Semantic Foundation of the Tagged Signal Model

1. Semantic Foundation of the Tagged Signal Model Xiaojun Liu Sun Microsystems, Inc. Chess Seminar February 21, 2006

2. Outline • Motivation • Heterogeneous embedded systems • The tagged signal model as a semantic metamodel • Tagged process networks • Timed process networks • Discrete event process networks • The metric structure of signals • Conclusion

3. Embedded Systems: “Invisible Computing Systems” GENERAL MOTORS YEAR 2000 STATEMENT, July 22, 1999 GM vehicles have long been equipped with microprocessors which today, depending on the vehicle, are used for • powertrain management • anti-lock braking systems • traction control • stability enhancement • supplemental inflatable • restraint systems • real-time damping • navigation systems • automatic climate control • remote keyless entry • head-up display • entertainment systems • entry control • ...

4. Safety-Critical Applications of Heterogeneous Embedded Systems G. Leen and D. Heffernan, “Expanding Automotive Electronic Systems,” IEEE Computer, January 2002.

5. Tools and Languages for Designing Heterogeneous Embedded Systems • Simulink and Stateflow (The MathWorks) • LabVIEW (National Instruments) • Modelica (Modelica Association) • CarSim (Mechanical Simulation Corp.) • System Studio (Synopsys, Inc.) • Advanced Design System (Agilent EEsof EDA) • SystemC (Open SystemC Initiative) • VHDL and Verilog with Analog and Mixed Signal Extensions • GME (Vanderbilt University) • Polis and Metropolis (UC Berkeley) • Ptolemy and Ptolemy II (UC Berkeley)

6. The Generic Modeling Environment Meta Object Facility (MOF) Specification, Version 1.4, April 2002, OMG. http://www.isis.vanderbilt.edu/Projects/gme/ Metropolis Meta Model CDIF (CASE Data Interchange Format) http://www.eigroup.org/cdif/intro.html http://embedded.eecs.berkeley.edu/metropolis/ The Metamodeling Approach

7. The Tagged Signal Model asa Semantic Metamodel • Proposed as a Framework for Comparing Models of Computation (Lee & ASV, 1997) • Opportunities Provided: • To compare certain properties of various models of computation, such as their notion of synchrony • To define formal relations among signals and process behaviors from different models of computation • To facilitate the cross-fertilization of results and proof techniques among models of computation

8. The Goal of This Presentation Establish a Semantic Foundation of the Tagged Signal Model (TSM) Explore the Opportunities Provided by the TSM Framework

9. Approach • A Three-Step Development Process • Study the properties of a mathematical structure of (sets of) signals • Use this structure to characterize the processes that are functions on signal sets, such as continuity and causality • Determine conditions under which the characterizations are compositional

10. Outline • Motivation • Heterogeneous embedded systems • The tagged signal model as a semantic metamodel • Tagged process networks • Timed process networks • Discrete event process networks • The metric structure of signals • Conclusion

11. Signals • A signal has a tag setT, which is a partially ordered set. • A down-set of T is a downward-closed subset of T. D(T ) is the set of down-sets of T. • A signal is a function from a down-set D2D(T ) to some value set V, signal:D!V 2 D([0, 1)) 0 1 2 3  D([0, 1))

12. 6 15 6 2 5 2 3 3 1 1 The Prefix Order on Signals • A signal s1:D1!V is a prefix of s2:D2!V, denoted s1¹s2, if and only if D1µD2, and s1(t) = s2(t), 8t2D1 ¹

13. The Order Structure of Signals • For any poset T of tags and set V of values, let S(T,V ) be the set of all signals from down-sets of T to V. • With the prefix order ¹, S(T,V ) is • a poset • a complete partial order (CPO) • a complete lower semilattice (i.e. any subset of signals have a “longest” common prefix)

14. x2S(T1,V1) y2S(T2,V2) u2S(T3,V3) v2S(T4,V4) P Q PµS(T1,V1)£S(T2,V2) Q: S(T3,V3)!S(T4,V4) Processes • Processes are relations or functions among signal sets • As functions among posets/CPOs, processes may be • Monotonic, r¹s)P(r) ¹P(s) • (Scott) Continuous, P(ÇD) = ÇP(D) • Maximal

15. (y,z) = F(x) where (y,z) is the least solution of the equations y = P(x,z) z = Q(y) x y P z Q F Tagged Process Networks • A direct generalization of Kahn process networks • If the processes P and Q are Scott-continuous, then F is Scott-continuous. Scott-continuity is compositional.

16. Outline • Motivation • Heterogeneous embedded systems • The tagged signal model as a semantic metamodel • Tagged process networks • Timed process networks • Discrete event process networks • The metric structure of signals • Conclusion

17. Timed Process Networks • Each signal in a tagged process network may have its own tag set. • In a timed process network, all signals share the same totally ordered tag set.

18. Timed Signals • Let T = [0, 1), and V =V[fg, where  represents the absence of value. S(T,V) is the set of timed signals. s(t) = 1 s(1-1/k) = 1, k = 1, 2, … s(k) = 1, k = 0, 1, 2, …

19. Timed Processes s1: D1 V s: D V add s2: D2 V s1: D1 V D = D1  D2 s(t) = s1(t) +s2(t) s: D V biased merge s2: D2 V D = D1  D2 s(t) = s1(t), when s1(t)V s2(t), otherwise s2: D2 V s1: D1 V delay by 1 D2 = D1{1}  [0, 1) s2(t)= s1(t 1), when t 1 , when t  [0, 1)

20. delay by 1 z y biased merge x A Timed Process Network Example

21. lookahead by 1 z: y: ;V ;V biased merge x A Non-Causal Process in the Network

22. Causality • A timed process P is causal if • it is monotonic, and • for all s:D1!V1, P(s):D2!V2, D1µD2 • A timed process P is strictly causal if • it is monotonic, and • for all s:D1!V1, P(s):D2!V2, D1½D2, or D2 =T

23. Causality and Continuity • Neither implies the other. • A process may be continuous but not causal, e.g. “lookahead by 1”. • A process may be causal but not continuous, e.g. one that produces an output event after counting an infinite number of input events.

24. (y,z) = F(x) where (y,z) is the least solution of the equations y = P(x,z) z = Q(y) x y P z Q F Causal Timed Process Networks • If processes P and Q are causal and continuous, and at least one of them is strictly causal, then F is causal and continuous. Causality + continuity is compositional.

25. Outline • Motivation • Heterogeneous embedded systems • The tagged signal model as a semantic metamodel • Tagged process networks • Timed process networks • Discrete event process networks • The metric structure of signals • Conclusion

26. dom(s) = [0, ) s(k) = 1, k = 0, 1, 2, … dom(s) = [0, ) s(1-1/k) = 1, k = 1, 2, … dom(s) = [0, 1) s(1-1/k) = 1, k = 1, 2, … Discrete Event Signals • A timed signal s:D!V is a discrete event signal if for all t2D, s is present at a finite number of times before t. DE, Non-Zeno Not DE DE, Zeno

27. The Order Structure of DE Signals • For any totally ordered set T of tags and set V of values, the set of all DE signals Sd(T,V ), with the prefix order ¹, is • a poset • a complete partial order • a complete lower semilattice (i.e. any subset of DE signals have a “longest” common prefix)

28. A DE Process Network Example delay by 1 z y biased merge x

29. (y,z) = F(x) where (y,z) is the least solution of the equations y = P(x,z) z = Q(y) x y P z Q F Discrete Event Process Networks • If DE processes P and Q are Scott-continuous, then F is a DE process and is Scott-continuous. Discreteness + Scott-continuity is compositional.

30. (y,z) = F(x) where (y,z) is the least solution of the equations y = P(x,z) z = Q(y) x y P z Q F Non-Zeno DE Process Networks • If DE processes P and Q are causal and Scott-continuous, and at least one of them is strictly causal, then F is a DE process and is causal and Scott-continuous. • F is non-Zeno in the sense that if x is non-Zeno, F(x) is non-Zeno. Discreteness + Causality + Scott-continuity is compositional.

31. Tagged Process Networks Timed Process Networks Causality Discreteness Non-Zenoness A Brief Recap Tagged Signal Model Kahn Process Networks

32. Outline • Motivation • Heterogeneous embedded systems • The tagged signal model as a semantic metamodel • Tagged process networks • Timed process networks • Discrete event process networks • The metric structure of signals • Conclusion

33. Generalized Ultrametric • Let X be a set and  be a poset with a minimum element 0. A function d:X£X ! is a generalized ultrametric if for all x, y, z2X and 2, d(x,y) = 0 ,x = y d(x,y) = d(y,x) d(x,y)· and d(y,z)·)d(x,z)·

34. A Generalized Ultrametric onTagged Signals • For any poset T of tags, the set of generalized ultrametric distances, T, is T = (D(T ), ¶) • For any tag set T and value set V, is a generalized ultrametric.

35. A Metric “Generator” • If the tag set T is totally ordered, T is also totally ordered. • For any totally ordered set  with a minimum element 0, and a function g:T! such that • g is monotonic • g() = 0, = 0 g±dds is a (generalized) ultrametric on S(T,V ).

36. Summary • The Order Structure of Tagged Signals and Tagged Process Networks • Causality, Discreteness, and Non-Zenoness in Timed Process Networks • A Generalized Ultrametric on Tagged Signals • A Formulation of Processes as Labeled Transition Systems for Comparing Implementations or Simulations of Tagged Processes

37. The End Questions?