Rudolf Mak [e-mail]

1
Periodic-Drop-Take Calculus forStream
Transformers(based on CS-Report 05-02)

• Rudolf Mak
• January 21, 2005

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.

3
Question
What does this system compute for various values
of k?
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Periodic Stream samplers
5
PDT-calculus

• Operators
• Drop operators
• Take operators
• Equational rules
• Drop expansion/contraction
• Drop exchange
• Complement
• Drop elimination/Introduction
• Take composition

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Drop operator
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Canonical forms

1. Period-consecutive
2. Rank-increasing
3. Primitive

X
8
Drop expansion/contraction rule
9
Example
(l1)-fold
q-fold
10
Drop exchange rule
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Completeness
12
Rewriting to canonical form
13
Take operator
14
Complement
15
Rules involving take operators

• Drop elimination/introduction
• Take composition

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Split component
17
Merge component
18
Block reverser
19
Split-merge systems
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The set of equations Esv
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Solving a single equation 1

• Arbitrary shape
• Canonical shape
• Period-aligned, pseudo-canonical shape

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Solving a single equation 2
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Example
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Esv theorem for SISO systems
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Split component
26
Emv theorem for SISO systems
27
Question revisited
What does this system compute for various values
of k?
28
Answer

• k 0, junk, irreparable deadlock
• k 1, 2-place buffer
• k 2, block reverser with block size 2

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.