Safe Stratified Datalog With Integer Order Does not Have Syntax

Stratified Datalog with integer (gap)-order (or Datalog :;!z ) is considered. A Datalog :;!z -program is said to be safe iff its bottom-up processing terminates on all valid inputs. We prove that safe Datalog :;!z - programs do not have effective syntax in the sense that there is no recursively enumerable set S of safe Datalog :;!z -programs such that every safe Datalog :;!z -program is equivalent to a program in S. Introduction We consider databases organized over the domain Z of integer numbers together with the standard linear order. Even if the initial database state is finite, answers to very simple first-order queries may be infinite. For domains that admit elimination of quantifiers in first-order formulas, Kanellakis et al. [5] suggested using finitely representable (f.r.) database states, where extensional database predicates are defined as quantifier-free firstorder formulas. Because of the elimination of quantifiers, answers to firstorder queries in f.r. states ar..