Bounded Implication for Existential Rules (Extended Abstract)

The problem This paper deals with the property of boundedness in rule languages. Boundedness is an important notion that formalizes the fact that a rule set Σ can be unfolded into a finite set Σ of acyclic (i.e., non-recursive) rules such that Σ and Σ are equivalent on every database: it is therefore a crucial property for optimizing the processing of rules. Such a property has been extensively studied, especially for the Datalog rule language [9,4], and, recently, for Answer Set Programming [16]. In Datalog, the (uniform) boundedness of a program P can be defined as the existence of an integer k such that, for every database D, the number of iterated applications (in a forward chaining manner) of P to D that are necessary to compute the minimal model of P and D is bounded by k. This definition of boundedness is equivalent to the existence of a finite, non-recursive program that is equivalent to P. Also, it is well-known that a Datalog query is bounded if and only if it is equivalent to a first-order sentence [1,12]. More recently, rule-based languages have been used in the context of ontology-based data access [15]. In this framework, the main focus is on the problem of answering conjunctive queries over an ontology expressed by a set of rules, and one of the most studied properties is the first-order rewritability of conjunctive queries (CQFO-rewritability) over an ontology, which corresponds to the above mentioned first-order expressibility in Datalog: an ontology O is CQFO-rewritable if every conjunctive query q over the ontology can be equivalently rewritten into a first-order query q , i.e., q is such that, for every database instance D, the evaluation of q over O and D coincides with the evaluation of q over D. Notably, in the case when the ontology is expressed as a set P of Datalog rules, the CQFO-rewritability of P and the boundedness of P are equivalent properties. Existential rules, which extend Datalog rules to the presence of existentially quantified variables and multiple atoms in rule heads, have been proposed and studied in the last years as a specification language for ontology-based data access [5,2,14]. An exis-tential rule (or simply rule) σ over a relational schema S is an expression of the form ∀x∀y(Φ(x, y, ¯ a) → ∃zΨ (x, z, ¯ b)), where Φ(x, y, ¯ a) (the body of σ) and Ψ (x, z, …