Prediction-hardness of acyclic conjunctive queries

A conjunctive query problem is a problem to determine whether or not a tuple belongs to the answer of a conjunctive query over a database. Here, a tuple, a conjunctive query and a database in relational database theory are regarded as a ground atom, a nonrecursive function-free definite clause and a finite set of ground atoms, respectively, in inductive logic programming terminology. An acyclic conjunctive query problem is a conjunctive query problem with acyclicity. Concerned with the acyclic conjunctive query problem, in this paper, we present the hardness results of predicting acyclic conjunctive queries from an instance with a j-database of which predicate symbol is at most j-ary. Also 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. We mainly show that, from both a simple and an extended instances, acyclic conjunctive queries are not polynomialtime predictable with j-databases (j ≥ 3) under the cryptographic assumptions, and if acyclic conjunctive queries are polynomial-time predictable with 2-databases, then so are DNF formulas. Hence, the acyclic conjunctive queries become a natural example that collapses the equivalence between subsumption-efficiency and efficient pac-learnability from both a simple and an extended instances.