A Denotational Semantics For Dataflow with Firing

1. Paper Discussion for A Denotational Semantics For Dataflow with Firing Edward A. Lee Jike Chong Wei Zheng

2. Kahn Process Network Definition • Functional Nodes • Unbounded FIFO • Reads block • Writes don’t block • Signals are streams

3. Data-flow network Definition • A collection of functional nodes • Connected and communicate over unbounded FIFO queues • Nodes are commonly called actors • The bits of information that are communicated over the queues are commonly called tokens Source: ee249 slides

4. Question from this paper: • What is a denotational semantic? • Why is Kahn Process Networks important? • Why do we need firing?

5. Question from this paper: • Denotational Semantic • Two approach of semantics • Denotational • Developed by Scott and Strachey • Meaning of the language is in terms of relation • Operational • Turing machine • Meaning of the language is in terms of actions taken by some abstract machine • Closer to implementation

6. Question from this paper: • Why is Kahn Process Networks important? • General • Solid semantics • Properties: • Functional Node • Unbounded FIFO • Blocking read • No total order (concept of partial order) • No over specification • May be difficult to determine order in the real world

7. Question from this paper: • Why do we need firing? • Basic element of dataflow networks • Provides notion of tokens passed from inputs to outputs • Enables scheduling of input to output event • Closer to real world scenarios Kahn Process Network + Firing data-flow network

8. Formal Semantic Model –Kahn Process Network • Provides a formal semantic model to construct rigorous proofs for Process Networks properties • Need firing concepts based on the interactions of input sequences and the Process Network functional nodes

9. Paper Outline • Kahn Process Network property • Prefix order • Complete partial order • Monotonic function • Continuous function • Composition of processes • Determinacy of processes • Least fixed point semantics • Higher Order Function Example

10. Paper Outline • Kahn process network with Firing • Dataflow actors • Dataflow processes • Extend Monotonic property to dataflow processes • Extend Continuous property to dataflow processes • Continuity of dataflow process • Commutative firings and compositionality

11. Prefix Order • Prefix order {$} is a prefix of {$, #} • Partially ordered set - poset {{$}, {$,#}, {#}} -- {$} is not a prefix of {#} • Empty signal l: the empty sequence • Chain – totally ordered set {{$}, {$,#},{$,#,%}}

12. Complete Partial Order • Upper bound • where every element is a prefix • LUB – least upper bound LUB({{$}, {$,#},{$,#,%}}) = {$,#,%} Union of all sets • Complete Partial Order A poset with a bottom s.t. every chain in the set has a LUB

13. Paper Outline • Kahn Process Network property • Prefix order • Complete partial order • Monotonic function • Continuous function • Composition of processes • Determinacy of processes • Least fixed point semantics • Higher Order Function Example

14. Monotonic Function • untimed notion of causality

15. Continuous Function • Response to an infinite input sequence is the limit of its response to the finite approximation of this input

16. Paper Outline • Kahn Process Network property • Prefix order • Complete partial order • Monotonic function • Continuous function • Composition of processes • Determinacy of processes • Least fixed point semantics • Higher Order Function Example

17. Composition of Processes And a collection

18. Determinate Composition • A composition is determinate if given the input sequences, all other sequences are determined.

19. Paper Outline • Kahn Process Network property • Prefix order • Complete partial order • Monotonic function • Continuous function • Composition of processes • Determinacy of processes • Least fixed point semantics • Higher Order Function Example

20. Least Fixed Point • Composition does not deal with feedback • Feedback loops may not have deterministic behavior • Introduce the Least fixed point

21. Least Fixed Point • A method to gain deterministic behavior in process networks with feedback topologies • Start with the empty sequence • Apply the (monotonic) function • Apply the function again to the result • Repeat forever

22. Higher Order Function • Definition • Functions that take functions as arguments and return functions • CPO over functions • Use a similar technique to study dataflow with firing • (sm-) ? Is a CPO • Proof • All chains in the set have LUB • Bottom element

23. Higher Order Function • Functionals • example

24. Higher Order Function Example • Sieve of Eratosthenese • Output prime numbers by recursive filtering • mapping functionsonto functions

25. Paper Outline • Kahn Process Network property • Kahn process network with Firing • Dataflow actors • Dataflow processes • Extend Monotonic property to dataflow processes • Extend Continuous property to dataflow processes • Continuity of dataflow process • Commutative firings and compositionality

26. Dataflow with firings • Based on continuous functionals on POSET of functions • Dennis Dataflow • Kahn process networks where processes are made up of a sequence of atomic computations called firings • Firing can be described as functions • Firing rule

27. Dataflow actors S needs to be finite Concatenation of the two tuples of sequences , JOIN is defined to be LUB of the two tuples. If the join exists, They are said to be joinable Dataflow actor

28. Dataflow processes • Question: • Function F exists? • Function F unique? • Any assumption?

29. Dataflow processes • F exists, and unique in the pointwise prefix order • We can prove that f is monotonic and continuous • least fixed point F such that f(F) = F, this F satisfied the definition in previous slides • Find a constructive procedure for finding F • F is OK to be the semantics of the dataflow process

30. Paper Outline • Kahn process network with Firing • Dataflow actors • Dataflow processes • Extend Monotonic property to dataflow processes • Extend Continuous property to dataflow processes • Continuity of dataflow process • Commutative firings and compositionality

31. Continuity of dataflow process • F is the least fixed point of the continuous functional f • F(s) is the LUB of the chain given by (19) • why do we care about the continuity? • guarantee the determinacy

32. Commutative firings and Compositionality How to solve this problem? Replace rule 4 with followings:

33. Redefine the functional

34. Conclusion • Bridge the gap • Sequences of firings define a continuous Kahn process as the Least Fixed Point of an appropriately constructed functional

35. References [1] A Denotational Semantics for Dataflow with Firing, Edward A. Lee, Technical Memorandum UCB/ERL M97/3, Electronics Research Laboratory, Berkeley [2] Models of Computation for Embedded System Design, L. Lavagno, A. Sangiovanni-Vincentelliand E. Sentovich, 1998 NATO ASI Proceedings on System Synthesis, Il Ciocco (Italy) 1998 [3] Dataflow Process Networks, Edward A. Lee and Thomas M. Parks, Proceedings of the IEEE, vol. 83, no. 5, pp. 773-801 May, 1995 [4] Lecture notes from ee249/ee290n.