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Chess ReviewMay 11, 2005Berkeley, CA
Discrete-Event Systems:Generalizing Metric Spaces and Fixed-Point Semantics
Adam CataldoEdward LeeXiaojun LiuEleftherios MatsikoudisHaiyang Zheng
Cataldo, CHESS, May 11, 2005 2
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 3
Introduction to DE Systems
• In DE systems, concurrent objects (processes) interact via signals
Process
Process
Values
Time
Event
Signal Signal
Cataldo, CHESS, May 11, 2005 4
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
Cataldo, CHESS, May 11, 2005 5
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
Cataldo, CHESS, May 11, 2005 6
Simultaneity in Hardware
• Simultaneity is common in synchronous circuits
• Example:
0
100
10
1This value changes instantly
0
1
Time
1
1 0
0
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Simultaneity in Physical Systems
[Biswas]
Cataldo, CHESS, May 11, 2005 8
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)
• Superdense time [Maler, Manna, Pnueli]
Time increases in this direction
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
Input SignalOutput Signal
Cataldo, CHESS, May 11, 2005 12
Genuinely Zeno
• A source of genuinely Zeno signals
Input SignalOutput Signal
DelayBy Input
Value
Merge
1/2
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Simple Processes
• Definition: Simple Process
ProcessNon-Zeno Signal 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
First time at which the two signals differ
1
“Distance” betweentwo signals
0 t
tssd21, 21
1s
2s
t
t
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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
t
n
First time at which the two signals differ
First sequence number at which the two signals differ
“Distance” between two signals:
tssd21, 211
nssd21, 212
ntssd
21,
21, 21
1s
2s
t
t
n
n
Cataldo, CHESS, May 11, 2005 17
Delta Causal
Process
Definition: Delta CausalInput signals agree up to time t implies output signals agree up to time t +
t
t
t
t
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What Delta Causal Means
Delta Causal
Process
• Signals which delay their response to input events by delta will have non-Zeno fixed points
tt 2t
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Extending Delta Causal
• The system should be allowed to chatter
• As long as time eventually advances by delta
Delta Causal
Process
01
2
2ttt
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Tetric Delta Causal
Process
Definition: Tetric Delta Causal1) Input signals agree up to time (t,n) implies output signals agree up to time (t, n + 1)
2) If n is large enough, this alsoimplies output signals agree up to time (t + ,0)
t
t
t
t
1n
1n
t
t
n
n
Cataldo, CHESS, May 11, 2005 21
Causal
Process
Definition: CausalIf input signals agree up to supertime (t, n) then the output signals agree up to supertime (t, n)
n
n
n
n
t
t t
t
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Result 1
• Every tetric delta causal process has a unique feedback solution
Delta CausalProcess
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
MergeNon-Zeno Input Signal
Non-ZenoOutput Signal
Delay
Cataldo, CHESS, May 11, 2005 24
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
• Edward Lee• Xiaojun Liu• Eleftherios Matsikoudis• Haiyang Zheng• Aaron Ames• Oded Maler• Marc Rieffel• Gautam Biswas