Flow Logic
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A flow network is a directed graph in which each edge has a capacity, bounding the amount of flow that can travel  through it. 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. This includes direct applications, namely a search for a maximal flow in networks, as well as less direct applications, like maximal matching or optimal scheduling. 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 {\em 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 {\em flow propositions}, like $> \gamma$ or $\geq \gamma$, for $\gamma \in \N$, 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. For example, the BFL* formula $\EXISTS ((\geq 100) \wedge AG({\it low} \rightarrow (\leq 20))$ states that there is a legal flow function in which the flow is above 100 and in all paths, the amount of flow that travels through vertices with low security is at most 20.

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 {\em 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 $P^NP$, 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.