YAI for Linear Inequalities over Integers

1. YAI for Linear Equalities and Inequalities over Integers Christopher Lynch and Yuefeng Tang Department of Mathematics and Computer Science Clarkson University Presenter: Jen-Chi Lin

2. Abstract • Given two inconsistent sets of integer inequalities A and B, an formula I can be computed from - 1. A contradictory equation - 2. Bounds - 3. Gomory cuts. • I satisfies only the first two conditions of interpolant: - 1. A implies I - 2. I and B are also inconsistent

3. Preliminaries (I)

4. Preliminaries (II) • Atom : a single variable • Term : a constant or an atom or cx where c is a constant and x is a variable • Expression : a summation of terms • Active bound : Let axbe a term in the expression t K : a minimal set of bounds

5. Preliminaries (II)

6. Quick overview for DPLL Success updating No All bounds are asserted? Yes Bound violation? Assert a bound Pivot and Update Start Fail Yes No UNSAT Generate a model UNSAT equation

7. YAI for Linear Equalities and Inequalities over Integers • Note : The method is incomplete since the based linear arithmetic solver is incomplete.

8. YAI for Linear Equalities and Inequalities over Integers

9. YAI for Linear Equalities and Inequalities over Integers

10. YAI for Linear Equalities and Inequalities over Integers

11. YAI for Linear Equalities and Inequalities over Integers • Ex: { 0 0 0 0 } { 0 0 0 0 } s1=0, y≧0, y≦1 initialize z =1 { -1 0 0 1 } Contradict to s1=0

12. YAI for Linear Equalities and Inequalities over Integers { -1 0 0 1 } { 0 1/5 0 1 } Swap s1 and x All bounds are asserted! Since the assignment of x is rational, which can be derived from 5x = y+z+s1 with y ≥ 0, z ≥ 1 and s1 = 0, a Gomory cut y+z ≥ 5 is generated from that equation with its bounds. Flattened again : { s1 = 5x – y - z, s2 = y + z } with bounds { s1 = 0, s2 ≥ 5, y ≥ 0, y ≤ 1, z ≥ 1, z ≤ 2 } cut_info(s2) = { z = 1 }

13. YAI for Linear Equalities and Inequalities over Integers initialize s1 s2 x y z Contradict to s2 ≥ 5 { 0 0 0 0 0 } Swap s2 and y : y = s2 - z s1 s2 x y z Contradict to s1 = 0 { -5 5 0 5 0 } Swap s1 and x : 5x = s1+y+z Contradict to y ≤ 1 s1 s2 x y z s1 s2 x y z { 0 5 1 5 0 } { 0 5 1 1 4 } Swap y and z : z = s2-y Contradict to y ≤ 1

14. YAI for Linear Equalities and Inequalities over Integers Contradict to y ≤ 1 s1 s2 x y z { 0 5 1 1 4 } To decrease z (where z = s2-y), it needs to decrease s2 or increase y. Since s2 has reached its minimum and y has reached to its maximum, it’s impossible to make z < 4. i.e. z = s2-y is an UNSAT Equation with z ≤ 2, y ≤ 1, s2 ≥ 5

15. YAI for Linear Equalities and Inequalities over Integers • Observation

16. Thanks for your attention!