BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-90-1342
       ENTRY:: September 14, 1994
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Modeling concurrency with geometry
        TYPE:: Technical Report
      AUTHOR:: Pratt, Vaughan
        DATE:: November 1990
       PAGES:: 13
    ABSTRACT:: The phenomena of branching time and true or noninterleaving
               concurrency find their respective homes in automata and
               schedules. But these two models of computation are formally
               equivalent via Birkhoff duality, an equivalence we expound on
               here in tutorial detail. So why should these phenomena prefer
               one over the other? We identify dimension as the culprit:
               1-dimensional automata are skeletons permitting only
               interleaving concurrency, whereas rrue n-fold concurrency
               resides in transitions of dimension n. The truly concurrent
               automaton dual to a schedule is not a skeletal distributive
               lattice but a solid one! We introduce true nondeterminism and
               define it as monoidal homotopy; from this perspective
               nondeterminism in ordinary automata arises from forking and
               joining creating nontrivial homotopy. The automaton dual to a
               poset schedule is simply connected whereas that dual to an
               event structure schedule need not be, according to monoidal
               homotopy though not to group homotopy. We conclude with a
               formal definition of higher dimensional automaton as an
               n-complex or n-category, whose two essential axioms are
               associativity of concatenation within dimension and an
               interchange principle between dimensions.
       NOTES:: [Adminitrivia V1/RAM/19940914]
         END:: STAN//CS-TR-90-1342