Given a relational specification between Boolean inputs and outputs, the goal of Boolean functional synthesis is to synthesize each output as a function of the inputs such that the specification is met. In this paper, we first show that unless some hard conjectures in complexity theory are falsified, Boolean functional synthesis must generate large Skolem functions in the worst-case. Given this inherent hardness, what does one do to solve the problem? We present a two-phase algorithm, where the first phase is efficient both in terms of time and size of synthesized functions, and solves a large fraction of benchmarks. To explain this surprisingly good performance, we provide a sufficient condition under which the first phase must produce correct answers. When this condition fails, the second phase builds upon the result of the first phase, possibly requiring exponential time and generating exponential-sized functions in the worst-case. Detailed experimental evaluation shows our algorithm to perform better than other techniques for a large number of benchmarks.
No abstract
Abstract-Complex SQL queries are widely used today, but it is rather difficult to check if a complex query has been written correctly. Formal verification based on comparing a specification with an implementation is not applicable, since SQL queries are essentially a specification without any implementation. Queries are usually checked by running them on sample datasets and checking that the correct result is returned; there is no guarantee that all possible errors are detected.In this paper, we address the problem of test data generation for checking correctness of SQL queries, based on the query mutation approach for modeling errors. Our presentation focuses in particular on a class of join/outer-join mutations, comparison operator mutations, and aggregation operation mutations, which are a common cause of error. To minimize human effort in testing, our techniques generate a test suite containing small and intuitive test datasets. The number of datasets generated, is linear in the size of the query, although the number of mutations in the class we consider is exponential. Under certain assumptions on constraints and query constructs, the test suite we generate is complete for a subclass of mutations that we define, i.e., it kills all non-equivalent mutations in this subclass.
Given a Boolean formula F (X, Y), where X is a vector of outputs and Y is a vector of inputs, the Boolean functional synthesis problem requires us to compute a Skolem function vector Ψ(Y) for X such that F (Ψ(Y), Y) holds whenever ∃X F (X, Y) holds. In this paper, we investigate the relation between the representation of the specification F (X, Y) and the complexity of synthesis. We introduce a new normal form for Boolean formulas, called SynNNF, that guarantees polynomialtime synthesis and also polynomial-time existential quantification for some order of quantification of variables. We show that several normal forms studied in the knowledge compilation literature are subsumed by SynNNF, although SynNNF can be super-polynomially more succinct than them. Motivated by these results, we propose an algorithm to convert a specification in CNF to SynNNF, with the intent of solving the Boolean functional synthesis problem. Experiments with a prototype implementation show that this approach solves several benchmarks beyond the reach of state-of-the-art tools.• We present a new sub-class of negation normal form, called SynNNF, that admits polynomial-time synthesis and quantifier elimination for a set of variables.
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