Flow networks have attracted a lot of research in computer science. Indeed,
many questions in numerous application areas can be reduced to questions about
flow networks. Many of these applications would benefit from a framework in
which one can formally reason about properties of flow networks that go beyond
their maximal flow. We introduce Flow Logics: modal logics that treat flow
functions as explicit first-order objects and enable the specification of rich
properties of flow networks. The syntax of our logic BFL* (Branching Flow
Logic) is similar to the syntax of the temporal logic CTL*, except that atomic
assertions may be flow propositions, like >γ or ≥γ, for
γ∈N, which refer to the value of the flow in a vertex, and
that first-order quantification can be applied both to paths and to flow
functions. We present an exhaustive study of the theoretical and practical
aspects of BFL*, as well as extensions and fragments of it. Our extensions
include flow quantifications that range over non-integral flow functions or
over maximal flow functions, path quantification that ranges over paths along
which non-zero flow travels, past operators, and first-order quantification of
flow values. We focus on the model-checking problem and show that it is
PSPACE-complete, as it is for CTL*. Handling of flow quantifiers, however,
increases the complexity in terms of the network to PNP, even
for the LFL and BFL fragments, which are the flow-counterparts of LTL and CTL.
We are still able to point to a useful fragment of BFL* for which the
model-checking problem can be solved in polynomial time. Finally, we introduce
and study the query-checking problem for BFL*, where under-specified BFL*
formulas are used for network exploration