0. Efficient Craig Interpolation For Subsets of Integer Linear Arithmetic
Himanshu Jain, CMU
Edmund M. Clarke, CMU
Orna Grumberg, Technion

1. Interpolants [Craig 1957] G(y,z) F(x,y)
Given formulas F, G such that F Æ G is unsatisfiable
An interpolant for (F,G) is a formula I:
1. F ) I
2. I Æ G is unsatisfiable
3. I contains only common variables of F and G
I(y)
G(y,z)
F(x,y)

2. Interpolants Example Example 1 (propositional logic):
F := p Æ q G := : q Æ r Æ s
I := q
Example 2 (linear arithmetic):
F := x + 2y ≤ 3 Æ –x - y ≤ -1
G := y ≥ 3
F ) y ≤ 2
I := y ≤ 2

3. Interpolants in Verification [McMillan 2003]
Useful in symbolic model checking
Interpolant
based image
Reach1(S) S
Computing Reach1(S) requires existential
quantification (costly using BDDs or SAT)

4. Interpolants in Verification [Jhala et al. 2004]
Useful for Property Directed Invariant Generation
Program P
Predicate
Abstraction
Invariants for
P expressible
in terms of S
Predicates S
Interpolants help in finding right set of predicates

5. How are Interpolants Obtained
proof of
unsatisfiability
of F Æ G
F Æ G
F, G
Interpolant for (F, G)

6. Existing Work on Computing Interpolants Pudlak, McMillan, Jhala et al
Existing Work on Computing Interpolants Pudlak, McMillan, Jhala et al., Yorsh et al., Kapur et al., Rybalchenko et al., Kroening et al., Cimatti et al., Beyer et al.
Can efficiently compute interpolants
For rational/real linear arithmetic
For equality with uninterpreted function symbols
Propositional logic (using SAT solvers)
No efficient interpolation algorithms for
Integer linear arithmetic
Bit-vector arithmetic
Decision problem for conjunctions is itself NP-hard
We make progress in this direction.

7. Difference between rational and integer linear arithmetic
Let H := x=2y Æ x=2z+1
If x, y, z are rational variables
H is satisfiable (take x=1,y=1/2, z=0)
If x, y, z are integer variables
H is unsatisfiable

8. Our results Polynomial time interpolation algorithms
For useful subsets of integer linear arithmetic
Integer (Diophantine) linear equations
E.g. x = 3y Æ 5x = 3z+u+2 Æ …
Integer linear congruences (modular equations)
E.g. 4x = 2y + 9 (mod 3) Æ 2z + 5x –y = 7 (mod 4) Æ …
Integer linear equations and disequations
E.g. : (4x + 5y = 8) Æ x = 3y Æ …

9. Outline Introduction Integer Linear Equations
Craig Interpolation
Related Work
Integer Linear Equations
Integer Linear Congruences
Integer Linear Equations+Disequations
Experimental results
New interpolation algorithms
We will only give intuition and examples in the talk. See paper for precise description of results.

10. Interpolation for Integer Linear Equations
F, G be conjuctions of integer linear equations
We show that interpolant for (F,G) is always:
An integer linear equation or
An integer linear congruence
F := (x = 2y) and G := (x=2z+1)
An interpolant is x = 0 (mod 2)

11. Interpolation Algorithm Step 1
Obtain a proof of unsatisfiability of F Æ G
(How to get a contradiction from F Æ G)
F := (30 x + 4y = 2)
G := ( y + 5z = 2)
[1/5, 1/5]
1/5 F + 1/5 G is equal to
6x+y+z=4/5 (Contradiction)

12. Interpolation Algorithm Step 2
Sum the equations from F according to the proof of unsatisfiability
F G
1/5 (30 x + 4y = 2) + 1/5 (y + 5z = 2)
6 x + 4/5 y = 2/5
Partial interpolant
We do not want x

13. Interpolation Algorithm Step 3
Remove variables not common to F and G
6 x + 4/5 y = 2/5
4/5 y -2/5 = -6x
) 4/5 y – 2/5 is divisible by 6
) 4/5 y – 2/5 = 0 (mod 6)
) 4y-2=0 (mod 30)
4y - 2 = 0 (mod 30) is an interpolant for (F, G)
We have proved the correctness of above algorithm

14. Complexity of the Algorithm
Obtain proof of unsatisfiability (step 1)
Polynomial time using Hermite Normal Form
Overall algorithm is polynomial time
Can also use modern SMT solvers
Multiple interpolants can be obtained

15. Multiple Interpolants
G := y+5z=2
F := 30x+4y=2
4y – 2 = 0 (mod 10)
4y – 2 = 0 (mod 30)
4y – 2 = 0 (mod 15)
4y – 2 = 0 (mod 5)

16. Outline Introduction Integer Linear Equations
Craig Interpolation
Related Work
Integer Linear Equations
Integer Linear Congruences
Integer Linear Equations+Disequations
Experimental results
New interpolation algorithms

17. Integer Linear Congruences
a = b (mod m) iff m divides (a-b)
a, b, m can be rational numbers
Integer Linear Congruence: i ai xi = b (mod m)
xi are integer variables
Example: 3x+2y+5z = 0 (mod 6)
SATISFIABLE (x=2,y=0,z=0)

18. Interpolation for Integer Linear Congruences
F, G be conjuctions of integer linear congruences
We show that interpolant for (F,G) is always:
An integer linear congruence
Basic steps same as before
Proof of unsatisfiability is more interesting

19. Proof of Unsatisfiability
Congruences may not hold with rational multipliers
9 = 5 (mod 2). But 9/4 ≠ 5/4 (mod 2)
We show get a proof of unsatisfiability
With integer multipliers for equations
Congruence hold with integer multipliers

20. Proof of Unsatisfiability for Congruences
2 (2x + 2y = 4) (mod 8) +
-4 (2x + y = 4) (mod 8) +
1 (4x = 4) (mod 8)
0 = -4 (mod 8)
2x + 2y = 4 (mod 8) Æ
2x + y = 4 (mod 8) Æ
4x = 4 (mod 8)
Both proofs of unsatisfiability and (multiple) interpolants
can be obtained in polynomial time

21. Outline Introduction Integer Linear Equations
Craig Interpolation
Related Work
Integer Linear Equations
Integer Linear Congruences
Integer Linear Equations+Disequations
Experimental results
New interpolation algorithms

22. Integer Linear Equations + Disequations
Example: (x+2y+z=1) Æ :(x=1) Æ …
All integer variables
Let F = Feq Æ Fneq
We show F has no integral solution iff
F has no rational solution, OR
Feq has no integral solution

23. Interpolation for Integer Linear Equations + Disequations
Given F = Feq Æ Fneq , G = Geq Æ Gneq , F Æ G is unsat
F Æ G has no rational solution
Interpolant as integer linear eqn/disequation
Feq Æ Geq has no integral solution
Interpolant as integer linear eqn/congruence

24. Outline Introduction Integer Linear Equations
Craig Interpolation
Related Work
Integer Linear Equations
Integer Linear Congruences
Integer Linear Equations+Disequations
Experimental results
New interpolation algorithms

25. Predicate Discovery Loop invariant: x+y is divisible by 4
void main() {
int x=0, y=0;
while(*)
x = x + 4*nondet();
y = y + 8*nondet();
assert(x+y != 1);
assert(x+y != 2);
assert(x+y != 3); }
Loop invariant:
x+y is divisible by 4
That is, x+y=0 (mod 4)
Such predicates can be found using our interpolation algorithms
C program

26. Predicate Discovery Experiments
Existing state-of-the-art tools such as
BLAST, SATABS, VCEGAR cannot verify
these programs.
With the help of predicates found by
our algorithms they can (VCEGAR).
Predicate Discovery Experiments
Example
Predicates/Interpolants
Time (secs)
ex1
y = 1 (mod 2)
2.72
ex2
x + y = 0 (mod 2)
0.83
ex4
x + y + z = 0 (mod 4)
0.95
ex5
x=0 (mod 4), y=0 (mod 4)
1.1
ex6
4x+2y+z = 0 (mod 8)
0.93
ex7
4x-2y+z = 0 (mod 222)
0.54
forb1
x + y = 0 (mod 3)
0.1

27. Conclusion Efficient Interpolation Algorithms Easy to implement
Integer linear equations
Integer linear congruences
Integer linear equations and disequations
Easy to implement
Proofs of unsatisfiability

28. Future Work Full integer linear arithmetic Bit-vector arithmetic
Cutting-plane proofs/Pudlak’s algorithm
Bit-vector arithmetic
Boolean Combinations using SMT

29. Questions