An Efficient Hybrid Constraint Solver for RTL Circuits

1
An Efficient Hybrid Constraint Solver for RTL
Circuits
2
Overview

• Boolean SAT Checking very fast, but
• Cost of RTL synthesis
• Poor performance on RTL with arithmetic
• Combine Boolean SAT with Arithmetic solving
• Enables direct SAT Checking on RTL
• Avoid costs of synthesis to Boolean domain
• Leverage RTL abstraction

3
Outline

• Prior Work
• DPLL
• Modeling
• HDPLL
• Conflict-based Learning
• Experimental Data
• Future Directions

4
Prior Work

• Barrett et. al SVC FMCAD 1996
• Implementation of Shostaks decision procedure
• Boolean, Ints, Arrays, Un-interpreted fns
• C. Liu et al. ICCAD, 2003
• CAMA SAT check of multi-valued logic
• Explicit representation impractical for
  arithmetic
• Fallah et al. DAC 1999, Huang et. al.DAC 2000
• Zeng et al. DATE 2001, M. Iyer ITC, 2003
• All have performance issues on RTL.

5
Modern Boolean DPLL

• while (Decide() ! Done) do
• while (bcp() conflict) do
• blevel analyzeconflicts()
• if (blevel 0) then
• return UNSATISFIABLE
• else
• backtrack(blevel)
• end if
• end while
• end while
• return SATISFIABLE

• Key Factors for Performance
• Fast BCP 2 watched literals Dynamic Decision
  Ordering based on learning Moskowicz et. al.
• Conflict-based learning Non-chronological
  backtracking Stallman, Marques-Silva et.al.

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Finite-Domain Constraints

• All RTL variables have finite domains
• Finite upper/lower-bounds
• eg. reg register70
• Boolean variables also have finite domains
• 0,1
• A constraint is defined as
• å ai . xi ? r , where, xi , r ? I
• Primitive Constraints have at most 3 variables
• RTL descriptions written as conjunct of
  constraints

7
Modeling RTL Operators

• Max Range L 2n 1, where n is bit-width of
  primitive
• Word operators can have Boolean outputs/inputs
• These Boolean variables are interface points

b
B
?
b2
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Modeling RTL Operators

• Model non-linear ops eg., concat, shifts by
• Bit-slicing Brinkmann et al. VLSI Design, 2001

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Hybrid DPLL
loop while (Decide() ! Done) while
(FdcpFme() conflict) blevel
hybrid_analyzeconflicts() if (blevel 0)
then return UNSATISFIABLE
else backtrack(blevel) end loop

• Decide() only on Boolean variables
• FdcpFME() does constraint propagation

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Consistency of Constraints

• Constraint locally consistent if some value in
  domain of each variable satisfies it
• a 2,...,3 b 2,..,3 ? 5 Consistent
  Constraint
• a 0,...,2 b 0,..,2 ? 5 Inconsistent
  Constraint
• Globally consistent if all constraints
  simultaneously consistent
• a 0,...,2 b 2,..,3 ? 5
  (1)
• a 0,...,2 c 0,..,1 ? 1
  (2)
• only a 2 satisfies (1),
• but that causes (2) to be inconsistent.

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Detecting Inconsistency

• Constraint propagation detects local
  inconsistency
• Bounds propagation very efficient
• Ensures bounds of variable consistent with
  constraint
• Caveat
• Even if all constraints are locally
  consistent, that does not mean that constraint
  set is globally consistent

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Finite-Domain Constraint Propagation

• Similar to Boolean Constraint Propagation(BCP)
• BCP has one rule to rule them all
• .. a clause shall imply only when all literals
  in the clause but one has gone to false.. The
  unit clause rule
• FDCP has many rules
• Efficiency suffers
• Solutions to efficiency problem
• Minimize number of variables in constraint
• Minimize evaluations for implication
• Evaluate when implication guaranteed

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Constraint Watching
5
a 2 b 2,..,7 c 1,2
?

RHS (r)
LHS
vg
vi
vi -

• D(vi ) is the domain of word variable vi
• Watch bounds of constraint

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Constraint Watching rules

• Constraint is satisfied if wsumc wsumlb ? r
• No additional implications
• Constraint is conflicting if wsumc wsumub lt r
• If wsumc lt r and wsumub- wsumlb- wsumc ? r
• Constraint currently inconsistent
• Can be made consistent by bound changes

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Completeness Issues

• Constraint Propagation cannot guarantee
  consistency under all conditions
• Can fix by deciding on all variables
• Eventually the conflict will be detected if it
  exists
• But we dont make decisions in data-path
• We use Omega Test for global consistency
• Implementation of Fourier-Motzkin elimination
• Complete decision procedure for integer arithmetic

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Global Consistency ? Use FME
3
b0 1

• Conflict not caught by constraint propagation
• No conflict does not guarantee satisfiability
• Need to ensure completeness
• So, use FME as global consistency check procedure

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HDPLL In Action
W1 5
5
i1 1
w5

W2 6
Proposition
6
i2 0
w7
S0 0

b0 1
decision
7
i3 1
W3 7

w6
W4 8
S1 1
decision

• Comparators are remodeled as two geqs each
• i1 modeled by b1 ? w7 ? 5, b2 ? 5 ? w7
• i2 modeled by b3 ? w7 ? 6, b4 ? 6 ? w7
• i3 modeled by b5 ? w7 ? 7, b6 ? 7 ? w7
• b1, .. , b6 are Boolean

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Hybrid Conflict-based learning

• Learning UIPs implies learning hybrid clauses
• Conflict analysis on implication graph is complex
• Compromise by learning only on interface points

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Experimental Results

• Solver is combinational
• Safety property checking on ITC benchmarks using
  BMC Biere et. al. DAC 1999
• Properties on b01, b02, b04 and b11
• Expanded by 10, 15 and 20 time-frames
• 2.4 GHz Pentium 4, 1 GB RDRAM
• Time-limit of 3600 CPU secs

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Benchmark suite characteristics
Number of Operators
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Relative size of Boolean/Arithmetic
Boolean to Arithmetic Ratio
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Effect of Constraint Propagation
100 CPU-time (secs)
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Comparison with SVC
100 CPU-time (secs)
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Conclusion

• HDPLL viable for RTL SAT checking
• Faster than SVC on benchmarks
• Strengths of paper
• Efficient integration of two decision procedures
• Efficient FDCP
• Conflict-based Learning
• Areas for improvement
• Current conflict-based learning inefficient
• Structure of data-path not cleverly used
• Not sequential
• Modeling issues remain

25
Future Work

• Hybrid UIP Learning
• Hybrid clauses has Boolean and Word variables
• Two-literal watching for hybrid clauses
• Resolution-graph based learning in FME
• Advanced learning and decision strategies
• Hybrid learning for sequential search
• HDPLL decision proc. for opt. engine

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The End
27
Prior Work In detail
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Hybrid Constraint Propagation
for all (ai ? impqueue) do if (fdcp (ai)
conflict) then return conflict end if end
for for all (ai ? impqueue and ai ? IPO IPI )
do if (fme () conflict) then return
conflict end if end for

• Impqueue holds event driven implications to be
  evaluated
• Fdcp(ai) processes one event
• Fme() checks all current values on interface
  points
• Result is complete

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Improvements on DAC techniques
100 CPU-time (secs)
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Future Work

• Decision making on all variables
• Make decisions on Word and Boolean variables
• Advanced learning and decision strategies
• Advanced static and dynamic learning tech.
• Auto. constraint extraction Integration of more
  theories
• Hybrid learning for sequential search
• HDPLL decision proc. for opt. engine