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Extending Kahn's Principle to Timed Systems: A Discrete Event Approach

Explore how Kahn's Principle can be applied to timed systems using discrete event signals, operational semantics, and prefix orders in this theoretical study.

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Extending Kahn's Principle to Timed Systems: A Discrete Event Approach

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  1. Kahn’s Principle and the Semantics of Discrete Event Systems Xiaojun Liu EE290N Class Project December 10, 2004

  2. Kahn Process Networks • An elegant model of computation for deterministic concurrent processes. • Processes communicate streams of data asynchronously through unbounded FIFO channels. • A denotational semantics based on complete partial orders (CPOs) and Scott continuous functions.

  3. Kahn’s Principle Can we extend this to timed systems? • The input-output function of any network is denoted by the least fixed point solution of the network’s recursion equations. z (y, z) = f(x) F(f)(x) = (p(x, 2(f(x))), q(1(f(x)))) f = fix(F) x p q y Robert Yates and Guang Gao, “A Kahn Principle for Networks of Nonmonotonic Real-Time Processes” http://www.sable.mcgill.ca/~hendren/ftp/acaps/memo60.ps.gz

  4. Previous Work: Yates & Gao • Timed stream: a sequence of time-stamped tokens. • The time stamps are strictly increasing and separated by a minimum time interval   0. • Processes are -causal.

  5. Previous Work, Continued • The principle novelty of the new functional (F) … is that it cuts off all outputs as soon as the earliest new output token appears. • The new functional lets the network “run” at each step only until the earliest new output token appears.

  6. Highlights of Approach • A direct extension of KPN to DE systems. • More general than the previous work outlined earlier. • Useful guidance to develop operational semantics and simulator.

  7. Prefix Order − Definition • Let T, a poset, be the set of all tags. Let L(T) be the set of lower sets of T. • A signal is a function from a lower set LL(T)to some value set V, signal: L V • A signal s1: L1 V is a prefix of s2: L2 V, denoted s1s2, if and only if L1  L2, and s1(t) = s2(t), tL1 • Let S(T, V) be the set of all signals from lower sets of T to V.

  8. Prefix Order − Examples • Given any set A, treated as a poset with the discrete order, S(A, B) is the set of all partial functions from A to B. • Let T = [0, ), and V = V  {}, where  represents the absence of value, S(T, V) is the set of DE signals.

  9. Prefix Order − Properties • For any poset T of tags and set V of values, S(T, V) with the prefix order is • a poset • a CPO • a complete lower semilattice (i.e. any subset of signals have a GLB) • For any network of processes that are Scott continuous functions from input to output, Kahn’s principle applies.

  10. Discrete Event Signals • The set of DE signals, S([0, ), V  {}) is a CPO. • Examples: clock: 0 1 2 3 … … zeno: 0 ½ 1 2 3 … … r-zeno: 0 ½ 1 2 3 …

  11. Discrete Event Processes s1: L1 V add s: L V s2: L2 V L = L1  L2 s(t) = s1(t) +s2(t), tL delay by 1 s1: L1 V s2: L2 V L2 = L1{1}  [0, 1) s2(t) = s1(t 1), when t 1 and , when t  [0, 1)

  12. Biased Merge or “Default” s1: L1 V biased merge s: L V s2: L2 V Let p = sup L2. If pL2, L = L1  L2 {q | q  p and t[p, q], s1(t)V } and if pL2, L = L1  L2 {q | q  p and t(p, q], s1(t)V } s(t) = s1(t), when s1(t)V s2(t), otherwise, tL

  13. 0 ½ 1 … … … … … … … … 0 ½ 1 2 3 … 0 ½ 1 2 3 … 0 ½ 1 … 0 ½ 1 2 0 ½ 1 2 A Discrete Event System delay by 1 biased merge zeno … 0 ½ 1 2 3 …

  14. YADES  lookahead by 1  biased merge r-zeno … 0 ½ 1 2 3 …

  15. Continuity and Causality • 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.

  16. Discrete Signals • A DE signal s: L V is discrete if • s-1(V) is order-isomorphic with a lower set of N and • s-1(V) is a finite set, or L = [0, sup s-1(V)). • Let D([0, ), V  {}) be the set of discrete signals. With the prefix order, it is a poset, CPO, and a complete lower semilattice.

  17. clock: 0 1 … … 0 0 ½ ½ 1 1 A Caveat biased merge ? zeno:

  18. Operational Semantics • Represent signals as timed streams • Processes map input timed streams to output timed streams • An alternative development of Chandy and Misra’s distributed DE simulation

  19. Representing Discrete Signals as Timed Streams • Only “present” events? (0.0, 1) many-to-one prefix not a prefix (0.0, 1), (1.0, 1)

  20. Null Events (0.0, 1), (1.0, ) (0.0, 1), (2.0, ) one-to-many (0.0, 1), (1.0, ), (2.0, ) (0.0, 1), (1.0, 1), (2.0, )

  21. Simultaneous Events • Extend the tag set to [0, )  N, with the lexicographical order. • All developments so far can be generalized. • Any continuous/causal DE process at a time t can be treated as a Scott continuous function from input sequences to output sequences. • A “snapshot” of a DE system at a time t is a KPN.

  22. Current Status • Substantial progress on theoretical development • Developing a plan for implementation • Introduce “end-of-stream” marker in KPN • Change event queue to contain sequences of events with the same time stamp and receiver • Key criterion for improvement: more efficient simulation due to less traffic into/out of the event queue

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