Correctness in Causal Systems

7th Biennial Ptolemy Miniconference

Berkeley, CAFebruary 13, 2007

Eleftherios Matsikoudis

UC Berkeley

Matsikoudis, Berkeley 2Ptolemy Miniconference, February 13, 2007

Causality (Informally)

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

Relevance

Modeling and Simulation

Synchronous Programming of Reactive Systems

Hardware Description

Correctness

Systems..

sSE fAsA

Systems..

sSE fAsA

as Fixed-Point Equations

sA = fA (sM ;sC )

sB = fB (sA ;sC )

sM = fM (sB ;sSE )

Signals

Prefix Order

s1 s2v

Generalized Ultrametric Distance

d(s1;s2)

Causal Functions

f (s1)

f (s2)

Existence of Fixed Points..?

f (s) def=

(fh¿;vig if ¿ 62 doms,; otherwise.

-Causal Functions

f (s1)

f (s2)

Construction of Fixed Points

limn! 1

f n(s)

Strictly Causal Functions

f (s1)

f (s2)½

f (f ( ))

f ( )f ( )

f (f ( ))

f (; )

f (f (; ))

Beyond Strict Causality..

Algebraic Loops

y(t) =x(t) =

x2(t) +u(t)

K y(t)

x2(t) +u(t)

K y(t)

Algebraic Loops

y(t) =x(t) =

Algebraic Loops

y(t) =x(t) =

14:9282

3:7321

… in Simulink

… in Ptolemy II

Functions Strictly Causal on Orbits

f (f (s))

f (f ( ))

f ( )f ( )

f (f ( ))

f (; )

f (f (; ))

Conclusion

Proceed with caution..

Correctness in Causal Systems

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