D³: Data-driven disjunctive abstraction

We address the problem of computing an abstraction for a set of examples, which is precise enough to separate them from a set of counterexamples. The challenge is to find an over-approximation of the positive examples that does not represent any negative example. Conjunctive abstractions (e.g., convex numerical domains) and limited disjunctive abstractions, are often insufficient, as even the best such abstraction might include negative examples. One way to improve precision is to consider a general disjunctive abstraction. We present D³, a new algorithm for learning general disjunctive abstractions. Our algorithm is inspired by widely used machine-learning algorithms for obtaining a classifier from positive and negative examples. In contrast to these algorithms which cannot generalize from disjunctions, D³ obtains a disjunctive abstraction that minimizes the number of disjunctions. The result generalizes the positive examples as much as possible without representing any of the negative examples. We demonstrate the value of our algorithm by applying it to the problem of data- driven differential analysis, computing the abstract semantic difference between two programs. Our evaluation shows that D³ can be used to effectively learn precise differences between programs even when the difference requires a disjunctive representation.