7th Biennial Ptolemy Miniconference

Berkeley, CAFebruary 13, 2007

Correctness in Causal Systems

Eleftherios Matsikoudis

UC Berkeley

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Causality (Informally)

… is the constraint that an effect cannot precede its cause.

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Relevance

Modeling and Simulation

Synchronous Programming of Reactive Systems

Hardware Description

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Correctness

?

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Systems..

sC

sSE fAsA

sBfB

fMsM

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Systems..

sC

sSE fAsA

sBfB

fMsM

as Fixed-Point Equations

sA = fA (sM ;sC )

sB = fB (sA ;sC )

sM = fM (sB ;sSE )

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Signals

T

V

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Prefix Order

s1 s2v

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Generalized Ultrametric Distance

d(s1;s2)

s1

s2

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Causal Functions

µ

s1

s2

f (s1)

f (s2)

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Existence of Fixed Points..?

f (s) def=

(fh¿;vig if ¿ 62 doms,; otherwise.

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-Causal Functions

R0

±

¸ ±

s1

s2

f (s1)

f (s2)

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Construction of Fixed Points

limn! 1

f n(s)

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Strictly Causal Functions

s1

s2

f (s1)

f (s2)½

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Zeno

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Construction of Fixed Points

f (f ( ))

f ( )f ( )

f (f ( ))

f (; )

f (f (; ))

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Beyond Strict Causality..

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Algebraic Loops

y(t) =x(t) =

x2(t) +u(t)

K y(t)

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x2(t) +u(t)

K y(t)

Algebraic Loops

y(t) =x(t) =

1:072

0:268

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1:072

0:268

Algebraic Loops

y(t) =x(t) =

14:9282

3:7321

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… in Simulink

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… in Ptolemy II

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Functions Strictly Causal on Orbits

½

s

f (s)

f (f (s))

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Construction of Fixed Points

f (f ( ))

f ( )f ( )

f (f ( ))

f (; )

f (f (; ))

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Conclusion

Proceed with caution..