1. Periodic-Drop-Take Calculus for Stream Transformers (based on CS-Report 05-02)
Rudolf Mak
January 21, 2005
24-Nov-18
Rudolf Mak
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.
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

3. Periodic Stream samplers
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

4. PDT-calculus Operators Equational rules Unit Drop operators
Take operators
Equational rules
Unit rule
Drop expansion/contraction
Drop exchange
Complement
Drop elimination/introduction
Take composition
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

5. X Drop operator 1 k l 24-Nov-18 Rudolf Mak r.h.mak@tue.nl1 k l X
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

6. X Canonical forms Period-consecutive Rank-increasing
Primitive (no repetitive rank-pattern) X
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

7. Transform to period-consecutive
q-fold
(l+1)-fold
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

8. Drop expansion/contraction rule
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

9. Transform to rank-increasingb e c d f a b f d e a b f c d a b d f
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

10. Drop exchange rule 24-Nov-18 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

11. Completeness 24-Nov-18 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

12. X Take operator 1 k l 24-Nov-18 Rudolf Mak r.h.mak@tue.nl1 k l X
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

13. Complement rule 24-Nov-18 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
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

15. Split component 24-Nov-18 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

16. Merge component 24-Nov-18 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

17. Block reverser design 24-Nov-18 Rudolf Mak r.h.mak@tue.nl
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

18. Split-merge systems DR 24-Nov-18 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
24-Nov-18
Rudolf Mak
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
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

21. Solving a single equation: 2
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

22. Esv theorem for SISO systems
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

23. Split component 24-Nov-18 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
24-Nov-18
Rudolf Mak
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?
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking

26. Solution k = 0, junk, irreparable deadlock k = 1, 2-place buffer
k = 2, block reverser with block size 2
Solution
suffers from reparable deadlock
24-Nov-18
Rudolf Mak
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.
24-Nov-18
Rudolf Mak
TU/e, Dept. of Math. and Comp. Sc., System Architecture and Networking