Liveness and Boundedness of Synchronous Data Flow Graphs

1
Liveness and Boundedness of Synchronous Data Flow
Graphs

• A.H. Ghamarian, M.C.W Geilen, T. Basten, B.
  Theelen, M. Mousavi, and S. Stuijk

2
Outline

• Introduction
• Synchronous Data Flow Graphs (SDFGs)
• Definition of Liveness and Boundedness
• Live and Bounded SDFGs
• Live and Strictly Bounded SDFGs
• Live and Self-timed Bounded SDFGs
• Conclusions

boundedness
intro
definitions
strict bndss
self-tmd bndss
conclusions
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Introduction

• Streaming Multimedia and DSP Applications
• Throughput
• Buffer sizes
• SDFG
• Modeling and analysis
• Single and multiprocessor platforms

Given an SDFG,is it realizable within bounded
memory while running forever ?
boundedness
intro
definitions
strict bndss
self-tmd bndss
conclusions
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Structure of a (Timed) SDFG
actor
Self-loop channel
channel (unbounded)
execution time
3
3
2
1
1
C,1
A,2
B,1
1
1
2
3
1
rate
token
boundedness
intro
definitions
strict bndss
self-tmd bndss
conclusions
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Behavior Firing an Actor
A,1
1
2
2
2
1
D,1
B, 1
2
2
5
3
3
1
4
1
C, 1
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Behavior Execution and Self-timed Execution

• An execution is a finite or infinite sequence of
  actor firings.
• An execution is maximal iff it is finite with no
  actor enabled in the final state or if it is
  infinite.
• A self-timed execution is an execution in which
  each actor fires as soon as it can fire.

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intro
definitions
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conclusions
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Consistency and Repetition Vector
3
3
2
C,1
A,2
B,1
1
1
2
3
1
1.q(A)1.q(A) 1.q(A)1.q(B) 2.q(B)3.q(C) 3.q(C)2
.q(B)
Repetition vector q(A,3), (B,3), (C,2)
An SDFG is consistent iff q(A) gt 0 for all actors
A
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intro
definitions
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conclusions
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Liveness and Deadlock

• An SDFG is live iff it has an execution in which
  all actors fire infinitely often.
• An SDFG has a deadlock iff it has a maximal
  execution of finite length.

Consistent, but neither live nor deadlocked !
boundedness
intro
definitions
strict bndss
self-tmd bndss
conclusions
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Boundedness

• An execution is bounded iff for all states of the
  execution the number of tokens in all channels is
  at most some bound.
• SDFG
• bounded iff it has a bounded execution.
  ((un-)timed)
• Strictly bounded iff all possible executions are
  bounded. ((un-)timed)
• Self-timed bounded iff self-timed execution is
  bounded. (timed)

Necessary and sufficient conditions for an SDFG
to be live in combination with each of the three
definitions of boundedness.
boundedness
intro
definitions
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conclusions
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Liveness and Boundedness

• A live SDFG is bounded iff it is consistent.
• A strongly connected SDFG is live iff it is
  deadlock free.
• If one strongly connected component (SCC) in an
  SDFG G deadlocks then either G deadlocks or it is
  unbounded.

An SDFG is live and bounded iff it is consistent
and all its SCCs are deadlock free.
3
3
2
(A,3), (B, 3), (C, 2)
C,1
A,2
B,1
1
1
2
3
1
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Liveness and Strict Boundedness

• A live SDFG is strictly bounded iff it is
  consistent and strongly connected.

3
3
2
C,1
B,1
5
3
2
1
An SDFG is live and strictly bounded iff it is
deadlock free, consistent and strongly connected.
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Throughput Definition

• Actor throughput
• The average number of firings of one actor per
  time unit in the self-timed execution.
• (Normalized) graph throughput (if SDFG is
  consistent)

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Local and Normalized Throughput
Each firing of a provides tokens for p/c firings
of b.
p
c
b
a
Normalized actor throughput of a for b
x
a,E
Local actor throughput
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Throughput Relation between Actors
Throughput of each actor equals the minimum of
its local throughput and normalized throughput of
its predecessors.
1/2
NTh(b,a)1/4NTh(c,a)4/3NTh(d,a)3
Th(b)1Th(c)2Th(d)3
1
b
1/4
4
a,1
3
2
c
4/3
LTh(a)1/2
2
d
3
2
Th(a)1/4
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Main Rule for Self-timed Boundedness
p
c
b
a
A channel between a and b is self-timed bounded
iff
Golden Rule
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Liveness and Self-timed Boundedness for Acyclic
SDFGs

• Liveness and self-timed boundedness for an
  acyclic SDFG
• Acyclic SDFGs are live ?
• Checking self-timed boundedness for acyclic SDFGs
• Calculate the throughput for all actors
• Sort the graph topologically
• Calculate the throughput for the source actors
  (actors without any predecessors) and propagate
  the calculation
• For each channel between actors a and b check the
  golden rule.

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intro
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Example for Acyclic SDFGs
1
b
1/4
4
a,1
1/4
3
Self-timed unbounded !!!
2
C
4/3
2
d
3
2
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Reduction
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Properties of the Reduced SDFG

• The reduction preserves
• Throughput
• Self-timed boundedness
• Liveness

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definitions
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Example for a General SDFG
3
3
2
C,1
A,2
B,1
1
1
2
3
1
x2,4
x1,6
1/6 lt 1/4
Self-timed bounded ?
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intro
definitions
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Conclusions

• Necessary and sufficient conditions for checking
• Liveness and boundedness of (timed) SDFGs
• Liveness and strict boundedness of (timed) SDFGs
• Liveness and self-timed boundedness of timed
  SDFGs
• Throughput calculations for general SDFGs (not
  necessarily strongly connected!)

boundedness
intro
definitions
strict bndss
self-tmd bndss
conclusions
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Questions
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