1. Chess Review May 11, 2005 Berkeley, CA Discrete-Event Systems: Generalizing Metric Spaces and Fixed-Point Semantics Adam Cataldo Edward Lee Xiaojun Liu Eleftherios Matsikoudis Haiyang Zheng

2. Cataldo, CHESS, May 11, 2005 2 Discrete-Event (DE) Systems Traditional Examples –VHDL –OPNET Modeler –NS-2 Distributed systems –TeaTime protocol in Croquet (two players vs. the computer)

3. Cataldo, CHESS, May 11, 2005 3 Introduction to DE Systems In DE systems, concurrent objects (processes) interact via signals Process Values Time Event Signal

4. Cataldo, CHESS, May 11, 2005 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. Cataldo, CHESS, May 11, 2005 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. Cataldo, CHESS, May 11, 2005 6 Simultaneity in Hardware Simultaneity is common in synchronous circuits Example: This value changes instantly Time

7. Cataldo, CHESS, May 11, 2005 7 Simultaneity in Physical Systems

8. Cataldo, CHESS, May 11, 2005 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. Cataldo, CHESS, May 11, 2005 9 Models of Time Time (real time) Superdense time [Maler, Manna, Pnueli] Time increases in this direction Supertime increases first in this direction (sequence time) then in this direction (real time)

10. Cataldo, CHESS, May 11, 2005 10 Zeno Signals Definition: Zeno Signal infinite events in finite real time Chattering Zeno [Ames] Genuinely Zeno [Ames]

11. Cataldo, CHESS, May 11, 2005 11 Source of Zeno Signals Feedback can cause Zeno Merge Input Signal Output Signal

12. Cataldo, CHESS, May 11, 2005 12 Genuinely Zeno A source of genuinely Zeno signals Input Signal Output Signal Delay By Input Value Merge 1/2

13. Cataldo, CHESS, May 11, 2005 13 Simple Processes Definition: Simple Process Process Non-Zeno Signal Merge is simple, but it has Zeno feedback solutions Merge When are compositions of simple processes simple?

14. Cataldo, CHESS, May 11, 2005 14 Cantor Metric for Signals First time at which the two signals differ 1 “Distance” between two signals 0 t

15. Cataldo, CHESS, May 11, 2005 15 Tetrics: Extending Metric Spaces Cantor metric doesn’t capture simultaneity Tetrics are generalized metrics We generalized metric spaces with “tetric spaces” Our tetric allows us to deal with simultaneity

16. Cataldo, CHESS, May 11, 2005 16 Our Tetric for signals t n First time at which the two signals differ First sequence number at which the two signals differ “Distance” between two signals:

17. Cataldo, CHESS, May 11, 2005 17 Delta Causal Process Definition: Delta Causal Input signals agree up to time t implies output signals agree up to time t + 

18. Cataldo, CHESS, May 11, 2005 18 What Delta Causal Means Delta Causal Process Signals which delay their response to input events by delta will have non-Zeno fixed points

19. Cataldo, CHESS, May 11, 2005 19 Extending Delta Causal The system should be allowed to chatter As long as time eventually advances by delta Delta Causal Process 0 1 2

20. Cataldo, CHESS, May 11, 2005 20 Tetric Delta Causal Process Definition: Tetric Delta Causal 1) Input signals agree up to time (t,n) implies output signals agree up to time (t, n + 1) 2) If n is large enough, this also implies output signals agree up to time (t + ,0)

21. Cataldo, CHESS, May 11, 2005 21 Causal Process Definition: Causal If input signals agree up to supertime (t, n) then the output signals agree up to supertime (t, n)

22. Cataldo, CHESS, May 11, 2005 22 Result 1 Every (extended) delta causal process has a unique feedback solution Delta Causal Process Unique feedback solution

23. Cataldo, CHESS, May 11, 2005 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 Merge Non-Zeno Input Signal Non-Zeno Output Signal Delay

24. Cataldo, CHESS, May 11, 2005 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. Cataldo, CHESS, May 11, 2005 25 Acknowledgements Edward Lee Xiaojun Liu Eleftherios Matsikoudis Haiyang Zheng Aaron Ames Oded Maler Marc Rieffel