Efficient Craig Interpolation For Subsets of Integer Linear Arithmetic

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Efficient Craig Interpolation For Subsets of
Integer Linear Arithmetic

• Himanshu Jain, CMU
• Edmund M. Clarke, CMU
• Orna Grumberg, Technion

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Interpolants Craig 1957
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)
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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
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Interpolants in Verification McMillan 2003

• Useful in symbolic model checking

Interpolant based image
S
Computing Reach1(S) requires existential
quantification (costly using BDDs or SAT)
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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
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How are Interpolants Obtained
proof of unsatisfiability of F Æ G
F Æ G
F, G
Interpolant for (F, G)
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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.
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Difference between rational and integer linear
arithmetic

• Let H x2y Æ x2z1
• If x, y, z are rational variables
• H is satisfiable (take x1,y1/2, z0)
• If x, y, z are integer variables
• H is unsatisfiable

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Our results

• Polynomial time interpolation algorithms
• For useful subsets of integer linear arithmetic
• Integer (Diophantine) linear equations
• E.g. x 3y Æ 5x 3zu2 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

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Outline

• Introduction
• Craig Interpolation
• Related Work
• Integer Linear Equations
• Integer Linear Congruences
• Integer Linear EquationsDisequations
• Experimental results

New interpolation algorithms
We will only give intuition and examples in the
talk. See paper for precise description of
results.
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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 (x2z1)
• An interpolant is x 0 (mod 2)

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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
6xyz4/5 (Contradiction)
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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
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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-20 (mod 30)
4y - 2 0 (mod 30) is an interpolant for (F,
G) We have proved the correctness of above
algorithm
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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

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Multiple Interpolants
G y5z2
F 30x4y2
4y 2 0 (mod 10)
4y 2 0 (mod 30)
4y 2 0 (mod 15)
4y 2 0 (mod 5)
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Outline

• Introduction
• Craig Interpolation
• Related Work
• Integer Linear Equations
• Integer Linear Congruences
• Integer Linear EquationsDisequations
• Experimental results

New interpolation algorithms
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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 3x2y5z 0 (mod 6)
• SATISFIABLE (x2,y0,z0)

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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

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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

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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
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Outline

• Introduction
• Craig Interpolation
• Related Work
• Integer Linear Equations
• Integer Linear Congruences
• Integer Linear EquationsDisequations
• Experimental results

New interpolation algorithms
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Integer Linear Equations Disequations

• Example (x2yz1) Æ (x1) 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

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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

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Outline

• Introduction
• Craig Interpolation
• Related Work
• Integer Linear Equations
• Integer Linear Congruences
• Integer Linear EquationsDisequations
• Experimental results

New interpolation algorithms
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Predicate Discovery
void main() int x0, y0 while()
x x 4nondet() y y 8nondet()
assert(xy ! 1) assert(xy ! 2)
assert(xy ! 3)
Loop invariant xy is divisible by 4 That is,
xy0 (mod 4)
Such predicates can be found using our
interpolation algorithms
C program
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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).
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 x0 (mod 4), y0 (mod 4) 1.1
ex6 4x2yz 0 (mod 8) 0.93
ex7 4x-2yz 0 (mod 222) 0.54
forb1 x y 0 (mod 3) 0.1
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Conclusion

• Efficient Interpolation Algorithms
• Integer linear equations
• Integer linear congruences
• Integer linear equations and disequations
• Easy to implement
• Proofs of unsatisfiability
• Integer linear congruences
• Integer linear equations and disequations

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Future Work

• Full integer linear arithmetic
• Cutting-plane proofs/Pudlaks algorithm
• Bit-vector arithmetic
• Boolean Combinations using SMT

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Questions