Containment of Monadic Datalog Programs via Bounded Clique-Width

Containment of monadic datalog programs over data trees (labelled trees with an equivalence relation) is undecidable. Recently, decidability was shown for two incomparable fragments: downward programs, which never move up from visited tree nodes, and linear child-only programs, which have at most one intensional predicate per rule and do not use descendant relation. As different as the fragments are, the decidability proofs hinted at an analogy. As it turns out, the common denominator is admitting bounded clique-width counter-examples to containment. This observation immediately leads to stronger decidability results with more elegant proofs, via decidability of monadic second order logic over structures of bounded clique-width. An argument based on two-way alternating tree automata gives a tighter upper bound for linear child-only programs, closing the complexity gap: the problem is

$$2\textsc {-ExpTime}$$

-complete. As a step towards these goals, complexity of containment over arbitrary structures of bounded clique-width is analysed: satisfiability and containment of monadic programs with stratified negation is in

$$3\textsc {-ExpTime}$$

, and containment of a linear monadic program in a monadic program is in

$$2\textsc {-ExpTime}$$

.