Directed path spaces via discrete vector fields

Higher Dimensional Automaton is a cubical complex with two distinguished vertices: the initial and the final state and some labeling of vertices. Higher Dimensional Automata are used to model concurrent programs; possible states (respectively possible executions) of a given program correspond to points (respectively directed paths) of the geometric realization of the underlying cubical complex.
For a given cubical complex K, I will construct a CW-complex which is homotopy equivalent to the space of directed paths on K. Next, I will construct a discrete vector field on this CW-complex and give a description of its critical cells. These provide an efficient method of calculating the homology of spaces of directed paths.