On the Relationship of Joint Acyclicity and Super-Weak Acyclicity

Rules that allow existential quantifiers in the head (conclusion) have been studied extensively in knowledge representation, databases, and logic programming (in the related form of logic programs with function symbols). In general, it is not decidable if a set of such rules admits a finite model, but a number of sufficient conditions have been proposed to detect situations where this is the case. Two such acyclicity conditions are joint acycliclity [8] and super-weak acyclicity [9]. This short technical note investigates the relationship between the two conditions. The main result is that, while super-weak acyclcity is a more general condition, the two conditions coincide on all sets of rules that do not contain multiple occurrences of the same variable in any body atom (Theorem 1). We also observe that this restriction does not reduce the expressivity of jointly acyclic rules (Theorem 2). We first introduce our basic notation (Section 2) and recall the idea of acyclicity using the case of weak acyclicity (Section 3). Thereafter, we introduce joint acyclicity (Section 4) and conmpare it with super-weak acyclicity (Section 5).