spqr - security and privacy research group - qbf-based … · 2014-05-20 · 2 march 2014 qbf...

QBF-BASED SYNTHESIS OF OPTIMAL WORD-SPLITTING IN APPROXIMATE MULTI-LEVEL CELLSDaniel E. Holcomb Kevin Fu University of Michigan

QBF Synthesis of Optimal Word-Splitting2 March 2014

Motivation

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Motivation• Multi-level cells (MLCs)

• Store 4 bits per cell using 16 discrete levels

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• Store 4 bits per cell using 16 discrete levels• Approximate computing

• Save resources by allowing errors [Sampson et al. ’13] • Errors between nearby levels • But still want to minimize impact of errors

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• Store 4 bits per cell using 16 discrete levels• Approximate computing

• Save resources by allowing errors [Sampson et al. ’13] • Errors between nearby levels • But still want to minimize impact of errors

• Design decisions • Data words don't fit in one MLC • How to split word across multiple MLCs?

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Example

MLCs:

Data Word:

??? ???

Write Operation Read Operation

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Example

MLCs:

Data Word:

Write Operation Read Operation

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Example

MLCs:

Data Word:

Write Operation Read Operationwrite 96

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0 1 1 0 0 0 0 0


Example

MLCs:

Data Word:


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write 3 write 00 0 1 1 0 0 0 0

0 1 1 0 0 0 0 0


Example

MLCs:

Data Word:


!3

write 3 read 4write 0 read 00 0 1 1 0 0 0 0 0 0 0 00 1 0 0

0 1 1 0 0 0 0 0


Example

MLCs:

Data Word:


!3

write 3 read 4

MLC error = 1

write 0 read 00 0 1 1 0 0 0 0 0 0 0 00 1 0 0

0 1 1 0 0 0 0 0


Example

MLCs:

Data Word:

Write Operation Read Operationwrite 96 read 16

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write 3 read 4

MLC error = 1

write 0 read 00 0 1 1 0 0 0 0 0 0 0 00 1 0 0

0 0 0 1 0 0 0 00 1 1 0 0 0 0 0


Example

MLCs:

Data Word:


word error = 80

!3

write 3 read 4

MLC error = 1

write 0 read 00 0 1 1 0 0 0 0 0 0 0 00 1 0 0

0 0 0 1 0 0 0 00 1 1 0 0 0 0 0


Example

MLCs:

Data Word:


word error = 80

!3

write 3 read 4

MLC error = 1

write 0 read 00 0 1 1 0 0 0 0 0 0 0 00 1 0 0

0 0 0 1 0 0 0 00 1 1 0 0 0 0 0

Assuming a fixed upper bound on MLC error, can we find a mapping between data word and MLCs that minimizes worst-case word error?


QBF Formulation of Word-Splitting

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DW

C1R C0R

SXBAR XBAR



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• QBF: SAT with alternating quantifiers• Formulate QBF-based synthesis by way

of combinational circuit

DW

C1R C0R

SXBAR XBAR



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of combinational circuit• Crossbars implement mapping

between data words and MLCs• 64 bits specify xbar connections• 8-bit written dataword • 4-bit value read from each MLC

DW

C1R C0R

SXBAR XBAR



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DW

C1R C0R

S

C1WC0W

DR

XBAR XBAR



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• Check word and MLC error bounds

DW

C1R C0R

S

C1WC0W

DR

XBAR XBAR


Results• QBF-based synthesis to find optimal word-splitting

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the probability of having a word error of 0 is less than25%, because the remapping M introduces a systematicerror even when the cells are error free.

Figure 7 shows the average word error as a function ofthe probability of a single-level error in each MLC. Whenthe cell error probability is 0, the only approach to have non-zero word-level error is the synthesized word-splitting withremapping. This error is on account of the remapping, whichintroduces a systematic error for the remapped levels. As theprobability of cell error increases, the synthesized result withremapping has the best average-case error, in addition to thebest worst-case error of 10.

0

0.2

0.4

0 20 40 60 80 100 120 140

pro

bab

ilit

y

word-level error

interleaved

0

0.2

0.4

0 20 40 60 80 100 120 140

pro

bab

ilit

y

word-level error

synthesized

0

0.2

0.4

0 20 40 60 80 100 120 140

pro

bab

ilit

y

word-level error

synthesized, with remapping

Figure 6. Probability distribution of word-level error(|DR � DW |) using the three approaches discussed in thepaper. The distribution is obtained from 500k random trialsusing Verilog simulation. The results are for the the inter-leaved word-splitting and the two synthesized word-splittingapproaches. The probability of a single-level error in eachcell is 50%, and the errors are independent for each cell. Thevertical lines are the average word-level error and the largesterror observed in the random trials.

0

0.25

0.5

0.75

1

1.25

0 0.02 0.04 0.06 0.08 0.1

avg

wo

rd-l

evel

err

or

probability of cell error

interleavedsynthesized

synthesized, with remapping

Figure 7. Observed average word-level error (|DR �DW |)as a function of the probability of a single-level error in eachMLC.

6. Future Work and ConclusionsThis work proposed the use of automated synthesis tech-niques for splitting data words across approximate MLCstorage, such that the worst-case error is minimized atthe word-level. The synthesis technique is based on QBF-solving, and the results are validated through random simu-lation.

While MLC errors will generally occur as transitionsbetween neighboring levels as assumed in this work, fu-ture work will explore an error model that is more closelymatched to the particular errors of a specific MLC technol-ogy. Other QBF solving techniques such as quantifier instan-tiation can be applied. To address the limitation of havingno preference among solutions with equivalent worst-caseword-level error bounds (Sec. 5.3), the synthesis techniquecan be extended to consider all solutions that guarantee theoptimal worst-case bound, and to choose among them basedon a metric such as average-case word-level error.Acknowledgement: This work was supported in part byC-FAR, one of six centers of STARnet, a SemiconductorResearch Corporation program sponsored by MARCO andDARPA, and NSF CNS-0845874. Any opinions, findings,conclusions, and recommendations expressed in these mate-rials are those of the authors and do not necessarily reflectthe views of the sponsors

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