periodic-drop-take calculus for stream transformers (based on cs-report 05-02)
DESCRIPTION
Periodic-Drop-Take Calculus for Stream Transformers (based on CS-Report 05-02). Rudolf Mak January 21, 2005. Motivation. For stream processing systems build in a LEGO- like fashion from a fixed set of building blocks we want to specify verify analyze - PowerPoint PPT PresentationTRANSCRIPT
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
1
Rudolf MakJanuary 21, 2005
Periodic-Drop-Take Calculus for
Stream Transformers(based on CS-Report 05-02)
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
2
MotivationFor stream processing systems build in a LEGO- like fashion from a fixed set of building blockswe want to
• specify• verify• analyze
their functional behavior. Moreover we want to• design
systems of specified functionality.
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
3
Question
What does this system compute for various values of k?
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
4
Periodic Stream samplers
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
5
PDT-calculus• Operators
– Drop operators– Take operators
• Equational rules– Drop expansion/contraction– Drop exchange– Complement– Drop elimination/Introduction– Take composition
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
6
Drop operator
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
7
Canonical forms
1. Period-consecutive
2. Rank-increasing
3. Primitive
X
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
8
Drop expansion/contraction rule
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
9
Example
(l+1)-foldq-fold
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
10
Drop exchange rule
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
11
Completeness
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
12
Rewriting to canonical form
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
13
Take operator
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
14
Complement
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
15
Rules involving take operators• Drop elimination/introduction
• Take composition
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
16
Split component
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
17
Merge component
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
18
Block reverser
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
19
Split-merge systems
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
20
The set of equations Esv
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
21
• Arbitrary shape
• Canonical shape
• Period-aligned, pseudo-canonical shape
Solving a single equation 1
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
22
Solving a single equation 2
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
23
Example
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
24
Esv theorem for SISO systems
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
25
Split component
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
26
Emv theorem for SISO systems
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
27
Question revisited
What does this system compute for various values of k?
Apr 22, 2023 Rudolf Mak [email protected]/e, Dept. of Math. and Comp. Sc., System Architecture and Networking
28
• k = 0, junk, irreparable deadlock
• k = 1, 2-place buffer
• k = 2, block reverser with block size 2
Answer
Apr 22, 2023 Rudolf Mak [email protected]/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.