Section: New Results

Foundations of AI

Various problems of databases and knowledge bases are closely related to foundational problems in artificial intelligence, since they are rooted in logic or graph theory.

Knowledge Compilation

Many problems in Artificial Intelligence boil down to the exploration of the solution set (called the models) of logical formulas. Such an exploration can be finding one model of the formula, counting the number of models or enumerating them all. However, even for simple quantifier-free formulas, those explorations are known be untractable (NP -hard).

Knowledge compilation encompasses methods that aim to change the representation of the set of models in order to get tractable algorithms for (some of) those tasks. A big computational cost is paid during the compilation time but then replying to queries become tractable on the new representation. More generally, the core of Knowledge compilation is the study of the trade-off between the size of the representation and the easiness of queries. This subject is of interest for both Artificial Intelligence and Database communities.

At STACS [15], Capelli, in cooperation with Mengel from CRIL (Lens), studied knowledge compilation techiques for quantified Boolean formulas. Deciding the existence of models for such formulas is known to climb arbitrarly high the polynomial time hierarchy. The authors provide an efficient compilation procedure for formulas having a bounded tree-width generalizing results from SAT solving.

Aggregation and Enumeration for Graphs

Aggregation and enumeration are not relevant for answer sets of database queries but equally for any kinds of sets, most typically defined by combinatoric problems on graphs.

In a paper published at ICALP [8], Paperman proposed (in cooperation with Amarilli from Telecom Paristech) to study the problem of finding so called topological sort satisfying constraints provided by regular expressions. Searching topological sort happens typically in situations where an order is uncertain. For instance, in relational database where users provides a partial preference order, or in concurrent and distributed programming where some tasks can be executed in an arbitrary order. A classical task in preferential query answering is to find a topological sort satisfying some global constrained. Typically, to find a total order satisfying all (or most) of the customers. The paper provides and proves sufficient conditions on the shape of the constraints to make the problem tractable (P-time) as well as sufficient condition to make the problem $\mathrm{𝙽𝙿}$-hard. They also prove a complete dichotomy for an adapted and well chosen version of the constrained topological sort problem.

In an article in JCSS [2], Capelli (with Bergougnoux and Kanté from Bordeaux and Clérmont-Ferrand) propose an algorithm for counting the number of transversal in some hypergraphs. Here, a hypergraph is a collection of sets – called hyperedges over a ground set and a traversal is a subset intersecting all hyperedges. In full generality, counting the number of minimal traversals in a hypergraph is a hard problem: it is known to be $\#P$-complete. They proved that under the assumptions of $\beta$-acyclicity, it is possible to count all the minimal traversals can be done in polynomial times.