The problem of learning logical rules from examples arises in diverse fields, including program synthesis, logic programming, and machine learning. Existing approaches either involve solving computationally difficult combinatorial problems, or performing parameter estimation in complex statistical models. In this paper, we present DIFFLOG, a technique to extend the logic programming language Datalog to the continuous setting. By attaching real-valued weights to individual rules of a Datalog program, we naturally associate numerical values with individual conclusions of the program. Analogous to the strategy of numerical relaxation in optimization problems, we can now first determine the rule weights which cause the best agreement between the training labels and the induced values of output tuples, and subsequently recover the classical discrete-valued target program from the continuous optimum. We evaluate DIFFLOG on a suite of 34 benchmark problems from recent literature in knowledge discovery, formal verification, and database query-byexample, and demonstrate significant improvements in learning complex programs with recursive rules, invented predicates, and relations of arbitrary arity.

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We propose an interactive approach to resolve static analysis alarms. Our approach synergistically combines a sound but imprecise analysis with precise but unsound heuristics, through user interaction. In each iteration, it solves an optimization problem to find a set of questions for the user such that the expected payoff is maximized. We have implemented our approach in a tool, Ursa, that enables interactive alarm resolution for any analysis specified in the declarative logic programming language Datalog. We demonstrate the effectiveness of Ursa on a state-of-the-art static datarace analysis using a suite of 8 Java programs comprising 41-194 KLOC each. Ursa is able to eliminate 74% of the false alarms per benchmark with an average payoff of 12× per question. Moreover, Ursa prioritizes user effort effectively by posing questions that yield high payoffs earlier.

We propose a general end-to-end deep learning framework Code2Inv, which takes a verification task and a proof checker as input, and automatically learns a valid proof for the verification task by interacting with the given checker. Code2Inv is parameterized with an embedding module and a grammar: the former encodes the verification task into numeric vectors while the latter describes the format of solutions Code2Inv should produce. We demonstrate the flexibility of Code2Inv by means of two small-scale yet expressive instances: a loop invariant synthesizer for C programs, and a Constrained Horn Clause (CHC) solver.