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Correctness in Causal Systems

Correctness in Causal Systems. Eleftherios Matsikoudis UC Berkeley. Causality (Informally). … is the constraint that an effect cannot precede its cause. Relevance. Modeling and Simulation Synchronous Programming of Reactive Systems Hardware Description. Correctness. ?. f. f.

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Correctness in Causal Systems

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  1. Correctness in Causal Systems Eleftherios Matsikoudis UC Berkeley

  2. Causality (Informally) … is the constraint that an effectcannot precede its cause.

  3. Relevance • Modeling and Simulation • Synchronous Programming of Reactive Systems • Hardware Description

  4. Correctness ?

  5. f f f s s s s s A C B M S A B M E Systems..

  6. f f f s s s s s C S M A B A M B E ( ) f s s s = A A M C ; ( ) f s s s = B B A C ; ( ) f s s s = M M B S E ; as Fixed-Point Equations Systems..

  7. Signals V T

  8. v s s 1 2 Prefix Order

  9. ( ) d s s s s 1 2 1 2 ; Generalized Ultrametric Distance

  10. ( ( ) ) f f µ s s s s 1 2 1 2 Causal Functions

  11. ( f h i g f 6 d i 2 ¿ v ¿ o m s d f ; , e ( ) f s = ; h i t o e r w s e . Existence of Fixed Points..?

  12. ( ( ) ) f f R ± ± ¸ s s s s 2 1 2 1 0 -Causal Functions

  13. ( ) n l f i m s n 1 ! Construction of Fixed Points

  14. ( ( ) ) f f ½ s s s s 1 2 1 2 Strictly Causal Functions

  15. Zeno

  16. ( ( ( ; ) ) ) ( ( ( ( ( ( ; ) ) ) ) ) ) f f f f f f f f f Construction of Fixed Points

  17. Beyond Strict Causality..

  18. 2 ( ( ) ) ( ) t t t + x y u = ( ( ) ) K t t y x = Algebraic Loops

  19. 2 ( ( ) ) ( ) 1 0 7 2 t t t + x y u = : ( ( ) ) K 0 2 6 8 t t y x = : Algebraic Loops

  20. ( ) 1 4 1 0 9 2 7 8 2 2 t y = : : ( ) 3 0 7 2 3 2 6 1 8 t x = : : Algebraic Loops

  21. … in Simulink

  22. … in Ptolemy II

  23. ( ( ) ( ) ) f f f ½ s s s Functions Strictly Causal on Orbits

  24. ( ( ( ; ) ) ) ( ( ( ( ( ( ; ) ) ) ) ) ) f f f f f f f f f Construction of Fixed Points

  25. Conclusion Proceed with caution..

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