1. 7th Biennial Ptolemy Miniconference Berkeley, CA February 13, 2007 Correctness in Causal Systems Eleftherios Matsikoudis UC Berkeley

2. Matsikoudis, Berkeley 2Ptolemy Miniconference, February 13, 2007 Causality (Informally) … is the constraint that an effect cannot precede its cause.

3. Matsikoudis, Berkeley 3Ptolemy Miniconference, February 13, 2007 Relevance Modeling and Simulation Synchronous Programming of Reactive Systems Hardware Description

4. Matsikoudis, Berkeley 4Ptolemy Miniconference, February 13, 2007 Correctness ?

5. Matsikoudis, Berkeley 5Ptolemy Miniconference, February 13, 2007 Systems.. s C s SE f A s A s B f B f M s M

6. Matsikoudis, Berkeley 6Ptolemy Miniconference, February 13, 2007 Systems.. s C s SE f A s A s B f B f M s M as Fixed-Point Equations s A = f A ( s M ; s C ) s B = f B ( s A ; s C ) s M = f M ( s B ; s SE )

7. Matsikoudis, Berkeley 7Ptolemy Miniconference, February 13, 2007 Signals T V

8. Matsikoudis, Berkeley 8Ptolemy Miniconference, February 13, 2007 Prefix Order s 1 s 2 v

9. Matsikoudis, Berkeley 9Ptolemy Miniconference, February 13, 2007 Generalized Ultrametric Distance d ( s 1 ; s 2 ) s 1 s 2

10. Matsikoudis, Berkeley 10Ptolemy Miniconference, February 13, 2007 Causal Functions µ s 1 s 2 f ( s 1 ) f ( s 2 )

11. Matsikoudis, Berkeley 11Ptolemy Miniconference, February 13, 2007 Existence of Fixed Points..? f ( s ) d e f = ( fh ¿ ; v ig i f ¿ 6 2 d oms, ; o t h erw i se.

12. Matsikoudis, Berkeley 12Ptolemy Miniconference, February 13, 2007 -Causal Functions R 0 ± ¸ ± s 1 s 2 f ( s 1 ) f ( s 2 )

13. Matsikoudis, Berkeley 13Ptolemy Miniconference, February 13, 2007 Construction of Fixed Points l i m n ! 1 f n ( s )

14. Matsikoudis, Berkeley 14Ptolemy Miniconference, February 13, 2007 Strictly Causal Functions s 1 s 2 f ( s 1 ) f ( s 2 ) ½

15. Matsikoudis, Berkeley 15Ptolemy Miniconference, February 13, 2007 Zeno

16. Matsikoudis, Berkeley 16Ptolemy Miniconference, February 13, 2007 Construction of Fixed Points f ( f ()) f () f () f ( f ()) f ( ; ) f ( f ( ; ))

17. Matsikoudis, Berkeley 17Ptolemy Miniconference, February 13, 2007 Beyond Strict Causality..

18. Matsikoudis, Berkeley 18Ptolemy Miniconference, February 13, 2007 Algebraic Loops y ( t ) = x ( t ) = x 2 ( t ) + u ( t ) K y ( t )

19. Matsikoudis, Berkeley 19Ptolemy Miniconference, February 13, 2007 x 2 ( t ) + u ( t ) K y ( t ) Algebraic Loops y ( t ) = x ( t ) = 1 : 072 0 : 268

20. Matsikoudis, Berkeley 20Ptolemy Miniconference, February 13, 2007 1 : 072 0 : 268 Algebraic Loops y ( t ) = x ( t ) = 14 : 9282 3 : 7321

21. Matsikoudis, Berkeley 21Ptolemy Miniconference, February 13, 2007 … in Simulink

22. Matsikoudis, Berkeley 22Ptolemy Miniconference, February 13, 2007 … in Ptolemy II

23. Matsikoudis, Berkeley 23Ptolemy Miniconference, February 13, 2007 Functions Strictly Causal on Orbits ½ s f ( s ) f ( f ( s ))

24. Matsikoudis, Berkeley 24Ptolemy Miniconference, February 13, 2007 Construction of Fixed Points f ( f ()) f () f () f ( f ()) f ( ; ) f ( f ( ; ))

25. Matsikoudis, Berkeley 25Ptolemy Miniconference, February 13, 2007 Conclusion Proceed with caution..