correctness in causal systems
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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..
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