Satisfiability Checking - Fourier–Motzkin Variable Elimination

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Adding the constraints

2. For each pair of a lower bound aln < 0 and

upper bound aun > 0, we have

βl ≤ xn ≤ βu

3. For each such pair, add the constraint

βl ≤ βu

Prof. Dr. Erika Ábrahám - Satisfiability Checking 8 / 10

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Adding the constraints 2. For each pair of a lower bound aln < 0 and upper bound aun > 0, we have βl ≤ xn ≤ βu 3. For each such pair, add the constraint βl ≤ βu Prof. Dr. Erika Ábrahám - Satisfiability Checking 8 / 10

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 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) Prof. Dr. Erika Ábrahám - Satisfiability Checking 9 / 10

Page 1 and 2: Satisfiability Checking Fourier-Mot
Page 3 and 4: Fourier-Motzkin Variable Eliminatio
Page 5 and 6: Removing unbounded variables Iterat
Page 7: Example for Upper and Lower Bounds

Satisfiability Checking - Fourier–Motzkin Variable Elimination

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