[PPT] - Automated Complexity Analysis of Rewrite Systems Florian Frohn RWTH PowerPoint Presentation

SLIDE 1

Automated Complexity Analysis of Rewrite Systems

Florian Frohn

RWTH Aachen University, Germany

December 11, 2018

Automated Complexity Analysis of Rewrite Systems 1

SLIDE 2

Automated Complexity Analysis of Rewrite Systems

Florian Frohn

RWTH Aachen University, Germany

December 11, 2018

Automated Complexity Analysis of Rewrite Systems 1

SLIDE 3

Automated Complexity Analysis of Rewrite Systems

Florian Frohn

RWTH Aachen University, Germany

December 11, 2018

Automated Complexity Analysis of Rewrite Systems 1

SLIDE 4

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 5

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z x loop iterations

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 6

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z x loop iterations 5 operations per iteration

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 7

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z x loop iterations 5 operations per iteration complexity 5 · x + 2

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 8

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z x loop iterations 5 operations per iteration complexity 5 · x + 2 upper bound: 7 · x + 4, ...

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 9

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z x loop iterations 5 operations per iteration complexity 5 · x + 2 upper bound: 7 · x + 4, ... lower bound: x + 1, ...

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 10

What is Complexity Analysis?

input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z x loop iterations 5 operations per iteration complexity 5 · x + 2 upper bound: 7 · x + 4, ... lower bound: x + 1, ... Goal: compute such bounds automatically

Automated Complexity Analysis of Rewrite Systems 2

SLIDE 11

Complexity Analysis with AProVE

AProVE

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 12

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 13

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 14

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph AProVE

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 15

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 16

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System AProVE CoFloCo LoAT . . .

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 17

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System AProVE CoFloCo LoAT . . . Upper Bound

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 18

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System AProVE CoFloCo LoAT . . . Upper Bound Lower Bound

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 19

Complexity Analysis with AProVE

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System AProVE CoFloCo LoAT . . . AProVE CoFloCo LoAT . . . Upper Bound Lower Bound

Automated Complexity Analysis of Rewrite Systems 3

SLIDE 20

What is Rewriting?

Rule-Based Representation of Programs

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 21

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 22

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 23

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 24

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 25

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 26

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 27

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 28

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 29

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 30

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Term Rewrite System (TRS)) length(cons(x, x′)) − → s(length(x′)) length(nil) − → input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 31

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Term Rewrite System (TRS)) length(cons(x, x′)) − → s(length(x′)) length(nil) − → input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

Automated Complexity Analysis of Rewrite Systems

4

SLIDE 32

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Term Rewrite System (TRS)) length(cons(x, x′)) − → s(length(x′)) length(nil) − → input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

1
s(0)

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 33

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Term Rewrite System (TRS)) length(cons(x, x′)) − → s(length(x′)) length(nil) − → input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

1
s(0)

2

s(s(0))

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 34

What is Rewriting?

Rule-Based Representation of Programs Example (Integer Transition System (ITS)) init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Term Rewrite System (TRS)) length(cons(x, x′)) − → s(length(x′)) length(nil) − → input: x, y ∈ N z = 0 while x > 0 do x = x − 1 z = z + y end return z

1
s(0)

2

s(s(0))

. . .

Automated Complexity Analysis of Rewrite Systems 4

SLIDE 35

Upper, Lower, Worst Case, Best Case, ...

input size runtime

Automated Complexity Analysis of Rewrite Systems 5

SLIDE 36

Upper, Lower, Worst Case, Best Case, ...

input size runtime

Automated Complexity Analysis of Rewrite Systems 5

SLIDE 37

Upper, Lower, Worst Case, Best Case, ...

input size runtime

Automated Complexity Analysis of Rewrite Systems 5

SLIDE 38

Upper, Lower, Worst Case, Best Case, ...

input size runtime

Automated Complexity Analysis of Rewrite Systems 5

SLIDE 39

Upper, Lower, Worst Case, Best Case, ...

input size runtime

Automated Complexity Analysis of Rewrite Systems 5

SLIDE 40

Outline

1 Introduction 2 Lower Bounds for ITSs 3 Lower Bounds for TRSs 4 Constant Upper Bounds for TRSs Automated Complexity Analysis of Rewrite Systems 6

SLIDE 41

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 42

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0)

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 43

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0]

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 44

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 45

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2)

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 46

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2]

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 47

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2] | = x > 0

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 48

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2] | = x > 0

5

− → while(0, 2, 4)

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 49

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2] | = x > 0

5

− → while(0, 2, 4) [x/0, y/2, z/4]

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 50

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2] | = x > 0

5

− → while(0, 2, 4) [x/0, y/2, z/4] | = x > 0

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 51

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2] | = x > 0

5

− → while(0, 2, 4) [x/0, y/2, z/4] | = x = 0

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 52

Evaluating ITSs

Example (ITS) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0] while(x, y, z)

1

− → return(z) [x = 0] Example (Evaluation) while(2, 2, 0) [x/2, y/2, z/0] | = x > 0

5

− → while(1, 2, 2) [x/1, y/2, z/2] | = x > 0

5

− → while(0, 2, 4) [x/0, y/2, z/4] | = x = 0

1

− → return(4)

Automated Complexity Analysis of Rewrite Systems 7

SLIDE 53

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 54

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x]

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 55

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 56

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 57

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 58

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations Example (Recurrence Equations) x(n) = x(n−1) − 1 y(n) = y(n−1) z(n) = z(n−1) + y(n−1)

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 59

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations Example (Recurrence Equations) x(n) = x(n−1) − 1 y(n) = y(n−1) z(n) = z(n−1) + y(n−1)

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 60

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations x: maximal number of loop iterations Example (Recurrence Equations) x(n) = x(n−1) − 1 y(n) = y(n−1) z(n) = z(n−1) + y(n−1)

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 61

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations x: maximal number of loop iterations

x 2, x − 1, . . . would be fine, too

Example (Recurrence Equations) x(n) = x(n−1) − 1 y(n) = y(n−1) z(n) = z(n−1) + y(n−1)

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 62

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations x: maximal number of loop iterations

x 2, x − 1, . . . would be fine, too

search for metering function

Example (Recurrence Equations) x(n) = x(n−1) − 1 y(n) = y(n−1) z(n) = z(n−1) + y(n−1)

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 63

Loop Acceleration

Example (Loop Acceleration) while(x, y, z)

5

− → while(x − 1, y, z + y) [x > 0]

while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] 5 · n: cost of n iterations x − n: value of x after n iterations z + y · n: value of z after n iterations x: maximal number of loop iterations

x 2, x − 1, . . . would be fine, too

search for metering function

Example (Recurrence Equations) x(n) = x(n−1) − 1 y(n) = y(n−1) z(n) = z(n−1) + y(n−1)

Automated Complexity Analysis of Rewrite Systems 8

SLIDE 64

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 65

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, z)

5·n

− − → while(x − n, y, z + y · n) [x > 0 ∧ 0 < n ≤ x] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 66

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·n

− − → while(x − n, y, 0 + y · n) [x > 0 ∧ 0 < n ≤ x] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 67

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·n

− − → while(x − n, y, y · n) [x > 0 ∧ 0 < n ≤ x] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 68

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·n

− − → while(x − n, y, y · n) [x > 0 ∧ 0 < n ≤ x] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 69

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(x − x, y, y · x) [x > 0 ∧ 0 < x ≤ x] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 70

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(0, y, y · x) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 71

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(0, y, y · x) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

init(x, y)

5·x+2

− − − → return(y · x) [x > 0 ∧ y ≥ 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 72

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(0, y, y · x) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

init(x, y)

5·x+2

− − − → return(y · x) [x > 0 ∧ y ≥ 0]

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 73

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(0, y, y · x) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

init(x, y)

5·x+2

− − − → return(y · x) [x > 0 ∧ y ≥ 0] whole program compressed into a single rule

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 74

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(0, y, y · x) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

init(x, y)

5·x+2

− − − → return(y · x) [x > 0 ∧ y ≥ 0] whole program compressed into a single rule lower bound 5 · x + 2 for all x > 0, y ≥ 0

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 75

Chaining

Example init(x, y)

1

− → while(x, y, 0) [x ≥ 0 ∧ y ≥ 0] while(x, y, 0)

5·x

− − → while(0, y, y · x) [x > 0] while(x, y, z)

1

− → return(z) [x = 0]

init(x, y)

5·x+2

− − − → return(y · x) [x > 0 ∧ y ≥ 0] whole program compressed into a single rule lower bound 5 · x + 2 for all x > 0, y ≥ 0 Works for arbitrary programs!

Automated Complexity Analysis of Rewrite Systems 9

SLIDE 76

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 77

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs key idea: loop acceleration

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 78

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs key idea: loop acceleration many other features, e.g.:

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 79

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs key idea: loop acceleration many other features, e.g.:

non-determinism

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 80

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs key idea: loop acceleration many other features, e.g.:

non-determinism asymptotic bounds

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 81

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs key idea: loop acceleration many other features, e.g.:

non-determinism asymptotic bounds recursive void functions SMT encodings . . .

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 82

Lower Bounds for ITSs – Wrap-Up

first technique to infer worst case lower bounds for ITSs key idea: loop acceleration many other features, e.g.:

non-determinism asymptotic bounds recursive void functions SMT encodings . . .

LoAT Best Upper Bound rc(n) Ω(1) Ω(n) Ω(n2) Ω(n3) Ω(n4) EXP ∞ O(1) (132) – – – – – – O(n) 37 126 – – – – – O(n2) 8 14 35 – – – – O(n3) 2 – 2 1 – – – O(n4) 1 – – – 2 – – EXP – – – – – 5 – ∞ 53 31 1 – – – 176

Automated Complexity Analysis of Rewrite Systems 10

SLIDE 83

Outline

1 Introduction 2 Lower Bounds for ITSs 3 Lower Bounds for TRSs 4 Constant Upper Bounds for TRSs Automated Complexity Analysis of Rewrite Systems 11

SLIDE 84

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y)))

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 85

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0)

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 86

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0) [y/0]

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 87

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0) [y/0] − → cons(0, inf(s(0)))

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 88

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0) [y/0] − → cons(0, inf(s(0))) [y/s(0)]

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 89

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0) [y/0] − → cons(0, inf(s(0))) [y/s(0)] − → cons(0, cons(s(0), inf(s(s(0)))))

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 90

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0) [y/0] − → cons(0, inf(s(0))) [y/s(0)] − → cons(0, cons(s(0), inf(s(s(0))))) [y/s(s(0))]

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 91

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Example (Evaluation) inf(0) [y/0] − → cons(0, inf(s(0))) [y/s(0)] − → cons(0, cons(s(0), inf(s(s(0))))) [y/s(s(0))] − → . . .

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 92

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Loop

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 93

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(y))) Loop = ⇒ Non-Termination

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 94

Evaluating TRSs

Example (TRS) inf(y) − → cons(y,inf(s(0))) Loop = ⇒ Non-Termination

Automated Complexity Analysis of Rewrite Systems 12

SLIDE 95

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′))

Automated Complexity Analysis of Rewrite Systems 13

SLIDE 96

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′)) Loop

Automated Complexity Analysis of Rewrite Systems 13

SLIDE 97

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′)) Decreasing Loop Loop

Automated Complexity Analysis of Rewrite Systems 13

SLIDE 98

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′)) Decreasing Loop = ⇒ Linear Lower Bound Loop

Automated Complexity Analysis of Rewrite Systems 13

SLIDE 99

Decreasing Loops

Example length(cons(x, x′)) − → s(length(nil)) Decreasing Loop

=

⇒ Linear Lower Bound Loop

Automated Complexity Analysis of Rewrite Systems 13

SLIDE 100

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′)) Decreasing Loop = ⇒ Linear Lower Bound Loop

Automated Complexity Analysis of Rewrite Systems 13

SLIDE 101

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′))

length(cons(x, x′), y)

− → length(x′, s(y))

Automated Complexity Analysis of Rewrite Systems 14

SLIDE 102

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′))

length(cons(x, x′), y)

− → length(x′, s(y)) Decreasing Loops Loops

Automated Complexity Analysis of Rewrite Systems 14

SLIDE 103

Decreasing Loops

Example length(cons(x, x′)) − → s(length(x′))

length(cons(x, x′), y)

− → length(x′, s(y)) Decreasing Loops Revised

Automated Complexity Analysis of Rewrite Systems 14

SLIDE 104

Decreasing Loops – Results

Theorem If a TRS has a decreasing loop, then its complexity is at least linear.

Automated Complexity Analysis of Rewrite Systems 15

SLIDE 105

Decreasing Loops – Results

Theorem If a TRS has a decreasing loop, then its complexity is at least linear. Theorem If a TRS has several compatible decreasing loops, then its complexity is at least exponential.

Automated Complexity Analysis of Rewrite Systems 15

SLIDE 106

Decreasing Loops – Results

Theorem If a TRS has a decreasing loop, then its complexity is at least linear. Theorem If a TRS has several compatible decreasing loops, then its complexity is at least exponential. Example (Compatible Decreasing Loops) fib(s(s(x))) − → plus(fib(s(x)), fib(x))

Automated Complexity Analysis of Rewrite Systems 15

SLIDE 107

Decreasing Loops – Results

Theorem If a TRS has a decreasing loop, then its complexity is at least linear. Theorem If a TRS has several compatible decreasing loops, then its complexity is at least exponential. Theorem Linear lower bounds are not semi-decidable. Example (Compatible Decreasing Loops) fib(s(s(x))) − → plus(fib(s(x)), fib(x))

Automated Complexity Analysis of Rewrite Systems 15

SLIDE 108

Decreasing Loops – Results

Theorem If a TRS has a decreasing loop, then its complexity is at least linear. Theorem If a TRS has several compatible decreasing loops, then its complexity is at least exponential. Theorem Linear lower bounds are not semi-decidable. Lemma Decreasing loops are incomplete. Example (Compatible Decreasing Loops) fib(s(s(x))) − → plus(fib(s(x)), fib(x))

Automated Complexity Analysis of Rewrite Systems 15

SLIDE 109

Lower Bounds for TRSs – Wrap-Up

first technique to infer worst case lower bounds for TRSs

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 110

Lower Bounds for TRSs – Wrap-Up

first technique to infer worst case lower bounds for TRSs key idea: generalize loops

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 111

Lower Bounds for TRSs – Wrap-Up

first technique to infer worst case lower bounds for TRSs key idea: generalize loops linear, exponential, infinite bounds

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 112

Lower Bounds for TRSs – Wrap-Up

first technique to infer worst case lower bounds for TRSs key idea: generalize loops linear, exponential, infinite bounds

quadratic, cubic, . . .?

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 113

Lower Bounds for TRSs – Wrap-Up

first technique to infer worst case lower bounds for TRSs key idea: generalize loops linear, exponential, infinite bounds

quadratic, cubic, . . .? “induction technique”

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 114

Lower Bounds for TRSs – Wrap-Up

first second technique to infer worst case lower bounds for TRSs key idea: generalize loops linear, exponential, infinite bounds

quadratic, cubic, . . .? “induction technique”

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 115

Lower Bounds for TRSs – Wrap-Up

first second technique to infer worst case lower bounds for TRSs key idea: generalize loops linear, exponential, infinite bounds

quadratic, cubic, . . .? “induction technique”

incomplete

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 116

Lower Bounds for TRSs – Wrap-Up

first second technique to infer worst case lower bounds for TRSs key idea: generalize loops linear, exponential, infinite bounds

quadratic, cubic, . . .? “induction technique”

incomplete AProVE Ω(1) Ω(n) EXP ∞ 66 600 143 90

Automated Complexity Analysis of Rewrite Systems 16

SLIDE 117

Outline

1 Introduction 2 Lower Bounds for ITSs 3 Lower Bounds for TRSs 4 Constant Upper Bounds for TRSs Automated Complexity Analysis of Rewrite Systems 17

SLIDE 118

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x)))

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 119

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching.

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 120

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y)

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 121

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y) [y/g(x)]

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 122

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y) [y/g(x)]

g(h(f(x)))

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 123

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y) [y/g(x)]

g(h(f(x)))

[x/a]

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 124

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y) [y/g(x)]

g(h(f(x)))

[x/a]

g(h(g(h(a))))

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 125

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y) [y/g(x)]

g(h(f(x)))

[x/a]

g(h(g(h(a))))

[x/h(a)]

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 126

Narrowing

Example (from TPDB) f(a) − → g(h(a)) h(g(x)) − → g(h(f(x))) Definition (Narrowing) Just like rewriting, but with unification instead of matching. Example (Evaluation) h(y) [y/g(x)]

g(h(f(x)))

[x/a]

g(h(g(h(a))))

[x/h(a)]

g(g(h(f(h(a)))))

Automated Complexity Analysis of Rewrite Systems 18

SLIDE 127

Constant Upper Bounds

Theorem The complexity of a TRS is constant iff narrowing terminates for all terms f(x1, . . . , xn).

Automated Complexity Analysis of Rewrite Systems 19

SLIDE 128

Constant Upper Bounds

Theorem The complexity of a TRS is constant iff narrowing terminates for all terms f(x1, . . . , xn). Lemma Constant complexity is semi-decidable.

Automated Complexity Analysis of Rewrite Systems 19

SLIDE 129

Constant Upper Bounds – Wrap-Up

semi-decision procedure for constant bounds

Automated Complexity Analysis of Rewrite Systems 20

SLIDE 130

Constant Upper Bounds – Wrap-Up

semi-decision procedure for constant bounds key idea: use narrowing

Automated Complexity Analysis of Rewrite Systems 20

SLIDE 131

Constant Upper Bounds – Wrap-Up

semi-decision procedure for constant bounds key idea: use narrowing AProVE can (dis-)prove constant complexity in almost all cases

Automated Complexity Analysis of Rewrite Systems 20

SLIDE 132

Constant Upper Bounds – Wrap-Up

semi-decision procedure for constant bounds key idea: use narrowing AProVE can (dis-)prove constant complexity in almost all cases Evaluation (TPDB – 899 examples) succeeds for all 57 TRSs with constant complexity

Automated Complexity Analysis of Rewrite Systems 20

SLIDE 133

Constant Upper Bounds – Wrap-Up

semi-decision procedure for constant bounds key idea: use narrowing AProVE can (dis-)prove constant complexity in almost all cases Evaluation (TPDB – 899 examples) succeeds for all 57 TRSs with constant complexity solves 6 examples that couldn’t be solved before

Automated Complexity Analysis of Rewrite Systems 20

SLIDE 134

What’s next?

integers and data structures

Automated Complexity Analysis of Rewrite Systems 21

SLIDE 135

What’s next?

integers and data structures floats, arrays, . . .

Automated Complexity Analysis of Rewrite Systems 21

SLIDE 136

What’s next?

integers and data structures floats, arrays, . . . underapproximating program transformations

Automated Complexity Analysis of Rewrite Systems 21

SLIDE 137

What’s next?

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System AProVE CoFloCo LoAT . . . Upper Bound Lower Bound

integers and data structures floats, arrays, . . . underapproximating program transformations

Automated Complexity Analysis of Rewrite Systems 21

SLIDE 138

What’s next?

Program (Java, C, . . .) AProVE SE-Graph AProVE Integer Transition System Term Rewrite System AProVE CoFloCo LoAT . . . Upper Bound Lower Bound

✘

integers and data structures floats, arrays, . . . underapproximating program transformations

Automated Complexity Analysis of Rewrite Systems 21

SLIDE 139

What else?

’15 • Induction Technique for TRSs RTA ’16 • Lower Bounds for ITSs IJCAR ’17 • Decreasing Loops for TRSs JAR ’17 • Upper Bounds for TRSs (full rewriting) LPAR ’17 • Upper Bounds for TRSs (innermost rewriting) FroCoS ’17 • Upper Bounds for Java iFM ’18 • Constant Upper Bounds for TRSs IPL

Automated Complexity Analysis of Rewrite Systems 22

SLIDE 140

What else?

’15 • Induction Technique for TRSs RTA ’16 • Lower Bounds for ITSs IJCAR ’17 • Decreasing Loops for TRSs JAR ’17 • Upper Bounds for TRSs (full rewriting) LPAR ’17 • Upper Bounds for TRSs (innermost rewriting) FroCoS ’17 • Upper Bounds for Java iFM ’18 • Constant Upper Bounds for TRSs IPL

Automated Complexity Analysis of Rewrite Systems 22

SLIDE 141

What else?

’15 • Induction Technique for TRSs RTA ’16 • Lower Bounds for ITSs IJCAR ’17 • Decreasing Loops for TRSs JAR ’17 • Upper Bounds for TRSs (full rewriting) LPAR ’17 • Upper Bounds for TRSs (innermost rewriting) FroCoS ’17 • Upper Bounds for Java iFM ’18 • Constant Upper Bounds for TRSs IPL

Automated Complexity Analysis of Rewrite Systems 22

SLIDE 142

What else?

’15 • Induction Technique for TRSs RTA ’16 • Lower Bounds for ITSs IJCAR ’17 • Decreasing Loops for TRSs JAR ’17 • Upper Bounds for TRSs (full rewriting) LPAR ’17 • Upper Bounds for TRSs (innermost rewriting) FroCoS ’17 • Upper Bounds for Java iFM ’18 • Constant Upper Bounds for TRSs IPL

Termination and Complexity Analysis for C AProVE Tool Papers IJCAR ’14, SEFM ’16, JAR ’17, JLAMP ’18 IJCAR ’14, JAR ’17

Automated Complexity Analysis of Rewrite Systems 22

SLIDE 143

What else?

’15 • Induction Technique for TRSs RTA ’16 • Lower Bounds for ITSs IJCAR ’17 • Decreasing Loops for TRSs JAR ’17 • Upper Bounds for TRSs (full rewriting) LPAR ’17 • Upper Bounds for TRSs (innermost rewriting) FroCoS ’17 • Upper Bounds for Java iFM ’18 • Constant Upper Bounds for TRSs IPL

Termination and Complexity Analysis for C AProVE Tool Papers IJCAR ’14, SEFM ’16, JAR ’17, JLAMP ’18 IJCAR ’14, JAR ’17

Automated Complexity Analysis of Rewrite Systems 22

Thank you!

SLIDE 144

AProVE at TermComp

points of AProVE points of best competitor

ITS 2016 2017 2018 w/ lower bounds 2.09 N/A 2.60 w/o lower bounds 0.66 N/A 0.92 TRS 2014 2015 2016 2017 2018 innermost 1.05 2.73 1.18 N/A 1.79 full – 2.94 1.23 N/A 1.70

Automated Complexity Analysis of Rewrite Systems 23

SLIDE 145

Recurrence Solving

Example (Recurrence Solving) y(v+1) = 1 − y(v) x(v+1) = x(v) + 1 − 3y(v) y(v) =

1 2 − 1 2(−1)v + (−1)vy

x(v) =

3 4 − 3 4(−1)v − 3 2y − 1 2v + 3 2(−1)vy + x

Automated Complexity Analysis of Rewrite Systems 24