Decision Procedures
An Algorithmic Point of View
Part V
Linear Arithmetic
Linear Arithmetic
D. Kroening
O. Strichman
ETH/Technion
Version 1.0, 2007
Fourier-Motzkin Variable Elimination
Fourier-Motzkin Variable Elimination
Outline
1
Goal: decide satisfiability of conjunction of linear constraints over
reals
^ X
ai,j xj ≤ bi
History
1≤i≤m 1≤j≤n
2
Linear Arithmetic over the Reals
3
Partitioning and Bounds
Discovered in 1826 by Fourier, re-discovered by Motzkin in 1936
4
Complexity
Basic idea of variable elimination:
Earliest method for solving linear inequalities
Pick one variable and eliminate it
Continue until all variables but one are eliminated
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Linear Arithmetic over the Reals
D. Kroening, O. Strichman (ETH/Technion)
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Removing unbounded variables
Iteratively remove variables that are not bounded in both ways
(and all the constraints that use them)
Input: A system of conjoined linear inequalities Ax ≤ b
The new problem has a solution iff the old problem has one!

m constraints
a11
a12 · · ·
.
a22 . .
···

 a21

 ..
..
 .
.
am1 a22 · · · · · ·
n variables
D. Kroening, O. Strichman (ETH/Technion)
Decision Procedures
a1n
..
.
..
.
amn






x1
..
.
..
.
xn


 
 
≤
 
 
b1
..
.
..
.
bn
Version 1.0, 2007






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8x
x
y
z
20
≥
≥
≥
≥
≥
7y
3
z
10
z
D. Kroening, O. Strichman (ETH/Technion)
−→
y ≥ z
z ≥ 10
20 ≥ z
Decision Procedures
−→
z ≥ 10
20 ≥ z
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Partitioning the Constraints
Example for Upper and Lower Bounds
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
X
Category?
(1)
(2)
(3)
(4)
ai,j · xj ≤ bi
j=1
⇒
⇒
ai,n · xn ≤ bi −
xn ≤
bi
−
ai,n
n−1
X
ai,j · xj
j=1
n−1
X ai,j
j=1
D. Kroening, O. Strichman (ETH/Technion)
x1 − x2 ≤ 0
x1 − x3 ≤ 0
−x1 + x2 + 2x3 ≤ 0
−x3 ≤ −1
ai,n
· xj
Upper bound
Upper bound
Lower bound
Assume we eliminate x1 .
=: βi
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Adding the constraints
D. Kroening, O. Strichman (ETH/Technion)
Decision Procedures
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Fourier-Motzkin: Example
Category?
2. For each pair of a lower bound al,n < 0 and
upper bound au,n > 0, we have
βl
≤
xn
≤
βu
(1)
(2)
(3)
(4)
x1 − x2 ≤ 0
x1 − x3 ≤ 0
−x1 + x2 + 2x3 ≤ 0
−x3 ≤ −1
(5)
(6)
2x3 ≤ 0
x2 + x3 ≤ 0
(from 1,3)
(from 2,3)
(7)
0 ≤ −1
(from 4,5)
Upper bound
Upper bound
Lower bound
Lower bound
we eliminate x1
3. For each such pair, add the constraint
Upper bound
Upper bound
we eliminate x3
βl
≤
βu
→ Contradiction (the system is UNSAT)
D. Kroening, O. Strichman (ETH/Technion)
Decision Procedures
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Complexity
Worst-case complexity:
m → m2 → (m2 )2 → . . . → m2
n
Heavy! So why is it so popular in verification?
The bottleneck: case-splitting
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