Minimizing Buffer Requirements of Synchronous Dataflow Graphs with Model Checking

1
Minimizing Buffer Requirements of Synchronous
Dataflow Graphs with Model Checking

• Marc Geilen, Twan Basten and
• Sander Stuijk

2
Introduction

• DSP software synthesis
• Single appearance schedule (SAS)
• Minimize code size
• Multi-processor systems-on-chip
• Kernels in multimedia applications
• Reduce cost and improve energy efficiency

3
Synchronous Dataflow Graphs Lee87
actor
channel
rate
token
4
Storage space
(a,2),(ß,0)
Tokens on channels must be stored in memory.

• Shared memory for all channels
• Separate memory for each channel
• Shared memory for sets of channels

5
Problem definition
(a,2),(ß,0)

• Problem
• Smallest size storage capacity for deadlockfree
  execution.
• Schedule that realizes a continuing, infinite
  execution within these bounds.
• Problem is NP-complete Bhatt96
• Heuristics to determine small but sufficient
  buffersizes exist.

6
SDF state space
(a,4),(ß,0)
(a,0),(ß,0)
(a,2),(ß,0)
(a,0),(ß,0)
(a,4),(ß,0)
(a,2),(ß,0)
(a,1),(ß,1)
7
Scheduling

• Schedules are (infinite) paths through theSDF
  state space
• Cycles are periodic schedules
• There exists a bounded schedule if and onlyif
  there is a cycle

8
A schedule and the storage space

• The buffer size of a schedule is the LUB of the
  numbersof tokens in the buffers during the
  schedule
• Shared maximum of the sum of the number of
  stored tokens
• Separate sum of the maximum numbers of stored
  tokens

9
Problem definition

• Problem
• Smallest size storage capacity for deadlock free
  execution.
• Schedule that realizes a continuing, infinite
  execution within these bounds.
• In terms of the state-space
• Find a cycle in the SDF state-
• space with the smallest bound

10
Model checking

• Model-checking
• Used for NP-complete scheduling problems
• Search complete state space to prove a property
• Use partial-order reduction to reduce state space
• Model-checker SPIN
• General-purpose model-checking tool
• State space defined in modeling language Promela
• State spaces of up to 108 states
• Partial search modes for larger spaces

11
Model-checking state space
(0,0)

• SDF state (a, ß)

• Model-checking state (a, ß, max a, max ß)

(0,0,0,0)
(2,0,2,0)
(4,0,4,0)
(1,1,4,1)
(3,1,4,1)
(0,2,4,2)
(0,0,4,2)
A
B
C
A
A
B
A
B
(2,0,4,2)
(3,1,4,2)
A
A
B
(4,0,4,2)
(1,1,4,2)
12
Computing minimum buffer size

• Verification Claim
• There exists no schedule of the graph which does
  not exceed the total buffer size b
• Counter example if the claim is not true(i.e.
  periodic schedule)
• Shared and separate memory case can be modeled
• Binary search to find the smallest b for which
  the claimis false

13
Problem space exploration

• Upper-bound (ub) using balance equations Lee87
• Lower-bound (lb) per channel Murthy96
• Binary search to find minimal buffer requirements

14
Encoding in Promela (separate memory)

• int ch2 Channel contents (a ch0, ß
  ch1)
• int sz2 Channel sizes

Claim always eventually SUMgtBOUND define p
SUMgtBOUND define BOUND 6 define SUM
sz0sz1
15
Encoding in Promela
define WAIT(c,n) chcgtn define CONSUME(c,n)
chc chc - n define PRODUCE(c,n) chc
chc n UPDATE(c) define UPDATE(c) if
chcgtszc -gt szc chc else fi

• proctype Actor_a()
• do
• atomic
• PRODUCE(0,2)
• od

• proctype Actor_b()
• do
• atomic
• WAIT(0,3) -gt
• CONSUME(0,3)
• PRODUCE(1,1)
• od

16
Experiment sample rate converter
17
Experiment modem
1
1
1
1
1
1
1
2
1
1
2
1
1
2
1
1
2
2
2
2
2
2
1
1
1
8
2
4
2
2
2
2
1
1
2
2
1
18
Experiment example graph
19
Experiment H.263 decoder
20
Conclusions and future work

• Exact minimum memory requirements.
• Different measures of storage considered.
• Experiments show encouraging results.
• Use additional partial-order reductions to
  reducemodel-checking state space.
• Explore throughput vs. memory trade-off.
• Extend the approach to BDF graphs.

21
Experimental results

• Separate memory

• Shared memory