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Periodic-Drop-Take Calculus for Stream Transformers (based on CS-Report 05-02)

Periodic-Drop-Take Calculus for Stream Transformers (based on CS-Report 05-02). Rudolf Mak January 21, 2005. Motivation for a calculus. For stream processing systems build in a LEGO r -like fashion from a fixed set of building blocks we want to specify verify analyze

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Periodic-Drop-Take Calculus for Stream Transformers (based on CS-Report 05-02)

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  1. Periodic-Drop-Take Calculus forStream Transformers(based on CS-Report 05-02) Rudolf Mak January 21, 2005 Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  2. Motivation for a calculus • For stream processing systems build in a LEGOr-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. Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  3. Periodic Stream samplers Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  4. PDT-calculus • Operators • Unit • Drop operators • Take operators • Equational rules • Unit rule • Drop expansion/contraction • Drop exchange • Complement • Drop elimination/introduction • Take composition Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  5. k 0 1 l Drop operator X Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  6. X X Canonical forms • Period-consecutive • Rank-increasing • Primitive (no repetitive rank-pattern) Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  7. (l+1)-fold q-fold Transform to period-consecutive Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  8. Drop expansion/contraction rule Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  9. a b c d e f a a b b d c e d f f a b d f Transform to rank-increasing Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

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

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

  12. k 0 1 l X X X X X Take operator Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  13. Complement rule Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  14. Rules involving take operators • Drop elimination/introduction • Take composition Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

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

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

  17. Block reverser design Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

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

  19. The set of equations Esv Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  20. Solving a single equation: 1 • Arbitrary shape • Canonical shape • Period-aligned, pseudo-canonical shape Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  21. Solving a single equation: 2 Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  22. Esv theorem for SISO systems Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

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

  24. Emv theorem for SISO systems Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  25. Analysis problem (cyclic system) What does this system compute for various values of k? Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  26. k = 0, junk, irreparable deadlock k = 1, 2-place buffer k = 2, block reverser with block size 2 Solution suffers from reparable deadlock Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

  27. Summary • 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 in the monoid). • Algorithm to solve linear equations in a single variable (solves the division problem in the monoid). • Functionality of arbitrary SISO-systems consisting of split and merge components can be analyzed. • Only partial correctness is addressed. Rudolf Mak r.h.mak@tue.nl TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

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