a

^∪

^b

f f^k

⊥

f^k

⊤

0

∅

U

1 b a

∪

b

(15)

Example 1:

^D

⁼

²^U^, ^{f x}

⁼

^x

^∩

^a

^∪

^b

f f^k

⊥

f^k

⊤

0

∅

U

1 b a

∪

b

2 b a

∪

b

(16)

Example 1:

^D

⁼

²^U^, ^{f x}

⁼

^x

^∩

^a

^∪

^b

f f^k

⊥

f^k

⊤

0

∅

U

1 b a

∪

b

2 b a

∪

b

Example 2:

^D

⁼

^N

^{∪ {}

^∞

^}

Assume f x

=

x

+

1. Then

fⁱ

⊥ =

fⁱ 0

=

i ⊏ _i

+

1

=

fⁱ⁺¹

⊥

(17)

Example 1:

^D

⁼

²^U^, ^{f x}

⁼

^x

^∩

^a

^∪

^b

f f^k

⊥

f^k

⊤

0

∅

U

1 b a

∪

b

2 b a

∪

b

Example 2:

^D

⁼

^N

^{∪ {}

^∞

^}

Assume f x

=

x

+

1. Then

fⁱ

⊥ =

fⁱ 0

=

i ⊏ _i

+

1

=

fⁱ⁺¹

⊥

==⇒ Ordinary iteration will never reach a fixpoint :-(

==⇒ Sometimes, transfinite iteration is needed :-)

(18)

Conclusion:

Systems of inequations can be solved through fixpoint iteration, i.e., by repeated evaluation of right-hand sides :-)

(19)

Conclusion:

Warning:

Naive fixpoint iteration is rather inefficient :-(

(20)

Conclusion:

Warning:

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗ y;

Pos(x > 1) Neg(x >1)

0 1 2 3 4 5

(21)

Conclusion:

Warning:

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗ y;

1

0 ∅

1 {1,x >1,x−1}

2 Expr

3 {1,x >1,x−1}

4 {1}

5 Expr

(22)

Conclusion:

Warning:

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗ y;

1 2

0 ∅ ∅

1 {1,x >1,x−1} {1}

2 Expr {1,x>1,x−1}

3 {1,x >1,x−1} {1,x>1,x−1}

4 {1} {1}

5 Expr {1,x>1,x−1}

(23)

Conclusion:

Warning:

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗ y;

1 2 3

0 ∅ ∅ ∅

1 {1,x >1,x−1} {1} {1}

2 Expr {1,x>1,x−1} {1,x >1}

3 {1,x >1,x−1} {1,x>1,x−1} {1,x>1,x−1}

4 {1} {1} {1}

5 Expr {1,x>1,x−1} {1,x >1}

(24)

Conclusion:

Warning:

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗ y;

1 2 3 4

0 ∅ ∅ ∅ ∅

1 {1,x >1,x−1} {1} {1} {1}

2 Expr {1,x>1,x−1} {1,x >1} {1,x >1}

3 {1,x >1,x−1} {1,x>1,x−1} {1,x>1,x−1} {1,x >1}

4 {1} {1} {1} {1}

5 Expr {1,x>1,x−1} {1,x >1} {1,x >1}

(25)

Conclusion:

Warning:

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗ y;

1 2 3 4 5

0 ∅ ∅ ∅ ∅

1 {1,x >1,x−1} {1} {1} {1}

2 Expr {1,x>1,x−1} {1,x >1} {1,x >1}

3 {1,x >1,x−1} {1,x>1,x−1} {1,x>1,x−1} {1,x >1} dito

4 {1} {1} {1} {1}

5 Expr {1,x>1,x−1} {1,x >1} {1,x >1}

(26)

Idea: Round Robin Iteration

Instead of accessing the values of the last iteration, always use the current values of unknowns :-)

(27)

Idea: Round Robin Iteration

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗y;

Pos(x > 1) Neg(x > 1)

0 1 2 3 4 5

(28)

Idea: Round Robin Iteration

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗y;

Pos(x > 1) Neg(x > 1)

1

0 ∅

1 {1} 2 {1,x > 1} 3 {1,x > 1} 4 {1} 5 {1,x > 1}

(29)

Idea: Round Robin Iteration

Example:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗y;

Pos(x > 1) Neg(x > 1)

1 2

0 ∅

1 {1} 2 {1,x > 1}

3 {1,x > 1} dito 4 {1}

5 {1,x > 1}

(30)

The code for Round Robin Iteration in Java looks as follows:

for

(

i

=

1;i

≤

n;i

++)

x_i

= ⊥

; do {

finished

=

^true;

for

(

i

=

1; i

≤

n; i

++)

{ new

=

f_i

(

x₁, . . . , x_n

)

; if (!

(

x_i

⊒

new

))

{

finished

=

^false; x_i

=

x_i

⊔

new;

} }

} while

(

!finished

)

;

(31)

Correctness:

Assume y⁽_i^d⁾ is the i-th component of F^d

⊥

.

Assume x⁽_i^d⁾ is the value of x_i after the d-th RR-iteration.

(32)

Correctness:

⊥

.

Assume x⁽_i^d⁾ is the value of x_i after the i-th RR-iteration.

One proves:

(1) y⁽_i^d⁾

⊑

x⁽_i^d⁾ :-)

(33)

Correctness:

⊥

.

One proves:

(1) y⁽_i^d⁾

⊑

x⁽_i^d⁾ :-)

(2) x_i⁽^d⁾

⊑

z_i for every solution

(

z₁, . . . , z_n

)

:-)

(34)

Correctness:

⊥

.

One proves:

(1) y⁽_i^d⁾

⊑

x⁽_i^d⁾ :-)

(2) x_i⁽^d⁾

⊑

z_i for every solution

(

z₁, . . . , z_n

)

:-) (3) If RR-iteration terminates after d rounds, then

(

x⁽₁^d⁾, . . . , x⁽n^d⁾

)

is a solution :-))

(35)

Warning:

The efficiency of RR-iteration depends on the ordering of the unknowns !!!

(36)

Warning:

Good:

→ u before v, if u

→

^∗ v;

→ entry condition before loop body :-)

(37)

Warning:

Good:

→ u before v, if u

→

^∗ v;

→ entry condition before loop body :-) Bad:

e.g., post-order DFS of the CFG, starting at start :-)

(38)

Good:

3 2

4 5

0

1

y = 1;

x = x−1;

y = x∗y;

Pos(x > ₁) Neg(x >₁)

Bad:

0

5

4

3 2 1

x = x−1;

y = x∗y;

Pos(x > 1) Neg(x > 1)

y = 1;

(39)

Inefficient Round Robin Iteration:

0

5

4

3 2 1

x = x−1;

y = x∗y;

Pos(x > ₁) Neg(x > ₁)

y =1;

0 1 2 3 4 5

(40)

Inefficient Round Robin Iteration:

0

5

4

3 2 1

x = x−1;

y = x∗y;

Pos(x > ₁) Neg(x > ₁)

y =1; ¹

0 Expr

1 {1}

2 {1,x−1,x >1}

3 Expr

4 {1}

5 ∅

(41)

Inefficient Round Robin Iteration:

0

5

4

3 2 1

x = x−1;

y = x∗y;

Pos(x > ₁) Neg(x > ₁)

y =1; ¹ ²

0 Expr {1,x >1}

1 {1} {1}

2 {1,x−1,x >1} {1,x−1,x> 1}

3 Expr {1,x >1}

4 {1} {1}

5 ∅ ∅

(42)

Inefficient Round Robin Iteration:

0

5

4

3 2 1

x = x−1;

y = x∗y;

Pos(x > ₁) Neg(x > ₁)

y =1; ¹ ² ³

0 Expr {1,x >1} {1,x >1}

1 {1} {1} {1}

2 {1,x−1,x >1} {1,x−1,x> 1} {1,x >1}

3 Expr {1,x >1} {1,x >1}

4 {1} {1} {1}

5 ∅ ∅ ∅

(43)

Inefficient Round Robin Iteration:

0

5

4

3 2 1

x = x−1;

y = x∗y;

Pos(x > ₁) Neg(x > ₁)

y =1; ¹ ² ³ ⁴

0 Expr {1,x >1} {1,x >1}

1 {1} {1} {1}

2 {1,x−1,x >1} {1,x−1,x> 1} {1,x >1} dito 3 Expr {1,x >1} {1,x >1}

4 {1} {1} {1}

5 ∅ ∅ ∅

==⇒ significantly less efficient :-)

(44)

... end of background on: Complete Lattices

(45)

... end of background on: Complete Lattices

Final Question:

Why is a (or the least) solution of the constraint system useful ???

(46)

... end of background on: Complete Lattices

Final Question:

For a complete lattice D, consider systems:

I [

start

] ⊒

d₀

I [

v

] ⊒ [[

k

]]

^♯

(I [

u

])

k

= (

u,_,v

)

edge where d₀

∈

D _{and all}

[[

k

]]

^♯ : D

→

D are monotonic ...

(47)

... end of background on: Complete Lattices

Final Question:

For a complete lattice D, consider systems:

I [

start

] ⊒

d₀

I [

v

] ⊒ [[

k

]]

^♯

(I [

u

])

k

= (

u,_,v

)

edge where d₀

∈

D _{and all}

[[

k

]]

^♯ : D

→

D are monotonic ...

==⇒ Monotonic Analysis Framework

(48)

Wanted: MOP

(Merge Over all Paths)

I

^∗

[

v

] =

^G

{[[

^π

]]

^♯ d₀

|

^π : start

→

^∗ v

}

(49)

Wanted: MOP

I

^∗

[

v

] =

^G

{[[

^π

]]

^♯ d₀

|

^π : start

→

^∗ v

}

Theorem

Kam, Ullman 1975

Assume

I

is a solution of the constraint system. Then:

I [

v

] ⊒ I

^∗

[

v

]

for every v

(50)

Jeffrey D. Ullman, Stanford

(51)

Wanted: MOP

I

^∗

[

v

] =

^G

{[[

^π

]]

^♯ d₀

|

^π : start

→

^∗ v

}

Theorem

Kam, Ullman 1975

Assume

I

is a solution of the constraint system. Then:

I [

v

] ⊒ I

^∗

[

v

]

for every v

In particular:

I [

v

] ⊒ [[

^π

]]

^♯ d₀ for every π _: _start

→

^∗ v

(52)

Proof:

Induction on the length of π_.

(53)

Proof:

Foundation: π

=

^ǫ (empty path)

(54)

Proof:

Foundation: π

=

^ǫ (empty path) Then:

[[

^π

]]

^♯ d₀

= [[

^ǫ

]]

^♯ d₀

=

d₀

⊑ I [

start

]

(55)

Proof:

Foundation: π

=

[[

^π

]]

^♯ d₀

= [[

^ǫ

]]

^♯ d₀

=

d₀

⊑ I [

start

]

Step: π

=

^π^′k for k

= (

u,_, v

)

edge.

(56)

Proof:

Foundation: π

=

[[

^π

]]

^♯ d₀

= [[

^ǫ

]]

^♯ d₀

=

d₀

⊑ I [

start

]

Step: π

=

^π^′k for k

= (

u,_, v

)

edge.

Then:

[[

^π^′

]]

^♯ d₀

⊑ I [

u

]

by I.H. for π

==⇒

[[

^π

]]

^♯ d₀

= [[

k

]]

^♯

([[

^π^′

]]

^♯ d₀

)

⊑ [[

k

]]

^♯

(I [

u

])

since

[[

k

]]

^♯ monotonic

⊑ I [

v

]

since

I

solution :-))

(57)

Disappointment:

Are solutions of the constraint system just upper bounds ???

(58)

Disappointment:

Answer:

In general: yes :-(

(59)

Disappointment:

Answer:

In general: yes :-(

With the notable exception when all functions

[[

k

]]

^♯ are distributive ... :-)

(60)

The function f : D₁

→

D₂ _{is called}

• distributive, if f

(

^F X

) =

^F

{

f x

|

x

∈

X

}

for all

∅ 6=

X

⊆

D_;

• strict, if f

⊥ = ⊥

.

• totally distributive, if f is distributive and strict.

(61)

→

D₂ _{is called}

(

^F X

) =

^F

{

f x

|

x

∈

X

}

for all

∅ 6=

X

⊆

D_;

• strict, if f

⊥ = ⊥

.

Examples:

• f x

=

x

∩

a

∪

b for a, b

⊆

U .

(62)

→

D₂ _{is called}

(

^F X

) =

^F

{

f x

|

x

∈

X

}

for all

∅ 6=

X

⊆

D_;

• strict, if f

⊥ = ⊥

.

Examples:

• f x

=

x

∩

a

∪

b for a, b

⊆

U .

Strictness: f

∅ =

a

∩ ∅ ∪

b

=

b =

∅

whenever b

= ∅

:-(

(63)

→

D₂ _{is called}

(

^F X

) =

^F

{

f x

|

x

∈

X

}

for all

∅ 6=

X

⊆

D_;

• strict, if f

⊥ = ⊥

.

Examples:

• f x

=

x

∩

a

∪

b for a, b

⊆

U .

Strictness: f

∅ =

a

∩ ∅ ∪

b

=

b =

∅

whenever b

= ∅

:-(

Distributivity:

f

(

x₁

∪

x₂

) =

a

∩ (

x₁

∪

x₂

) ∪

b

=

a

∩

x₁

∪

a

∩

x₂

∪

b

= f x₁

∪

f x₂ :-)

(64)

• D₁

=

D₂

=

N

∪ {

^∞

}

, inc x

=

x

+

1

(65)

• D₁

=

D₂

=

N

∪ {

^∞

}

, inc x

=

x

+

1

Strictness: f

⊥ =

^inc 0

=

1 6=

⊥

:-(

(66)

• D₁

=

D₂

=

N

∪ {

^∞

}

, inc x

=

x

+

1

Strictness: f

⊥ =

^inc 0

=

1 6=

⊥

:-(

Distributivity: f

(

^F X

)

= ^F

{

x

+

1

|

x

∈

X

}

for

∅ 6=

X :-)

(67)

• D₁

=

D₂

=

N

∪ {

^∞

}

, inc x

=

x

+

1

Strictness: f

⊥ =

^inc 0

=

1 6=

⊥

:-(

(

^F X

)

= ^F

{

x

+

1

|

x

∈

X

}

for

∅ 6=

X :-)

• D₁

= (

N

∪ {

^∞

})

², D₂

=

N

∪ {

^∞

}

, f

(

x₁, x₂

) =

x₁

+

x₂

(68)

• D₁

=

D₂

=

N

∪ {

^∞

}

, inc x

=

x

+

1

Strictness: f

⊥ =

^inc 0

=

1 6=

⊥

:-(

(

^F X

)

= ^F

{

x

+

1

|

x

∈

X

}

for

∅ 6=

X :-)

• D₁

= (

N

∪ {

^∞

})

², D₂

=

N

∪ {

^∞

}

, f

(

x₁, x₂

) =

x₁

+

x₂ : Strictness: f

⊥ =

0

+

0 = 0 :-)

(69)

• D₁

=

D₂

=

N

∪ {

^∞

}

, inc x

=

x

+

1

Strictness: f

⊥ =

^inc 0

=

1 6=

⊥

:-(

(

^F X

)

= ^F

{

x

+

1

|

x

∈

X

}

for

∅ 6=

X :-)

• D₁

= (

N

∪ {

^∞

})

², D₂

=

N

∪ {

^∞

}

, f

(

x₁, x₂

) =

x₁

+

x₂ : Strictness: f

⊥ =

0

+

0 = 0 :-)

Distributivity:

f

((

1, 4

) ⊔ (

4, 1

)) =

f

(

4, 4

) =

8

6= 5

=

f

(

1, 4

) ⊔

f

(

4, 1

)

:-)