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

Discrete-Event Systems: Generalizing Metric Spaces and Fixed-Point Semantics. Adam Cataldo Edward Lee Xiaojun Liu Eleftherios Matsikoudis Haiyang Zheng. Discrete-Event (DE) Systems. Traditional Examples VHDL OPNET Modeler NS-2 Distributed systems TeaTime protocol in Croquet.

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

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  1. Discrete-Event Systems:Generalizing Metric Spaces and Fixed-Point Semantics Adam Cataldo Edward Lee Xiaojun Liu Eleftherios Matsikoudis Haiyang Zheng

  2. Discrete-Event (DE) Systems • Traditional Examples • VHDL • OPNET Modeler • NS-2 • Distributed systems • TeaTime protocol in Croquet (two players vs. the computer)

  3. Introduction to DE Systems • In DE systems, concurrent objects (processes) interact via signals Process Values Event Signal Signal Time Process

  4. 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

  5. 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

  6. Simultaneity in Hardware • Simultaneity is common in synchronous circuits • Example: This value changes instantly Time

  7. Simultaneity in Physical Systems [Biswas]

  8. 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

  9. 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)

  10. Zeno Signals • Definition: Zeno Signal infinite events in finite real time Chattering Zeno [Ames] Genuinely Zeno [Ames]

  11. Merge Source of Zeno Signals • Feedback can cause Zeno Output Signal Input Signal

  12. Merge Genuinely Zeno • A source of genuinely Zeno signals 1/2 Delay By Input Value Output Signal Input Signal

  13. Merge Simple Processes • Definition: Simple Process Process Non-Zeno Signal Non-Zeno Signal • Merge is simple, but it has Zeno feedback solutions • When are compositions of simple processes simple?

  14. Cantor Metric for Signals “Distance” between two signals 1 First time at which the two signals differ 0 t

  15. 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

  16. Our Tetric for signals “Distance” between two signals: First time at which the two signals differ t First sequence number at which the two signals differ n

  17. Delta Causal Definition: Delta Causal Input signals agree up to timetimplies output signals agree up to timet + Process

  18. What Delta Causal Means • Signals which delay their response to input events by delta will have non-Zeno fixed points Delta Causal Process

  19. Extending Delta Causal • The system should be allowed to chatter Delta Causal Process 2 1 0 • As long as time eventually advances by delta

  20. 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) Ifnis large enough, this also implies output signals agree up to time (t + ,0)

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

  22. Result 1 • Every tetric delta causal process has a unique feedback solution Delta Causal Process Unique feedback solution

  23. 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 Output Signal Merge Non-Zeno Input Signal

  24. 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

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

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