Weak bisimulations for labelled transition systems weighted over semirings

Weighted labelled transition systems are LTSs whose transitions are given weights drawn from a commutative monoid. WLTSs subsume a wide range of LTSs, providing a general notion of strong (weighted) bisimulation. In this paper we extend this framework towards other behavioural equivalences, by considering semirings of weights. Taking advantage of this extra structure, we introduce a general notion of weak weighted bisimulation. We show that weak weighted bisimulation coincides with the usual weak bisimulations in the cases of non-deterministic and fully-probabilistic systems; moreover, it naturally provides a definition of weak bisimulation also for kinds of LTSs where this notion is currently missing (such as, stochastic systems). Finally, we provide a categorical account of the coalgebraic construction of weak weighted bisimulation; this construction points out how to port our approach to other equivalences based on different notion of observability

Lusztig Factorization Dynamics of the Full Kostant-Toda Lattices

We study extensions of the classical Toda lattices at several different space-time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant-Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel Lie algebras. Our study brings into play factorizations of Loewner-Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space-time scales. Along the way we derive a novel, minimal box-ball system for Full Toda that doesn't involve any capacities or colorings, as well as an extension of O'Connell's ODEs to Full Toda