1. 7th Biennial Ptolemy Miniconference Berkeley, CA February 13, 2007 Causality Interfaces for Actor Networks Ye Zhou and Edward A. Lee University of California, Berkeley

2. 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

3. 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?

4. 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.

5. 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

6. Zhou, Berkeley 6Ptolemy Miniconference, February 13, 2007 The Tagged Signal Model (Cont’d)

7. Zhou, Berkeley 7Ptolemy Miniconference, February 13, 2007 Causality Interfaces A causality interface for an actor a with input ports P i and output ports P o is a function where D is a partially ordered set with elements called dependencies.

8. 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 ( ).

9. 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):

10. Zhou, Berkeley 10Ptolemy Miniconference, February 13, 2007 Dependency Algebra Axioms (Cont’d) Ordering Axiom:

11. 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.

12. Zhou, Berkeley 12Ptolemy Miniconference, February 13, 2007 Interpretation Recall that an actor function is The projection of F a 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.

13. Zhou, Berkeley 13Ptolemy Miniconference, February 13, 2007 Feedforward Compositions Use for serial compositions and for parallel compositions. Example:

14. Zhou, Berkeley 14Ptolemy Miniconference, February 13, 2007 Feedback Compositions The gain of a cyclic path c = (p 1, p 2, …, p n, p 1 ) is: Productivity order

15. 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.

16. Zhou, Berkeley 16Ptolemy Miniconference, February 13, 2007 Example – Adaptive Filtering

17. 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.

18. Zhou, Berkeley 18Ptolemy Miniconference, February 13, 2007 Thank You!