  We study the interaction of views, queries, and background knowledge in the
form of existential rules. The motivating questions concern monotonic
determinacy of a query using views w.r.t. rules, which refers to the ability to
recover the query answer from the views via a monotone function. We study the
decidability of monotonic determinacy, and compare with variations that require
the ``recovery function'' to be in a well-known monotone query language, such
as conjunctive queries or Datalog. Surprisingly, we find that even in the
presence of basic existential rules, the borderline between well-behaved and
badly-behaved answerability differs radically from the unconstrained case. In
order to understand this boundary, we require new results concerning entailment
problems involving views and rules.
