  We study solutions to systems of stream inclusions of the form 'f in T(f)',
where the nondeterministic transformer 'T' on omega-infinite streams is assumed
to be causal in the sense that elements in output streams are determined by a
finite prefix of inputs. We first establish a correspondence between
logic-based causality and metric-based contraction. Based on this
causality-contraction connection we then apply fixpoint principles to the
spherically complete ultrametric space of streams to construct solutions of
stream inclusions. The underlying fixpoint iterations induce fixpoint induction
principles to reason about these solutions.In addition, the fixpoint
approximation provides an anytime algorithm with which finite prefixes of
solutions can be calculated. These developments are illustrated for some
central concepts of system design.
