Discrete spacetime: classical causality,prediction, retrodiction and the   mathematical arrow of time
  A mathematical definition of classical causality over discrete spacetime dynamics is formulated. The approach is background free and permits a definition of causality in a precise way whenever the spacetime dynamics permits. It gives a natural meaning to the concepts of cosmic time, spacelike hypersurfaces and timelike or lightlike flows without assuming the notion of a background metric. The concepts of causal propagators and the speed of causality are discussed. In this approach the concepts of spacetime and dynamics are linked in an essential and inseparable whole, with no meaning to either on its own. 
George Jaroszkiewicz (School of Mathematical Sciences, University of   Nottingham, UK)