# Directed branching bisimulation via apartness and positive logic

Branching bisimulation is a relation on states of a labelled transition system, closely connected to Hennessy-Milner Logic with Until (HMLU): two states are branching bisimilar if they validate the same HMLU formulas. In this paper, we introduce a directed notion of branching bisimulation, and show that for a particular class of good, positive HMLU formulas, a state p is directed branching bisimilar to a state q if and only if every good, positive HMLU formula true in p is also true in q. We also introduce a novel logic, Positive Hennessy-Milner Logic with Until (PHMLU), and show that PHMLU formulas can play the role of good, positive HMLU formulas in the above construction. As part of these constructions, we make extensive use of notions of branching apartness, the complement of branching bisimilarity. We introduce a notion of directed branching apartness, and present a novel proof that if two states are distinguished by a HMLU formula then they are branching apart. This proof proceeds by induction on the distinguishing formula.

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