On the Monotonicity of (LDL) Logic Programs with Set

LDL is one of the recently proposed logical query languages, which incorporate set, for data and knowledge base systems. Since LDL programs can simulate negation, they are not monotonic in general. On the other hand, there are monotonic LDL programs. This paper addresses the natural question of “When are the generally nonmonotonic LDL programs monotonic?” and investigates related topics such as useful applications for monotonicity. We discuss four kinds of monotonicity, and examine two of them in depth. The first of the two, called “ω-monotonicity”, is shown to be undecidable even when limited to single-stratum programs. The second, called “uniform monotonicity”, is shown to imply ω-monotonicity. We characterize the uniform monotonicity of a program (i) by a relationship between its Bancilhon-Khoshafian semantics and its LDL semantics, and (ii) with a useful property called subset completion independence. Characterization (ii) implies that uniformly monotonic programs can be evaluated more efficiently by discarding dominated facts. Finally, we provide some necessary and/or sufficient, syntactic conditions for uniform monotonicity. The conditions pinpoint (a) enumerated set terms, (b) negations of membership and inclusion, and (c) sharing of set terms as the main source for nonuniform monotonicity