Structural Search for RTL with Predicate Learning

1
Structural Search for RTL with Predicate Learning

• G. Parthasarathy
• M.K. Iyer
• K.-T. Cheng
• F. Brewer
• Department of ECE
• University of California Santa Barbara
• Santa Barbara, CA

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SAT on RTL Some History

• Satisfiability Checking (SAT) on RTL attractive
• RTL abstracts data operations
• Leverage abstraction for speed and ease of use
• SAT has problems on large data-path circuits
• Brinkmann 02, Strichmann,03

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HDPLL Fundamentals

• HDPLL combines Boolean and Arithmetic theories
• Branch and Bound decision procedure
• Decision-Making on Boolean Variables
• Omega-Test for arithmetic feasibility checking
• Combined Interval-Boolean constraint propagation
• Parthasarathy et. al., DAC04
• Modeling
• Data Path Finite-domain Linear Arith.
  constraints
• Parthasarathy et. al., DAC04
• Boolean control AND-INV graph (AIG)
• Kuehlmann et. al., DAC01
• Conflict-based learning on combined theories
• Iyer et. al., DATE05

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Interval Constraints

• All RTL variables have finite domains
• Finite upper/lower-bounds e.g., reg A70
• Boolean variables have finite domains 0,1
• Max Range L 2n 1, where n is bit-width
• An interval constraint is defined as
• å ai . xi ? r , where, xi , r ? I
• RTL modeled as Boolean functions over conjuncts
  of interval constraints

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Hybrid DPLL - Algorithm

• Input
• Query ? on model ?
• Returns
• SAT or UNSAT

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Interval Constraint Propagation (ICP)

• deduce () uses ICP and BCP
• Interval Constraint is defined as
• å ai . xi ? r , where, xi , r ? I
• ai integer coefficients
• xi integer interval variables
• å is interval summation
• x1lb , x1ub x2lb , x2ub x1lb x2lb ,
  x1ub x2ub
• x1lb , x1ub - x2lb , x2ub x1lb - x2ub ,
  x1ub - x2lb
• Projection Operation proj on variable xi
• xi ? integer (1/ai (r - å aj . xj ) j ? i) for
  all constraints
• proj(xi) xilb, xiub must satisfy the
  inequality

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An example of ICP by projection

• 3y 4x ? 7
• x, y are 3 bit variables with range 0,..,7
• Assume change on y 0,..,7?0,..,4
• Implied change on x by proj(x)
• -x ? (¼ (7 30,..,4 ))
• x ? integer( -1.25,..,1.75 )
• Real Intervals inwardly rounded to integer
  interval.
• x ? -1,..,1
• Result xub reduced to 1.
• xlb does not change
• -1 is already less than xlb 0

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Recursive Learning on Boolean Circuits

• Proposed by Kunz and Pradhan, ITC02
• Based on case and Boolean implication analysis
• Basic Idea
• If all ways to satisfy (vi vali) implies (vj
  valj)
• Then (vi vali ) always implies (vj valj )
• Very effective in learning surprise relations
• Recursion depth can be unbounded
• Is then complete, but typically very costly
• Typically, recursion depth bounded for efficiency

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Recursive Learning Example
All ways to satisfy o21 implies o1 1. Hence,
o21 implies o1 1!
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Recursive learning on RTL Basics

• Key components of recursive learning
• Case analysis Decisions
• Boolean constraint propagation (BCP)
• Key property BCP is bijective
• Problems with RTL
• Decisions on words exponential in word-size
• ICP is neither onto nor not bijective
• For example, assume (wi 0..3) ? (wk 0..3
  )
• This doesnt mean that (wi 1) ? (wk 1)
• Solution
• Make decisions on Boolean Predicates
• Use ICP for implications across predicates

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Recursive learning on RTL Details

• Pick predicate functions as RL points
• Perform RL on RL points in topological order
• Use combined BCP-ICP for finding implications
• Use conflict-based learning for conflict analysis
• All RL relations and conflict-clauses stored for
  use in future RL steps
• Stop after cut-off

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RTL Recursive learning Example

• b5 0
• b1 0 ? w1 0, b2 0,b6 0
• b0 0 ? b6 0
• Therefore, b5 ? b6
• Learn (b5 ?b6)
• b6 0
• b2 0 ? w1 0, b1 0,b5 0
• b0 0 ? b5 0
• Therefore, b6 ? b5
• Learn (b6 ?b5)
• Bi-implication
• b5 ? b6 and b6 ? b5
• b6 b5
• Similarly b8 b9

0
0
0
0
0
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Experimental Analysis

• Compare effect of predicate learning with
  base-line
• BMC of safety properties in ITC99 benchmarks
• b01, b02, b04, b11, b13
• Cut off of 2500 relations for predicate learning
• Cut off of 1200 cpu seconds for overall solving

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Effect of Predicate Learning
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Analysis of approach

• Complete RL on RTL by this method is infeasible
• Limited RL can yield high performance gains
• Key result
• Leveraging Predicate correlation is important
• Question
• How do we do this efficiently ?
• Answer
• Look back to ATPG on Boolean circuits
• Structure of RTL can guide search process

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Structural Search on Circuits

• Key Concepts
• All operators in circuit obey rules of
  justification
• Unjustified if output value not implied by inputs
• Justified otherwise
• Data-structure called J-frontier guides search
• Similar to ATPG techniques
• Key property of J-frontier
• Query is SAT if and only if all operators in
  support of query are justified

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Justification on RTL Circuits

• Justification status of RTL operators
• Atomic Boolean ops
• Same as for Boolean gates
• If the output value cannot be uniquely determined
  by current values on inputs
• Non-Arithmetic RTL ops eg. ITE, CONCAT
• Operator has Boolean input(s)
• Output interval cant be implied from current
  values
• Pure Arithmetic ops Not justifiable

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Justification in RTL an Example

• 3-bit word variables
• Values 0..7
• Proposition b7 1
• Implications
• w2 6,7
• b40, b50, b61, w4 5
• J-Frontier w4
• w4 ? w3
• Decide b1 0
• Implications
• w3 5
• J-Frontier w3
• w3 ? w1
• Decide b2 0
• Implications
• w1 5
• J-Frontier ?

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Conflicts during justification
b3
b1
7
6
7
b21
w1
6
5
6

• Proposition b7 1
• Assume b2 1
• w3 6 w2 6,7
• No J-Path for w4 5
• J-conflict !
• Analyze implication graph to learn causes for
  conflict
• Causes for conflict
• b6 1 and b2 1.
• Learned clause
• (? b6 ? b2)

w2
w3
w4 5
b50
b61
b40
b7 1
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Additional Issues

• Currently decision variables are only Boolean
• This is why pure arithmetic ops are unjustifiable
• If word variables are also decision candidates
  then justification applies to all operators
• Key Problem No efficient decision strategy on
  word-variables (yet..)

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Experimental Analysis

• Compare effect of new methods with base-case
• BMC of safety properties in ITC99 benchmarks
• b01, b02, b04, b11, b13
• Cut off of 2500 relations for predicate learning
• Cases
• No predicate learning or structural search
• Structural search only
• Predicate learning is used to weight structural
  search decision points for breaking ties.
• Cut off of 1200 cpu seconds for overall solving

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Effects of Learning and Structural search
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Comparison with other solvers
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Conclusions

• Structural analysis possible on RTL circuits
• Can give good performance advantages
• Demonstrated recursive learning and justification
• Leverage existing work done on gate-level
• Structural analysis has additional applications
• Equivalence checking
• Predicate learning can aid abstraction
• Future work will explore these issues

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• The End

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Hybrid DPLL - Algorithm

• Input Boolean query ? on model ?
• Output Boolean, Satisfiable or Unsatisfiable.
• Set blevel ? 0
• Find unique Implications of ? on ?
• while ( True ) do
• while ( decide () ? Done) do
• Set deduction ? Null
• while ((deduction ? deduce ()) ? Conflict) do
• blevel ? analyze_conflicts()
• if (blevel ? 0) then
• return Unsatisfiable
• else backtrack (blevel)
• if (deduction ? SAT) then
• return Satisfiable