CAMA: A MultiValued Satisfiability Solver

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CAMA A Multi-Valued Satisfiability Solver

• Cong Liu1
• Andreas Kuehlmann1,2
• Matthew Moskewicz1
• Alberto Sangiovanni-Vincentelli1

1 University of California, Berkeley
2 Cadence Berkeley Labs
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Outline

• Motivation
• Preliminaries
• Solver Architecture
• MV Decision Heuristic
• MV Watch Literal Scheme
• MV Learning
• Results
• Conclusions

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Motivation

• MV logic can describe many CAD decisions
  naturally
• Choose one out of n, e.g. technology mapping
• Application of binary SAT solver requires
  encoding
• One-hot encoding one binary variable represents
  one value of a MV variable
• e.g.
• Binary encoding
• Additional clauses to exclude invalid values

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Motivation

• MV formulations can describe binary SAT problems
  more compactly
• E.g. by clustering Boolean circuits
• 7 Binary variables, 43 clauses
• 5 MV variables (y1 4-value, y2 8-value), 32 MV
  clauses
• Preprocess cluster

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Preliminaries

• Let xi denote a multi-valued variable with domain
  Pi 0,1,,Pi-1.
• xi is assigned to a non-empty value set vi, if xi
  can take any value from vi Í Pi but no value
  from Pi \ vi
• if vi 1, completely assigned, e.g. x
  2
• otherwise incompletely assigned, e.g. x 0,2
• A multi-valued literal is the Boolean
  function defined by
• where is the literal value
  set

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Preliminaries

• A multi-valued clause is a logical disjunction of
  one or more MV literals.
• A formula in multi-valued conjunctive normal form
  (MV-CNF) is the logical conjunction of a set of
  multi-valued clauses.
• A MV SAT problem given in MV-CNF is
• Satisfiable if there exists a set of complete
  assignments to all variables such that the
  formula evaluates to true.
• It is unsatisfiable if no such assignment exists.

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Resolution

• Recall Binary clause resolution
• variable x is eliminated
• MV resolution provides generalized case
• take intersection of literal value sets of the
  resolving variable x

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General Solver Flow
Algorithm DPLL () while (decide ()
SUCCESS) while (deduce () CONFLICT)
if (analyze_conflict () FAILURE)
return UNSAT
return SAT
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Decision Heuristic

• Binary case either 0 or 1 is assigned
• MV Case (2P-2) possible assignments initially
• E.g. P0,1,2 0, 1, 2, 0,1, 0,2,
  1,2
• Large decision scheme
• Pick one value from value set of selected
  variable
• Max depth of decision stack variables n
• Learning by contra-positive only forbids one
  value
• Small decision scheme
• Exclude one value
• Max depth of decision stack

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Watch-literal Scheme

• Similar to Chaff-class solvers
• Watch two literals and swap them with non-false
  literals as soon as they become false
• If no such literal exist ? implication
• No implication as long as watch literals not
  false
• Differences
• Only one linked list for each MV variable
• Process clause only when literal evaluates to
  false

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Boolean Constraint Propagation
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Conflict Analysis

• Grasp Chaff wisdom we can learn the
  contra-positive for any cut assignment
• Grasp Chaff suggest to use cut with exactly one
  literal on current decision level
• Unique implication point (UIP)

Assignment(Cuti) Þ Conflict Not(Conflict) Þ
Not(Assignment(Cuti))
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Conflict Analysis

• Cut at x3 and x4 as first UIP
• By contra-positive we learn

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However

• Learned clauses are actually generated by a
  sequence of resolution steps
• IG is processed in reverse implication order
  starting from the conflicting clause as
  accumulator clause.
• At each step the accumulator clause is resolved
  with the clause causing the implication, using
  the implied variable as the resolving variable.
• Binary case
• The result of the resolution sequence is
  identical to the contra-positive of the cut
  assignments
• For MV-SAT this is not the case

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MV-Conflict Analysis
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MV-Conflict Analysis
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MV-Conflict Analysis

• The learned clause is strictly stronger than
  the cut clause
• Cut clause forbids
• Learned clause forbids
• As if decisions were

Never visited before
Bigger search space
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Implementation Details

• Minimum Value Set (MVS) algorithm builds learned
  clause in one traversal of IG
• Original Assignment Storage (OAS) facilitates
  undo assignment
• Value sets are represented by sets of bits
• Positional notation
• Allow efficient bit-parallel operation
• Flexibility of processing incomplete assignments
• A decision-level stack for each variable

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Results

• Compare Cut-negation learning with MVS learning

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Results

• Equivalence checking of a MV function before and
  after applying MVSIS

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Conclusions

• Many CAD decision problems can be modeled as MV
  SAT problems
• Avoid encoding and additional constraint clauses
• Specialized MV SAT solver has advantages
• Bit-parallel processing of value sets
• Learning based on MV-resolution can prune search
  space never visited before
• Flexibility by using incomplete assignments
• Corresponding speedup observed when comparing
  with binary solver