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Rudolf Mak January 21, 2005 15-Jan-19 Rudolf Mak

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Presentation on theme: "Rudolf Mak January 21, 2005 15-Jan-19 Rudolf Mak"— Presentation transcript:

1 Periodic-Drop-Take Calculus for Stream Transformers (based on CS-Report 05-02)
Rudolf Mak January 21, 2005 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

2 Motivation For stream processing systems build in a LEGO-
like fashion from a fixed set of building blocks we want to specify verify analyze their functional behavior. Moreover we want to design systems of specified functionality. 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

3 Question What does this system compute for various values of k?
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

4 Periodic Stream samplers
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

5 PDT-calculus Operators Equational rules Drop operators Take operators
Drop expansion/contraction Drop exchange Complement Drop elimination/Introduction Take composition 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

6 Drop operator 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

7 X Canonical forms Period-consecutive Rank-increasing Primitive
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

8 Drop expansion/contraction rule
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

9 Example (l+1)-fold q-fold 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

10 Drop exchange rule 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

11 Completeness 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

12 Rewriting to canonical form
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

13 Take operator 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

14 Complement 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

15 Rules involving take operators
Drop elimination/introduction Take composition 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

16 Split component 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

17 Merge component 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

18 Block reverser 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

19 Split-merge systems 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

20 The set of equations Esv
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

21 Solving a single equation 1
Arbitrary shape Canonical shape Period-aligned, pseudo-canonical shape 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

22 Solving a single equation 2
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

23 Example 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

24 Esv theorem for SISO systems
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

25 Split component 15-Jan-19 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

26 Emv theorem for SISO systems
15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

27 Question revisited What does this system compute for various values of k? 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

28 Answer k = 0, junk, irreparable deadlock k = 1, 2-place buffer
k = 2, block reverser with block size 2 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

29 Conclusions PDT-calculus is a simple calculus to reason about periodically sampled streams. PDT-calculus is sound and complete. Semantic model in the form of a monoid. Algorithm to determine canonical forms (solves the word problem). Algorithm to solve linear equations in a single variable (solves the division problem). Functionality of arbitrary SISO-systems can be analyzed. Only partial correctness is addressed. 15-Jan-19 Rudolf Mak TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

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