Interpolants as ClassifiersRahul SharmaJoint work with Aditya Nori (MSR India) and Alex Aiken (Stanford)

Interpolants

If then an interpolant satisfies:

contains only the variables common to and

An interpolant is a simple proof

Quantifier free formulas in linear arithmetic have quantifier free interpolants.

Example

x

y

Binary Classification

Input: labeled examples Positive examples: Negative examples:

Find true,false For all positives, true For all negatives, false

Verification and ML

Interpolant: separates from

Classifier: separates positive examples from negative examples

Is there a connection?

Yes!

Consider the common variables of and :

Interpolant if If :

separates ’s from ’s

Our Contributions

Main result: view interpolants as classifiers as positive example as negative example

Use state-of-the-art classification algorithms (SVMs) for computing interpolants

Encouraging empirical results

Parallels b/w Verification and ML

Unroll the loops Find interpolants Get general proofs (loop invariants)

Get positive and negative examples Find a classifier Get a predicate which generalizes

Support Vector Machine (SVM)

positive examples

negative examples

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+

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+

+

Support Vector Machine (SVM)

Separators

Separators

Separators

Separators

All separators are good candidates for interpolants

+

++

+

+

Support Vector Machine (SVM)

Optimal Margin Classifier+

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+

+

Example

x = y = 0;while(*) x++; y++;while(x != 0) x--; y--;assert (y == 0);

x = y = 0;if(*) x++; y++;p:if(x != 0) x--; y--;if(x == 0) assert (y == 0);

Interpolants by SVM

x

y

(0,0)+

+ (1,1)

Basic Algorithm

Let be the common variables of and

Generate from satisfying assignments of

Generate from satisfying assignments of

Call SVM with and

Return predicate containing

Two Problems

Data is not linearly separable

The candidate interpolant is not an interpolant

x

y

+

+

No separating inequality

For each SVM( )

return

Cannot generate linear separators with both and .

x

y

(0,0)+

+ (1,1)(0,1)

(1,0)

Candidate is not an Interpolant

while(true) { Find candidate interpolant if ( ) Add to and continue;

if ( ) Add to and continue;

break; Exit if interpolant found } return ;

Theorem: terminates iff output is an interpolant between and

Example

x

y

(0,0)+

+ (1,1)

Interpolant!

Evaluation

1000 lines of C++ LIBSVM for SVM queries Z3 theorem prover

Experiments

Related Work

Interpolants used in tools BLAST, IMPACT …

Interpolants from proofs Krajícek[97], Pudlák[97], McMillan[05],

…

Interpolants from constraint solving ARMC, Rybalchenko et al. [07]

Conclusion

Connect interpolants and classifiers

A sound interpolation procedure

Future work: non-linear interpolants

Integrate with a verification tool EUF, arrays, bit-vectors, etc.