Bifurcation Logic: Separation Through Ordering
Didier Galmiche, Timo Lang, Daniel Méry, David Pym
TL;DR
BL addresses the challenge of integrating modal and substructural reasoning by defining a bifurcation-based semantics for $A * B$ and its implication, built on a modal ordering. The main approach includes a relational, Routley-Meyer–style semantics with a ternary relation $RelS$ over a tree of worlds and a labelled tableaux calculus, with soundness and completeness results. A key finding is that the standard finite model property fails for BL, but a finite representation via back-links suffices to establish decidability when wand is absent and $*$ is present. The work demonstrates a generic multi-agent access-control modelling example, highlighting BL's applicability to knowledge representation and reasoning in distributed systems.
Abstract
We introduce Bifurcation Logic, BL, which combines a basic classical modality with separating conjunction * together with its naturally associated multiplicative implication, that is defined using the modal ordering. Specifically, a formula A*B is true at a world w if and only if each of A,B holds at worlds that are each above w, on separate branches of the order, and have no common upper bound. We provide a labelled tableaux calculus for BL and establish soundness and completeness relative to its relational semantics. The standard finite model property fails for BL. However, we show that, in the absence of multiplicative implication, but in the presence of *, every model has an equivalent finite representation and that this is sufficient to obtain decidability. We illustrate the use of BL through an example of modelling multi-agent access control that is quite generic in its form, suggesting many applications.