Learning from BDDs in SATbased Bounded Model Checking

1
Learning from BDDs in SAT-based Bounded Model
Checking

• Aarti Gupta, NEC Labs America
• Malay Ganai, NEC Labs America
• Chao Wang, University of Colorado
• Zijiang (James) Yang, NEC Labs America
• Pranav Ashar, NEC Labs America

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Boolean Satisfiability (SAT)

• Given a Boolean formula
• find an assignment to the variables such
  thatformula evaluates to true
• or, prove that no such assignment exists
• Decision Procedure
• Davis Logemann Loveland (DLL/DPLL) algorithm
• Branch-and-backtracking search
• Applications in VLSI CAD
• ATPG, timing analysis, FPGA routing,
• Verification
• equivalence checking
• bounded model checking
• image computation
• unbounded model checking

3
Bounded Model Checking (BMC)

• BMC Given a design, and a correctness
  property,search for a bounded depth
  counterexample
• by unrolling time frames of design
• by specifying constraints on design signals

• BMC problem for a given depth k
• translated to a Boolean formula, where formula
  issatisfiable if and only if a counterexample
  exists
• SAT Check performed by a backend SAT solver
• Typically, use BMC for increasing k
• For deeper searches, efficiency of SAT solver is
  critical

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SAT DLL Decision Procedure
SAT_Solve() if (preprocess() CONFLICT)
return UNSAT while (1) if
(decide_next_branch()) while (deduce()
CONFLICT) blevel analyze_conflict()
if (blevel lt 0) return UNSAT else
backtrack(blevel) else return SAT

• Main Engines
• Decision for choosing which variable to branch
  on,and what value
• Deduction for performing BCP and checking
  conflicts
• Diagnosis for conflict-driven learning and
  backtracking
• Modern SAT Solvers Grasp, SATO, Chaff, BerkMin,
  Hybrid

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Motivation

• Improve SAT Solvers performance by Learning
• What is Learning (in SAT Solvers) ?
• Adds new but redundant clauses to the given
  problem
• Though redundant, they are useful in pruning
  search
• Learning Techniques in Modern SAT Solvers
• Conflict-driven learning key feature Silva 96
• Structural circuit simplifications Kuehl 01
• Recursive learning Kunz 93
• Stalmarcks algorithm Stalm 89
• Our Contribution Efficient learning from BDDs

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Related Work

• Combining BDDs and SAT both explore search space
  
• Equivalence checking Jain95, Kunz95, Gup98,
  Burch98,
• Image computation Gupta00
• Model checking Williams00, Abdulla00
• BMC Anderson02
• Combining BDDs and SAT in BMC framework
• Goal simplification Anderson02
• BDD sweeping for simplifying transition relation
  Kuehl97
• BDD reachability constraints Cabodi03, Gupta03
• BDD Learning is orthogonal to such use, and can
  be also combined with clause replication
  Strich01
• Contrast
• Dam03 Apply CNF-based SAT techniques to BDDs
• This talk Add information from BDDs into
  CNF-based SAT

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Outline

• Background
• SAT-based BMC
• SAT Decision Procedure
• Motivation and Related Work
• BDD Learning
• Review Conflict-driven learning
• BDD Learning Technique
• BDD Learning Engine
• Integration with a Standard SAT Solver
• Experimental Results
• Conclusions

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Review Conflict-Driven Learning

• x1 x4
• x1 x3 x8
• x1 x8 x12
• x2 x11
• x7 x3 x9
• x7 x8 x9
• x7 x8 x10
• x7 x10 x12

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Review Conflict-Driven Learning

• Benefits of CDL
• Allows non-chronological backtracking
• Avoids same conflict in future
• Decision heuristics using CDL information are
  more effective

x1
x2
x3
x3
x4
x5
x5
Conflict clause x1 x3 x5
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BDDs and Conflicts

• Each path to 0 in a BDD denotes a conflict on its
  variables
• A BDD captures all conflicts
• Each conflict can be avoided by adding a learned
  clause
• a b c d
• a b e
• Learning
• No need to add each clause
• Select clauses to add

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BDD Learning Technique

• Given a SAT problem
• Build multiple BDDs for parts of the problem
• For each BDD, paths to 0 represent conflicts
• Add learned clauses to avoid selected paths

• Tradeoff between usefulness and overhead
• Useful multiple conflicts are handled
  simultaneously
• Overhead too many learned clauses slow down BCP
• Strategy Effective and Lightweight BDD Learning

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Effective and Lightweight BDD Learning

• Global BDD learning for every circuit node
• Impractical, wasteful
• Targeted BDD Learning for selected circuit nodes
  (Seeds)
• Two Learning Paradigms
• Static BDD Learning
• Seeds are selected statically
• Learned clauses are added statically before
  decisions
• Dynamic BDD Learning
• Seeds are selected dynamically
• Learned clauses are added dynamically during
  decisions
• Heuristics are targeted at achieving a good
  balance between usefulness and overheads

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BDD Learning Engine

• 3 main components
• Seed selection
• Creation of a BDD for the seed
• Generation of learned clauses from the BDD

BDD_Learning_Engine() update_engine_info()
if (ready_for_learning) node
select_a_seed() bdd create_a_bdd(node)
cl_list generate_learned_clauses(bdd)
return (cl_list)
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Seed Selection

• Strategy preempt conflicts in the SAT Solver
• Seed Selection Heuristics (SSH)based on
  Decision heuristics of the SAT Solver
• Next decision
• Past decisions, ranked back from current decision
• Most frequent decisions
• Decisions at back-leap levels Pil 02
• Most frequent backtrack levels

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Creation of BDDs

• Region heuristic
• Fanin cone of chosen seed
• Region around the chosen seed (faninfanout)

• Region across a few logic levels (typically 5-10)
• Avoids BDD blowup, very fast
• Keeps BDD paths short, giving short learned
  clauses
• More useful for implications than long clauses
• Should not duplicate gate constraints

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Generation of Learned Clauses

• Simple cube enumeration of negated BDD
• Only a fixed number of cubes enumerated
• To avoid exponential enumeration
• Maximum clause length enforced, typically 5-10
• Shorter clauses are more useful for implication
• Fixed-depth cube enumeration with Approximation
• Shorter paths to 1 are enumerated (in negated
  BDD)
• Longer paths are approximated to 0
• Some heuristically chosen variables
  areuniversally quantified
• Corresponds to resolution on learned clauses
• Can result in shorter and fewer learned clauses

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Integration with SAT Solver
SAT_Solve() if (preprocess() CONFLICT)
return UNSAT while (1) cl_list
bdd_learning_engine() if (add_clauses(cl_list
) UNSAT) return UNSAT if
(decide_next_branch()) while (deduce()
CONFLICT) blevel analyze_conflict()
if (blevel 0) return UNSAT
else backtrack(blevel) else return SAT

• BDD Learning is in addition to standard CDL
• BDD Learning engine is invoked before a decision
• Static one time only, after preprocess
• Dynamic before each decision, but learning is
  performed every N backtracks (e.g. N100)
• A clause added due to BDD Learning is treated as
  a conflict clause
• affects future implications and decisions

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Dynamic Addition of Learned Clauses

• Effect of added clauses on SAT Solver invariants
• Conflict clause is conflicting/unit when added
• may not be the case for externally learned
  clauses
• Action in SAT Solver depends on clause status
• Conflicting add it immediately
• Unsatisfied, with ? 2 free literals add it
  immediately
• Unit Choice between
• Add immediately, and backtrack to implication
  level
• Add later, when decision level is the implication
  level
• Heuristic add it immediately if (d_level
  imp_level) lt threshold
• Pseudo-satisfied (should have been an
  implicationon the true literal) like a unit
  clause
• Satisfied add it immediately

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Filtering Learned Clauses

• Additional Filters
• Non-local literals assigned at different
  decision levels
• Avoid local learning, inferred directly from
  circuit
• Relevant sum of true/unassigned literals lt
  threshold
• Unrelevant clauses unlikely to cause implications
• Similar to conflict clause deletion criteria in
  SAT Solvers
• Three Levels of Learning Projected usefulness
• Level 1 conflicting 1-literal unit
• Level 2 (Level 1) unit pseudo-satisfied
• Level 3 (Level 2) all filtered clauses

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Prototype DiVer Verification System
DiVer
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Experiments

• Setup
• Six large industrial designs
• BMC used to check safety properties
• 3 hour time limit (2.2 GHz Xeon, 900 MHz Sun
  220R)
• Basic DiVer BMC (No BDD Learning)

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Results Static BDD Learning

• Chose top 20 variables ranked by decision
  heuristic

Mixed Results No Clear Winner
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Results Dynamic BDD Learning

• Significant Improvements with Level 3 Learning

D1 15 clauses, 3 seeds D5 89 clauses, 32
seeds SSH Past Decision (3/6)
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Results Dynamic BDD Learning

• Effect of Maximum Clause Length
• Max length 6 gave best empirical results
• May be related to BDD creation parameters

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Conclusions

• BDD Learning Technique
• Creates BDDs on-the-fly for selected seeds
• Adds useful learned clauses for conflicts to SAT
  Solver
• Static and Dynamic BDD Learning in SAT Solver
• Heuristics for Lightweight and Effective Learning
• Seed selection heuristics
• Can also be applied to selective replication
• Clause filtering heuristics
• Can also be applied to other external learning
  techniques
• Improved search in BMC with Dynamic BDD Learning
• Upto 73 reduction in time for same depth
• Upto 39 more (60 more) time frames searched

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SAT Example
Conjunctive Normal Form (CNF)
Clause
Literal