Discrete-Event Systems:
Generalizing Metric Spaces and Fixed-Point Semantics
Adam Cataldo
Edward Lee
Xiaojun Liu
Eleftherios Matsikoudis
Haiyang Zheng
Chess Review
May 11, 2005
Berkeley, CA
Discrete-Event (DE) Systems
• Traditional Examples
– VHDL
– OPNET Modeler
– NS-2
• Distributed systems
– TeaTime protocol in Croquet
(two players vs. the computer)
Cataldo, CHESS, May 11, 2005 2
Introduction to DE Systems
• In DE systems, concurrent objects
(processes) interact via signals
Values
Process
Signal
Event
Signal
Time
Process
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What is the semantics of DE?
• Simultaneous events may occur in a model
– VHDL Delta Time
• Simultaneity absent in traditional
formalisms
– Yates
– Chandy/Misra
– Zeigler
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Time in Software
• Traditional programming language
semantics lack time
• When a physical system interacts with
software, how should we model time?
• One possiblity is to assume some
computations take zero time, e.g.
– Synchronous language semantics
– GIOTTO logical execution time
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Simultaneity in Hardware
• Simultaneity is common in synchronous
circuits
• Example:
1 0
1 0
1 0
0 1
0
01
1
0
This value changes
instantly
Time
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Simultaneity in Physical Systems
[Biswas]
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Our Contributions
• We generalize DE semantics to handle
simultaneous events
• We generalize metric space concepts to
handle our model of time
• We give uniqueness conditions and
conditions for avoidance of Zeno behavior
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Models of Time
• Time (real time)
Time increases in this direction
• Superdense time [Maler, Manna, Pnueli]
Superdense time increases
first in this direction
(sequence time)
then in this direction
(real time)
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Zeno Signals
• Definition: Zeno Signal
infinite events in finite real time
Chattering Zeno [Ames]
Genuinely Zeno [Ames]
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Source of Zeno Signals
• Feedback can cause Zeno
Merge
Output Signal
Input Signal
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Genuinely Zeno
• A source of genuinely Zeno signals
1/2
Merge
Delay
By Input
Value
Output Signal
Input Signal
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Simple Processes
• Definition: Simple Process
Non-Zeno Signal
Process
Non-Zeno Signal
• Merge is simple, but it has Zeno feedback
solutions
Merge
• When are compositions of simple processes
simple?
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Cantor Metric for Signals
s1
“Distance” between
two signals
t
1
s2
t
1
d s1 , s2   t
2
0
t
First time at which
the two signals differ
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Tetrics: Extending Metric Spaces
• Cantor metric doesn’t capture simultaneity
• We can capture simultaneity with a tetric
• Tetrics are generalized metrics
• We generalized metric spaces with “tetric
spaces”
• Our tetric allows us to deal with simultaneity
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Our Tetric for signals
“Distance” between two signals:
1 1 
d s1 , s2    t , n 
2 2 
1
d1 s1 , s2   t
2
n
t
s2
n
First time at which
the two signals differ
t
s1
t
1
d 2 s1 , s2   n
2
n
First sequence number at
which the two signals differ
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Delta Causal
Definition: Delta Causal
Input signals agree up to time t implies
output signals agree up to time t + 
t
t
Process
t 
t 
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What Delta Causal Means
• Signals which delay their response to input
events by delta will have non-Zeno fixed
points
Delta
Causal
Process
t
t   t  2
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Extending Delta Causal
• The system should be allowed to chatter
Delta
Causal
Process
1
0
2
t
t   t  2
• As long as time eventually advances by
delta
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Tetric Delta Causal
Definition: Tetric Delta Causal
1) Input signals agree up to time (t,n)
implies output signals agree up to time (t, n + 1)
n 1
n
t
Process
t
t 
t
t 
n 1
n
t
2) If n is large enough, this also
implies output signals agree up to time (t + ,0)
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Causal
Definition: Causal
If input signals agree up to supertime (t, n) then
the output signals agree up to supertime (t, n)
n
n
t
n
t
Process
n
t
t
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Result 1
• Every tetric delta causal process has a
unique feedback solution
Delta Causal
Process
Unique feedback solution
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Result 2
• Every network of simple, causal processes
is a simple causal process, provided in each
cycle there is a delta causal process
• Example
Delay
Non-Zeno
Input Signal
Merge
Non-Zeno
Output Signal
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Conclusions
• We broadened DE semantics to handle superdense
time
• We invented tetric spaces to measure the
distance between DE signals
• We gave conditions under which systems will have
unique fixed-point solutions
• We provided sufficient conditions under which
this solution is non-Zeno
• http://ptolemy.eecs.berkeley.edu/papers/05/DE_Systems
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Acknowledgements
•
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•
•
•
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Edward Lee
Xiaojun Liu
Eleftherios Matsikoudis
Haiyang Zheng
Aaron Ames
Oded Maler
Marc Rieffel
Gautam Biswas
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