[PPT] - Decision Procedures An Algorithmic Point of View Linear Arithmetic PowerPoint Presentation

SLIDE 1

Decision Procedures

An Algorithmic Point of View Linear Arithmetic

D. Kroening
O. Strichman

ETH/Technion

Version 1.0, 2007

SLIDE 2

Part V Linear Arithmetic

SLIDE 3

Fourier-Motzkin Variable Elimination

Outline

1 History 2 Linear Arithmetic over the Reals 3 Partitioning and Bounds 4 Complexity

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 3 / 11

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Fourier-Motzkin Variable Elimination Goal: decide satisfiability of conjunction of linear constraints over reals

1≤i≤m
1≤j≤n

ai,jxj ≤ bi

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 4 / 11

SLIDE 5

Fourier-Motzkin Variable Elimination Goal: decide satisfiability of conjunction of linear constraints over reals

1≤i≤m
1≤j≤n

ai,jxj ≤ bi Earliest method for solving linear inequalities Discovered in 1826 by Fourier, re-discovered by Motzkin in 1936

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 4 / 11

SLIDE 6

Fourier-Motzkin Variable Elimination Goal: decide satisfiability of conjunction of linear constraints over reals

1≤i≤m
1≤j≤n

ai,jxj ≤ bi Earliest method for solving linear inequalities Discovered in 1826 by Fourier, re-discovered by Motzkin in 1936 Basic idea of variable elimination:

Pick one variable and eliminate it Continue until all variables but one are eliminated

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 4 / 11

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Linear Arithmetic over the Reals Input: A system of conjoined linear inequalities Ax ≤ b m constraints       a11 a12 · · · · · · a1n a21 a22 ... . . . . . . ... . . . am1 a22 · · · · · · amn             x1 . . . . . . xn       ≤       b1 . . . . . . bn       n variables

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 5 / 11

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Removing unbounded variables Iteratively remove variables that are not bounded in both ways (and all the constraints that use them) The new problem has a solution iff the old problem has one! 8x ≥ 7y x ≥ 3 y ≥ z z ≥ 10 20 ≥ z

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 6 / 11

SLIDE 9

Removing unbounded variables Iteratively remove variables that are not bounded in both ways (and all the constraints that use them) The new problem has a solution iff the old problem has one! 8x ≥ 7y x ≥ 3 y ≥ z z ≥ 10 20 ≥ z

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 6 / 11

SLIDE 10

Removing unbounded variables Iteratively remove variables that are not bounded in both ways (and all the constraints that use them) The new problem has a solution iff the old problem has one! 8x ≥ 7y x ≥ 3 y ≥ z z ≥ 10 20 ≥ z − → y ≥ z z ≥ 10 20 ≥ z

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 6 / 11

SLIDE 11

Removing unbounded variables Iteratively remove variables that are not bounded in both ways (and all the constraints that use them) The new problem has a solution iff the old problem has one! 8x ≥ 7y x ≥ 3 y ≥ z z ≥ 10 20 ≥ z − → y ≥ z z ≥ 10 20 ≥ z

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 6 / 11

SLIDE 12

Removing unbounded variables Iteratively remove variables that are not bounded in both ways (and all the constraints that use them) The new problem has a solution iff the old problem has one! 8x ≥ 7y x ≥ 3 y ≥ z z ≥ 10 20 ≥ z − → y ≥ z z ≥ 10 20 ≥ z − → z ≥ 10 20 ≥ z

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 6 / 11

SLIDE 13

Partitioning the Constraints

1. When eliminating xn, partition the constraints according to the

coefficient ain:

ai,n > 0: upper bound βi ai,n < 0: lower bound βi

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 7 / 11

SLIDE 14

Partitioning the Constraints

1. When eliminating xn, partition the constraints according to the

coefficient ain:

ai,n > 0: upper bound βi ai,n < 0: lower bound βi

n

j=1

ai,j · xj ≤ bi

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 7 / 11

SLIDE 15

Partitioning the Constraints

1. When eliminating xn, partition the constraints according to the

coefficient ain:

ai,n > 0: upper bound βi ai,n < 0: lower bound βi

n

j=1

ai,j · xj ≤ bi ⇒ ai,n · xn ≤ bi −

n−1

j=1

ai,j · xj

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 7 / 11

SLIDE 16

Partitioning the Constraints

1. When eliminating xn, partition the constraints according to the

coefficient ain:

ai,n > 0: upper bound βi ai,n < 0: lower bound βi

n

j=1

ai,j · xj ≤ bi ⇒ ai,n · xn ≤ bi −

n−1

j=1

ai,j · xj ⇒ xn ≤ bi ai,n −

n−1

j=1

ai,j ai,n · xj =: βi

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 7 / 11

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Example for Upper and Lower Bounds

Category?

(1) x1 − x2 ≤ 0 (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1 Assume we eliminate x1.

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 8 / 11

SLIDE 18

Example for Upper and Lower Bounds

Category?

(1) x1 − x2 ≤ 0 Upper bound (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1 Assume we eliminate x1.

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 8 / 11

SLIDE 19

Example for Upper and Lower Bounds

Category?

(1) x1 − x2 ≤ 0 Upper bound (2) x1 − x3 ≤ 0 Upper bound (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1 Assume we eliminate x1.

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 8 / 11

SLIDE 20

Example for Upper and Lower Bounds

Category?

(1) x1 − x2 ≤ 0 Upper bound (2) x1 − x3 ≤ 0 Upper bound (3) −x1 + x2 + 2x3 ≤ 0 Lower bound (4) −x3 ≤ −1 Assume we eliminate x1.

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 8 / 11

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Adding the constraints

2. For each pair of a lower bound al,n < 0 and

upper bound au,n > 0, we have

βl ≤ xn ≤ βu

3. For each such pair, add the constraint

βl ≤ βu

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 9 / 11

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Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 23

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 Upper bound (2) x1 − x3 ≤ 0 Upper bound (3) −x1 + x2 + 2x3 ≤ 0 Lower bound (4) −x3 ≤ −1

we eliminate x1

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 24

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 Upper bound (2) x1 − x3 ≤ 0 Upper bound (3) −x1 + x2 + 2x3 ≤ 0 Lower bound (4) −x3 ≤ −1

we eliminate x1

(5) 2x3 ≤ 0 (from 1,3)

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 25

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 Upper bound (2) x1 − x3 ≤ 0 Upper bound (3) −x1 + x2 + 2x3 ≤ 0 Lower bound (4) −x3 ≤ −1

we eliminate x1

(5) 2x3 ≤ 0 (from 1,3) (6) x2 + x3 ≤ 0 (from 2,3)

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 26

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1

we eliminate x1

(5) 2x3 ≤ 0 (from 1,3) (6) x2 + x3 ≤ 0 (from 2,3)

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 27

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1

we eliminate x1

(5) 2x3 ≤ 0 (from 1,3) (6) x2 + x3 ≤ 0 (from 2,3)

we eliminate x3

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 28

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1 Lower bound

we eliminate x1

(5) 2x3 ≤ 0 (from 1,3) Upper bound (6) x2 + x3 ≤ 0 (from 2,3) Upper bound

we eliminate x3

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

SLIDE 29

Fourier-Motzkin: Example

Category?

(1) x1 − x2 ≤ 0 (2) x1 − x3 ≤ 0 (3) −x1 + x2 + 2x3 ≤ 0 (4) −x3 ≤ −1 Lower bound

we eliminate x1

(5) 2x3 ≤ 0 (from 1,3) Upper bound (6) x2 + x3 ≤ 0 (from 2,3) Upper bound

we eliminate x3

(7) 0 ≤ −1 (from 4,5) → Contradiction (the system is UNSAT)

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 10 / 11

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Complexity Worst-case complexity: m → m2

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 11 / 11

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Complexity Worst-case complexity: m → m2 → (m2)2

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 11 / 11

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Complexity Worst-case complexity: m → m2 → (m2)2 → . . . → m2n

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 11 / 11

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Complexity Worst-case complexity: m → m2 → (m2)2 → . . . → m2n Heavy! So why is it so popular in verification?

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 11 / 11

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Complexity Worst-case complexity: m → m2 → (m2)2 → . . . → m2n Heavy! So why is it so popular in verification? The bottleneck: case-splitting

D. Kroening, O. Strichman (ETH/Technion)

Decision Procedures Version 1.0, 2007 11 / 11