Rahul Sharma Joint work with Aditya Nori (MSR India) and Alex Aiken (Stanford)

x y

Is there a connection?

 Unroll the loops  Find interpolants  Get general proofs (loop invariants)  Get positive and negative examples  Find a classifier  Get a predicate which generalizes

positive examples negative examples

Separators All separators are good candidates for interpolants

Optimal Margin Classifier

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);

x y (0,0) + + (1,1)

 Data is not linearly separable  The candidate interpolant is not an interpolant x y + +

x y (0,0) + + (1,1)(0,1) (1,0)

x y (0,0) + + (1,1) Interpolant!

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

 Interpolants used in tools  BLAST, IMPACT …  Interpolants from proofs  Krajícek[97], Pudlák[97], McMillan[05], …  Interpolants from constraint solving  ARMC, Rybalchenko et al. [07]

 Connect interpolants and classifiers  A sound interpolation procedure  Future work: non-linear interpolants  Integrate with a verification tool  EUF, arrays, bit-vectors, etc.