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					Discrete-Event Systems:
Generalizing Metric Spaces and Fixed-Point Semantics



Adam Cataldo
Edward Lee
Xiaojun Liu
Eleftherios Matsikoudis
Haiyang Zheng


Chess Review
May 11, 2005
Berkeley, CA
Discrete-Event (DE) Systems

  • Traditional Examples
    – VHDL
    – OPNET Modeler
    – NS-2
  • Distributed systems
    – TeaTime protocol in Croquet




       (two players vs. the computer)   Cataldo, CHESS, May 11, 2005 2
Introduction to DE Systems

   • In DE systems, concurrent objects
     (processes) interact via signals


         Process            Values
                                     Event


Signal             Signal



                                                   Time
         Process



                                     Cataldo, CHESS, May 11, 2005 3
What is the semantics of DE?

  • Simultaneous events may occur in a model
    – VHDL Delta Time




  • Simultaneity absent in traditional
    formalisms
    – Yates
    – Chandy/Misra
    – Zeigler                            Cataldo, CHESS, May 11, 2005 4
Time in Software

  • Traditional programming language
    semantics lack time

  • When a physical system interacts with
    software, how should we model time?

  • One possiblity is to assume some
    computations take zero time, e.g.
    – Synchronous language semantics
    – GIOTTO logical execution time


                                        Cataldo, CHESS, May 11, 2005 5
Simultaneity in Hardware

  • Simultaneity is common in synchronous
    circuits

  • Example:

                                This value changes
                                instantly




                                                         Time

                                     Cataldo, CHESS, May 11, 2005 6
Simultaneity in Physical Systems

                [Biswas]




                                   Cataldo, CHESS, May 11, 2005 7
Our Contributions

  • We generalize DE semantics to handle
    simultaneous events



  • We generalize metric space concepts to
    handle our model of time



  • We give uniqueness conditions and
    conditions for avoidance of Zeno behavior

                                    Cataldo, CHESS, May 11, 2005 8
Models of Time

  • Time (real time)

                                              Time increases in this direction



 • Superdense time [Maler, Manna, Pnueli]

Superdense time increases
first in this direction
(sequence time)




                            then in this direction
                            (real time)
                                                        Cataldo, CHESS, May 11, 2005 9
Zeno Signals

  • Definition: Zeno Signal
    infinite events in finite real time

                          Chattering Zeno [Ames]




                          Genuinely Zeno [Ames]




                                                  Cataldo, CHESS, May 11, 2005 10
Source of Zeno Signals

  • Feedback can cause Zeno



                      Merge   Output Signal

       Input Signal




                                Cataldo, CHESS, May 11, 2005 11
Genuinely Zeno

   • A source of genuinely Zeno signals
                                 Delay
                         1/2    By Input
                                 Value


                 Merge                             Output Signal
 Input Signal




                                       Cataldo, CHESS, May 11, 2005 12
Simple Processes

 • Definition: Simple Process

     Non-Zeno Signal    Process     Non-Zeno Signal



 • Merge is simple, but it has Zeno feedback
   solutions

                       Merge




 • When are compositions of simple processes
   simple?
                                    Cataldo, CHESS, May 11, 2005 13
Cantor Metric for Signals


“Distance” between
    two signals


   1




                            First time at which
       0                    the two signals differ
                 t
                            Cataldo, CHESS, May 11, 2005 14
Tetrics: Extending Metric Spaces

 • Cantor metric doesn’t capture simultaneity

 • We can capture simultaneity with a tetric

 • Tetrics are generalized metrics

 • We generalized metric spaces with “tetric
   spaces”

 • Our tetric allows us to deal with simultaneity

                                      Cataldo, CHESS, May 11, 2005 15
Our Tetric for signals

 “Distance” between two signals:




                             First time at which
     t                       the two signals differ




                                          First sequence number at
                                          which the two signals differ
      n                                               Cataldo, CHESS, May 11, 2005 16
Delta Causal

       Definition: Delta Causal
       Input signals agree up to time t implies
       output signals agree up to time t + D




                        Process




                                          Cataldo, CHESS, May 11, 2005 17
What Delta Causal Means

 • Signals which delay their response to input
   events by delta will have non-Zeno fixed
   points

                        Delta
                       Causal
                       Process




                                     Cataldo, CHESS, May 11, 2005 18
Extending Delta Causal

  • The system should be allowed to chatter
                        Delta
                       Causal
                       Process




                   2
               1
              0


  • As long as time eventually advances by
    delta
                                    Cataldo, CHESS, May 11, 2005 19
Tetric Delta Causal
       Definition: Tetric Delta Causal
       1) Input signals agree up to time (t,n)
       implies output signals agree up to time (t, n + 1)




                        Process




       2) If n is large enough, this also
       implies output signals agree up to time (t + D,0)
                                            Cataldo, CHESS, May 11, 2005 20
Causal

    Definition: Causal
    If input signals agree up to supertime (t, n) then
    the output signals agree up to supertime (t, n)




                        Process




                                            Cataldo, CHESS, May 11, 2005 21
Result 1

  • Every tetric delta causal process has a
    unique feedback solution



                     Delta Causal
                       Process




                Unique feedback solution
                                           Cataldo, CHESS, May 11, 2005 22
Result 2

  • Every network of simple, causal processes
    is a simple causal process, provided in each
    cycle there is a delta causal process
  • Example


                       Delay




                                      Non-Zeno
                        Merge         Output Signal
        Non-Zeno
        Input Signal

                                      Cataldo, CHESS, May 11, 2005 23
Conclusions

  • We broadened DE semantics to handle superdense
    time

  • We invented tetric spaces to measure the
    distance between DE signals

  • We gave conditions under which systems will have
    unique fixed-point solutions

  • We provided sufficient conditions under which
    this solution is non-Zeno

  • http://ptolemy.eecs.berkeley.edu/papers/05/DE_Systems
                                              Cataldo, CHESS, May 11, 2005 24
Acknowledgements

  •   Edward Lee
  •   Xiaojun Liu
  •   Eleftherios Matsikoudis
  •   Haiyang Zheng
  •   Aaron Ames
  •   Oded Maler
  •   Marc Rieffel
  •   Gautam Biswas



                                Cataldo, CHESS, May 11, 2005 25

				
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