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Understanding Periodic Drop Take Calculus for Stream Transformers

Discover the PDT calculus for stream processing systems built from modular blocks. This calculus helps specify, verify, design, and analyze the functional behavior of stream systems. Explore operators, equational rules, and canonical forms for transforming and analyzing stream data.

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Understanding Periodic Drop Take Calculus for Stream Transformers

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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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