## Section: New Results

### Provenance Circuits

We are interested in obtaining efficiently compact representation of the provenance of a query over a database.

In [28], we generalize three existing graph algorithms to compute the provenance of regular path queries over graph databases, in the framework of provenance semirings – algebraic structures that can capture different forms of provenance. Each algorithm yields a different trade-off between time complexity and generality, as each requires different properties over the semiring. Together, these algorithms cover a large class of semirings used for provenance (top-k, security, etc.). Experimental results suggest these approaches are complementary and practical for various kinds of provenance indications, even on a relatively large transport network.

In [16], we showcase ProvSQL, an open-source module for the PostgreSQL database management system that adds support for computation of provenance and probabilities of query results. A large range of provenance formalisms are supported, including all those captured by provenance semirings, provenance semirings with monus, as well as where-provenance. Probabilistic query evaluation is made possible through the use of knowledge compilation tools, in addition to standard approaches such as enumeration of possible worlds and Monte-Carlo sampling. ProvSQL supports a large subset of non-aggregate SQL queries.

Finally, in [20], [35], we focus on knowledge compilation, which can be used to obtain compact circuit-based representations of (Boolean) provenance. Some width parameters of the circuit, such as bounded treewidth or pathwidth, can be leveraged to convert the circuit to structured classes, e.g., deterministic structured NNFs (d-SDNNFs) or OBDDs. We show how to connect the width of circuits to the size of their structured representation, through upper and lower bounds. For the upper bound, we show how bounded-treewidth circuits can be converted to a d-SDNNF, in time linear in the circuit size. Our bound, unlike existing results, is constructive and only singly exponential in the treewidth. We show a related lower bound on monotone DNF or CNF formulas, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable). Specifically, any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth; and the same holds for pathwidth when compiling to OBDDs. Our lower bounds, in contrast with most previous work, apply to any formula of this class, not just a well-chosen family. Hence, for our language of DNF and CNF, pathwidth and treewidth respectively characterize the efficiency of compiling to OBDDs and (d-)SDNNFs, that is, compilation is singly exponential in the width parameter.