7th biennial ptolemy miniconference berkeley, ca february 13, 2007 causality interfaces for actor...
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7th Biennial Ptolemy Miniconference
Berkeley, CAFebruary 13, 2007
Causality Interfaces for Actor Networks
Ye Zhou and Edward A. Lee
University of California, Berkeley
Zhou, Berkeley 2Ptolemy Miniconference, February 13, 2007
Introduction
Actor receives tokens from input ports and reacts to these tokens by producing tokens on the output ports.
What flows in the connectors are signals (streams of tokens).
actor
output port
connector
input port
Zhou, Berkeley 3Ptolemy Miniconference, February 13, 2007
Introduction (Cont’d)
Any actor network can be treated as a feedback system. We assume all actors are (Scott) continuous and use the
least fixed point semantics as the behavior of the network. Question: Will the network deadlock? Is it possible to do
static analysis?
Zhou, Berkeley 4Ptolemy Miniconference, February 13, 2007
Goal: Causality Interfaces
Use an interface approach to capture the causality properties of an actor.
Develop a mathematical structure to algebraically compose these interfaces.
Determine whether an actor network is live under certain models of computation.
Zhou, Berkeley 5Ptolemy Miniconference, February 13, 2007
The Tagged Signal Model
[Lee and Sangiovanni-Vincentelli, 1998] A signal is a set of (tag, value) pairs.
[Liu, 2005] The tag set is a partial order. A signal is defined on a down-set of .
t
v
Zhou, Berkeley 7Ptolemy Miniconference, February 13, 2007
Causality Interfaces
A causality interface for an actor a with input ports Pi and output ports Po is a function
where D is a partially ordered set with elements called de
pendencies.
,: DPP oia
Zhou, Berkeley 8Ptolemy Miniconference, February 13, 2007
How to compose dependencies?
Serial connection Parallel connection
We need two operators, one for serial ( ), and one for parallel ( ).
Zhou, Berkeley 9Ptolemy Miniconference, February 13, 2007
Dependency Algebra Axioms
Dependency set D is a partially ordered set with two binary operators (for parallel) and (for serial) that satisfies the following axioms: Associativity:
Commutativity and Idempotence (for only):
122121 ,, ddddDdd
)()(,,, 321321321 ddddddDddd
)()(,,, 321321321 ddddddDddd
dddDd ,
Zhou, Berkeley 10Ptolemy Miniconference, February 13, 2007
Dependency Algebra Axioms (Cont’d)
Ordering Axiom:
DdD that such 1 0, ,
11
00
0
ddd
d
dd
.,, 12121 dddDdd
Zhou, Berkeley 11Ptolemy Miniconference, February 13, 2007
Causality Interface Model
The dependency set D is a set of functions:
where is the set of all down sets of .
computes the greatest lower bound of two functions.
is function composition.
Zhou, Berkeley 12Ptolemy Miniconference, February 13, 2007
Interpretation
Recall that an actor function is
The projection of Fa onto the tag set is
which reflects the dependency of output ports on input ports. is the projection of such dependency on a pair of (input, output) ports. In general, appropriate projection onto pairs of (input, output) ports are not always possible.
Zhou, Berkeley 13Ptolemy Miniconference, February 13, 2007
Feedforward Compositions
Use for serial compositions and for parallel compositions.
Example:
))],(),((),([),(),( 4267425121 2321ppppppppqq aaaab
Zhou, Berkeley 14Ptolemy Miniconference, February 13, 2007
Feedback Compositions
The gain of a cyclic path c = (p1, p2, …, pn, p1) is:
Productivity order
Zhou, Berkeley 15Ptolemy Miniconference, February 13, 2007
Liveness Condition
Theorem:
This theorem applies to synchronous languages, discrete-event models, continuous-time models, and dataflow models.
Zhou, Berkeley 16Ptolemy Miniconference, February 13, 2007
Example – Adaptive Filtering
),,,,( 9711109 pppppc
IIIc dddppppg ),(),( 711109
Zhou, Berkeley 17Ptolemy Miniconference, February 13, 2007
Conclusion
We presented an interface theory for causality interface of actors and their compositions. We gave an algebraic procedure to determine whether an actor network is live under certain models of computation. We showed that causality analysis only needs to be performed for each simple communication cycle. Reference:Ye Zhou and Edward A. Lee. "Causality Interfaces for Actor Networks," EECS Department, University of California, Berkeley, UCB/EECS-2006-148, November 16, 2006.
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