Variable Ordering Using aBDD Based Sampling

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Variable Ordering Using aBDDBased Sampling

• Yuan Lu
• Electrical and Computer Engineering
• Carnegie Mellon University

Co-author Jawahar Jain, Fujitsu Lab of
America Edmund Clarke, Carnegie Mellon
University Masahiro Fujita, University of Tokyo
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Outline

• Motivation
• Related Work
• Cube-based Sampling
• Window-based Sampling
• Experiments
• Conclusion and Future work

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Motivation

• BDDs are widely used in synthesis and
  verification
• Good variable ordering is essential
• Existing techniques are not adequate
• static ordering is fast, but lacks power
• dynamic reordering is slower, but efficient
• Obtaining a better order may be a hit-and-trial
  process
• numerous experiments are performed
• Sampling based approaches are promising

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Related Work

• Static Variable Ordering (Fujita88, Malik88)
• Dynamic Reordering (Fujita91,Rudell93,Panda95)
• reasonably powerful
• slower, local minimum
• Function Sampling-based Ordering (Jain98)
• application driven, forward look-ahead
• faster, sometimes very powerful
• Implemented using cube-based sampling
• large deviation in BDD size unpredictable
• BDD Manager based-sampling (Meinel98)
• Lacks forward look-ahead Not the focus of this
  paper

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Cube-based Sampling (Jain98)

• Given a Boolean function F(x1, x2, x3, ...), it
  is likely that a good variable order for F(0,1,
  x3,) is also a good order for F.
• Assume cube c!x1 x2, F(0,1,) F(x1,x2,) c
• Obtaining good orders for smaller function is
  simpler than original functions.
• Sample the Boolean space using random cubes
• Select the best from the obtained orders
• Unsolved puzzles
• Which cube to be chosen?
• How to average the effects from different orders?

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Window-based Sampling

• A cube may not represent the behavior of the full
  function, e.g. F(0,1, x3,) 1
• Instead of using one cube, more than one cube
  should be used w c1 c2 ...
• How to generate good windows?

Cube-based sampling
Window-based sampling
Boolean space
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Abstract BDDs

• Given abstraction function h, abstract BDT H(f)
  of f is constructed as follows.
• Nodes are classified into equivalence classes
• One representative v is chosen from each class
• Let , if
• Abstract BDT
• Different definitions for abstract BDDs are
  possible

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Abstract BDDs Example

• Assume
• Abstract BDT for is

0
0
1
1
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Abstract BDDs Properties

• Distribution property
• Let c(v) be the cube corresponding to v,

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Window-based Sampling Using aBDDs

• Use aBDDs to generate window samples
• Advantages
• Averaging effects of multiple cubes
• Produce more stable ordering than cube-based
  approaches
• Procedure
• Choose variables to be abstracted
• Generate window samples
• Select best samples

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Variable Ordering Methodology
Abstraction
estimation
candidate order select
functions
orders
circuit filter
evolution filter
orders
final initial order
Similar methodology as cube-based sampling
(Jain98)
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Experiments

• Experiments on combinational circuits
• Experiment I static ordering for single-output
  functions
• Experiment II dynamic ordering for single-output
  functions
• Experiment III dynamic ordering for
  multiple-output circuits
• Experiment on sequential circuits
• generate initial ordering for model checking PCI
  bus protocol

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Experiment I Static Ordering
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Experiment I Static Ordering
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Experiments II Dynamic Reordering
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Experiment II Dynamic Ordering
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Experiments IIIMultiple Outputs
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Variable Ordering for Model Checking

• Variable ordering is essential for model checking
• Current dynamic reordering approaches do not have
  the forward look-ahead capability
• Abstraction-based variable ordering
• Given a set of abstraction functions, the system
  automatically build abstract Kripke structure.
• CTL properties are preprocessed on the abstract
  Kripke structure.
• The obtained order is used as initial order for
  the final model checking process.

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Experiments IVModel Checking
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Conclusion

• Our approach overcomes the large deviation
  problem of the cube-based sampling approach.
• Window-based sampling techniques are efficient to
  obtain good variable orders.
• Window-based sampling can use other forms instead
  of aBDDs, e.g. decomposed BDDs.
• We are currently investigating how to incorporate
  similar ideas inside BDD packages instead of
  exploring the circuit structures.