A Performance Study of BDD-Based Model Checking

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A Performance Study of BDD-Based Model Checking

• Bwolen Yang

Randal E. Bryant, David R. OHallaron, Armin
Biere, Olivier Coudert, Geert Janssen Rajeev K.
Ranjan, Fabio Somenzi
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Motivation for Studying Model Checking (MC)

• MC is an important part of formal verification
• digital circuits and other finite state systems
• BDD is an enabling technology for MC
• Not well studied
• Packages are tuned using combinational circuits
  (CC)
• Qualitative differences between CC and MC
  computations
• CC build outputs, constant time equivalence
  checking
• MC build model, many fixed-points to verify
  the specs
• CC BDD algorithms are polynomial
• MC key BDD algorithms are exponential

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Outline

• BDD Overview
• Organization of this Study
• participants, benchmarks, evaluation process
• Experimental Results
• performance improvements
• characterizations of MC computations
• BDD Evaluation Methodology
• evaluation platform
• various BDD packages
• real workload
• metrics

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BDD Overview

• BDD
• DAG representation for Boolean functions
• fixed order on Boolean variables
• Set Representation
• represent set as Boolean function
• an elements value is true ltgt it is in the set
• Transition Relation Representation
• set of pairs (current to next state transition)
• each state variable is split into two copies
• current state and next state

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BDD Overview (Contd)

• BDD Algorithms
• dynamic programming
• sub-problems (operations)
• recursively apply Shannon decomposition
• memoization computed cache
• Garbage Collection
• recycle unreachable (dead) nodes
• Dynamic Variable Reordering
• BDD graph size depends on the variable order
• sifting based
• nodes in adjacent levels are swapped

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Organization of this StudyParticipants
Armin Biere ABCD Carnegie Mellon / Universität
Karlsruhe Olivier Coudert TiGeR Synopsys /
Monterey Design Systems Geert Janssen
EHV Eindhoven University of Technology Rajeev K.
Ranjan CAL Synopsys Fabio Somenzi
CUDD University of Colorado Bwolen Yang
PBF Carnegie Mellon
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Organization of this StudySetup

• Metrics 17 statistics
• Benchmark 16 SMV execution traces
• traces of BDD-calls from verification of
• cache coherence, Tomasulo, phone, reactor, TCAS
• size
• 6 million - 10 billion sub-operations
• 1 - 600 MB of memory
• Evaluation platform trace driver
• drives BDD packages based on execution trace

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Organization of this StudyEvaluation Process
Phase 1 no dynamic variable reordering Phase 2
with dynamic variable reordering
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Phase 1 Results Initial / Final
speedups gt 100 6 10 - 100 16 5 - 10
11 2 - 5 28
Conclusion collaborative efforts have led to
significant performance improvements
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Phase 1Hypotheses / Experiments

• Computed Cache
• effects of computed cache size
• amounts of repeated sub-problems across time
• Garbage Collection
• reachable / unreachable
• Complement Edge Representation
• work
• space
• Memory Locality for Breadth-First Algorithms

• Computed Cache
• effects of computed cache size
• amounts of repeated sub-problems across time
• Garbage Collection
• reachable / unreachable
• Complement Edge Representation
• work
• space
• Memory Locality for Breadth-First Algorithms

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Phase 1Hypotheses / Experiments (Contd)

• For Comparison
• ISCAS85 combinational circuits (gt 5 sec, lt 1GB)
• c2670, c3540
• 13-bit, 14-bit multipliers based on c6288
• Metrics depends only on the trace and BDD
  algorithms
• machine-independent
• implementation-independent

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Computed CacheRepeated Sub-problems Across Time

• Source of Speedup
• increase computed cache size
• Possible Cause
• many repeated sub-problems are far apart in time
• Validation
• study the number of repeated sub-problems across
• user issued operations (top-level operations).

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HypothesisTop-Level Sharing

• Hypothesis
• MC computations have a large number of repeated
• sub-problems across the top-level operations.
• Experiment
• measure the minimum number of operations with
• GC disabled and complete cache.
• compare this with the same setup, but
• cache is flushed between top-level operations.

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Results on Top-Level Sharing
flush cache flushed between top-level
operations Conclusion large cache is more
important for MC
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Garbage CollectionRebirth Rate

• Source of Speedup
• reduce GC frequency
• Possible Cause
• many dead nodes become reachable again (rebirth)
• GC is delayed till the number of dead nodes
• reaches a threshold
• dead nodes are reborn when they are part of the
• result of new sub-problems

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HypothesisRebirth Rate
Hypothesis MC computations have very high rebirth
rate. Experiment measure the number of deaths and
the number of rebirths
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Results on Rebirth Rate

• Conclusions
• delay garbage collection
• triggering GC should not base only on of dead
  nodes
• delay updating reference counts

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BF BDD Construction
Two packages (CAL and PBF) are BF based.
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BF BDD Construction Overview

• Level-by-Level Access
• operations on same level (variable) are
  processed together
• one queue per level
• Locality
• group nodes of the same level together in memory
• Good memory locality due to BF gt
• of ops processed per queue visit must be high

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Average BF Locality
Conclusion MC traces generally have less BF
locality
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Average BF Locality / Work
Conclusion For comparable BF locality, MC
computations do much more work.
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Phase 1Some Issues / Open Questions

• Memory Management
• space-time tradeoff
• computed cache size / GC frequency
• resource awareness
• available physical memory, memory limit, page
  fault rate
• Top-Level Sharing
• possibly the main cause for
• strong cache dependency
• high rebirth rate
• better understanding may lead to
• better memory management
• higher level algorithms to exploit the pattern

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Phase 2Dynamic Variable Reordering

• BDD Packages Used
• CAL, CUDD, EHV, TiGeR
• improvements from phase 1 incorporated

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Why is Variable Reordering Hard to Study

• Time-space tradeoff
• how much time to spent to reduce graph sizes
• Chaotic behavior
• e.g., small changes to triggering / termination
  criteria
• can have significant performance impact
• Resource intensive
• reordering is expensive
• space of possible orderings is combinatorial
• Different variable order gt different
  computation
• e.g., many dont-care space optimization
  algorithms

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Phase 2Experiments

• Quality of Variable Order Generated
• Variable Grouping Heuristic
• keep strongly related variables adjacent
• Reorder Transition Relation
• BDDs for the transition relation are used
  repeatedly
• Effects of Initial Variable Order
• with and without variable reordering

• Quality of Variable Order Generated
• Variable Grouping Heuristic
• keep strongly related variables adjacent
• Reorder Transition Relation
• BDDs for the transition relation are used
  repeatedly
• Effects of Initial Variable Order
• with and without variable reordering

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Variable Grouping HeuristicGroup Current / Next
Variables

• Current / Next State Variables
• for transition relation, state variable is split
  into two
• current state and next state
• Hypothesis
• Grouping the corresponding current- and
  next-state variables is a good heuristic.
• Experiment
• for both with and without grouping, measure
• work ( of operations)
• space (max of live BDD nodes)
• reorder cost ( of nodes swapped with their
  children)

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Results onGrouping Current / Next Variables
All results are normalized against no variable
grouping. Conclusion grouping is generally
effective
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Effects of Initial Variable OrderExperimental
Setup

• For each trace,
• find a good variable ordering O
• perturb O to generate new variable orderings
• fraction of variables perturbed
• distance moved
• measure the effects of these new orderings
• with and without dynamic variable reordering

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Effects of Initial Variable OrderPerturbation
Algorithm

• Perturbation Parameters (p, d)
• p probability that a variable will be perturbed
• d perturbation distance
• Properties
• in average, p fraction of variables is perturbed
• max distance moved is 2d
• (p 1, d infinity) gt completely random
  variable order
• For each perturbation level (p, d)
• generate a number (sample size) of variable orders

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Effects of Initial Variable OrderParameters

• Parameter Values
• p (0.1, 0.2, , 1.0)
• d (10, 20, , 100, infinity)
• sample size 10
• gt for each trace,
• 1100 orderings
• 2200 runs (w/ and w/o dynamic reordering)

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Effects of Initial Variable OrderSmallest Test
Case

• Base Case (best ordering)
• time 13 sec
• memory 127 MB
• Resource Limits on Generated Orders
• time 128x base case
• memory 500 MB

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Effects of Initial Variable Order Result
of unfinished cases
At 128x/500MB limit, no reorder finished
33, reorder finished
90. Conclusion dynamic reordering is effective
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Phase 2Some Issues / Open Questions

• Computed Cache Flushing
• cost
• Effects of Initial Variable Order
• determine sample size
• Need New Better Experimental Design

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BDD Evaluation Methodology

• Trace-Driven Evaluation Platform
• real workload (BDD-call traces)
• study various BDD packages
• focus on key (expensive) operations
• Evaluation Metrics
• more rigorous quantitative analysis

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BDD Evaluation MethodologyMetrics Time
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BDD Evaluation MethodologyMetrics Space
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Importance of Better MetricsExample Memory
Locality
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Summary

• Collaboration Evaluation Methodology
• significant performance improvements
• up to 2 orders of magnitude
• characterization of MC computation
• computed cache size
• garbage collection frequency
• effects of complement edge
• BF locality
• reordering heuristic
• current / next state variable grouping
• effects of reordering the transition relation
• effects of initial variable orderings
• other general results (not mentioned in this
  talk)
• issues and open questions for future research

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Conclusions

• Rigorous quantitative analysis can lead to
• dramatic performance improvements
• better understanding of computational
  characteristics
• Adopt the evaluation methodology by
• building more benchmark traces
• for IP issues, BDD-call traces are hard to
  understand
• using / improving the proposed metrics for
  future evaluation

For data and BDD traces used in this
study, http//www.cs.cmu.edu/bwolen/fmcad98/
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(No Transcript)
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Phase 1Hypotheses / Experiments

• Computed Cache
• effects of computed cache size
• amounts of repeated sub-problems across time
• Garbage Collection
• reachable / unreachable
• Complement Edge Representation
• work
• space
• Memory Locality for Breadth-First Algorithms

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Phase 2Experiments

• Quality of Variable Order Generated
• Variable Grouping Heuristic
• keep strongly related variables adjacent
• Reorder Transition Relation
• BDDs for the transition relation are used
  repeatedly
• Effects of Initial Variable Order
• for both with and without variable reordering

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Benchmark Sizes
Min Ops minimum number of sub-problems/operations
(no GC and complete cache) Max Live Nodes
maximum number of live BDD nodes
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Phase 1 Before/After Cumulative
Speedup Histogram
6 packages 16 traces 96 cases
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Computed Cache Size Dependency

• Hypothesis
• The computed cache is more important for MC than
  for CC.
• Experiment
• Vary the cache size and measure its effects on
  work.
• size as a percentage of BDD nodes
• normalize the result to minimum amount of work
• necessary i.e., no GC and complete cache.

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Effects of Computed Cache Size
of ops normalized to the minimum number of
operations cache size of BDD nodes
Conclusion large cache is important for MC
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Death Rate
Conclusion death rate for MC can be very high
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Effects of Complement Edge
Work
Conclusion complement edge only affects work for
CC
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Effects of Complement Edge
Space
Conclusion complement edge does not affect
space Note maximum of live BDD nodes would be
better measure
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Phase 2 ResultsWith / Without Reorder
4 packages 16 traces 64 cases
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Variable Reordering Effects of Reordering
Transition Relation Only
All results are normalized against variable
reordering w/ grouping.
nodes swapped number of nodes swapped with their
children
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Quality of the New Variable Order

• Experiment
• use the final variable ordering as the new
  initial order.
• compare the results using the new initial order
  with
• the results using the original variable order.

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Results on Quality of the New Variable Order
Conclusion qualities of new variable orders are
generally good.
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No Reorder (gt 4x or gt 500Mb)
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gt 4x or gt 500Mb
Conclusions For very low perturbation,
reordering does not work well.
Overall, very few cases get finished.
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gt 32x or gt 500Mb
Conclusion variable reordering worked rather well
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Memory Out (gt 512Mb)
Conclusion memory intensive at highly perturbed
region.
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Timed Out (gt 128x)
diagonal band from lower-left to upper-right
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Issues and Open QuestionsCross Top-Level Sharing
Potential experiment identify how far apart these
repetitions are
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Issues and Open QuestionsInconsistent
Cross-Platform Results
Probable Cause memory hierarchy This may shed
some light on the memory locality issues.