Title: Intuitionistic Layered Graph Logic: Semantics and Proof Theory,
Abstract: Models of complex systems are widely used in the physical and social
sciences, and the concept of layering, typically building upon graph-theoretic
structure, is a common feature. We describe an intuitionistic substructural
logic called ILGL that gives an account of layering. The logic is a bunched
system, combining the usual intuitionistic connectives, together with a
non-commutative, non-associative conjunction (used to capture layering) and its
associated implications. We give soundness and completeness theorems for a
labelled tableaux system with respect to a Kripke semantics on graphs. We then
give an equivalent relational semantics, itself proven equivalent to an
algebraic semantics via a representation theorem. We utilise this result in two
ways. First, we prove decidability of the logic by showing the finite
embeddability property holds for the algebraic semantics. Second, we prove a
Stone-type duality theorem for the logic. By introducing the notions of ILGL
hyperdoctrine and indexed layered frame we are able to extend this result to a
predicate version of the logic and prove soundness and completeness theorems
for an extension of the layered graph semantics . We indicate the utility of
predicate ILGL with a resource-labelled bigraph model.