1 / 25

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.

walt
Télécharger la présentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related