Rudolf Mak [e-mail]

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

• Rudolf Mak
• January 21, 2005

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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.

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Periodic Stream samplers
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Periodic Stream samplers
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PDT-calculus

• Operators
• Unit
• Drop operators
• Take operators
• Equational rules
• Unit rule
• Drop expansion/contraction
• Drop exchange
• Complement
• Drop elimination/introduction
• Take composition

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Drop operator
X
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Canonical forms

1. Period-consecutive
2. Rank-increasing
3. Primitive (no repetitive rank-pattern)

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Transform to period-consecutive
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Drop expansion/contraction rule
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Transform to rank-increasing
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Drop exchange rule
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Completeness
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Rewriting to canonical form
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Take operator
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Complement rule
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Rules involving take operators

• Drop elimination/introduction
• Take composition

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Split component
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Merge component
19
Block reverser design
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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
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Emv theorem for SISO systems
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Analysis problem (cyclic system)
What does this system compute for various values
of k?
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Solution

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

suffers from reparable deadlock
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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.