On the Hardness of Learning Acyclic Conjunctive Queries

A conjunctive query problem in relational database theory is a problem to determine whether or not a tuple belongs to the answer of a conjunctive query over a database. Here, a tuple and a conjunctive query are regarded as a ground atom and a nonrecursive function-free definite clause, respectively. While the conjunctive query problem is NP-complete in general, it becomes efficiently solvable if a conjunctive query is acyclic. Concerned with this problem, we investigate the learnability of acyclic conjunctive queries from an instance with a j-database which is a finite set of ground unit clauses containing at most j-ary predicate symbols. We deal with two kinds of instances, a simple instance as a set of ground atoms and an extended instance as a set of pairs of a ground atom and a description. Then, we show that, for each j ≥ 3, there exist a j-database such that acyclic conjunctive queries are not polynomially predictable from an extended instance under the cryptographic assumptions. Also we show that, for each n > 0 and a polynomial p, there exists a p(n)database of size O(2) such that predicting Boolean formulae of size p(n) over n variables reduces to predicting acyclic conjunctive queries from a simple instance. This result implies that, if we can ignore the size of a database, then acyclic conjunctive queries are not polynomially predictable from a simple instance under the cryptographic assumptions. Finally, we show that, if either j = 1, or j = 2 and the number of element of a database is at most l (≥ 0), then acyclic conjunctive queries are paclearnable from a simple instance with j-databases.