Applying Case Based Reasoning to Boolean Satisfiability

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Applying Case Based Reasoning to Boolean
Satisfiability

• Chris Cole
• 21 April 2008

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Outline of Presentation

• What is Boolean Satisfiability?
• Why might CBR be useful?
• A basic backtracking algorithm.
• Backtracking with a Case-base.
• Similarity and Adaptation.
• Design Details

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Satisfiability Problem

• A boolean expression is satisfiable if there is
  an assignment to each of the boolean variables in
  the expression such that the expression evaluates
  to true.
• The Satisfiability Problem is to determine if an
  expression is satisfiable.
• Tied to the question P NP by Cook-Levin
  theorem. See Sipser, 2006.

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Satisfiability Example 1

• (x ? y ? z) ? (x ? y ? z)
• Set z true, x and y to anything, and the
  expression evaluates to true.
• Therefore, the expression is satisfiable.

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Satisfiability Example 2

• (x ? y ? z) ? (x ? y ? z) ? y ? z
• y and z must be false by the last conjuncts. Then
  x must be false by the second conjunct. The first
  conjunct evaluates to false. There is no
  assignment of the variables to make the
  expression true.
• Therefore, the expression is not satisfiable.

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Why CBR Might Work

• Many useful problems, such as factorization, can
  be expressed as a satisfiability problem.
• Problems drawn from such domains may contain
  statistical structure some sub-problems are much
  more likely than others.
• If the most likely sub-problems are stored in a
  case-base, then they might be used to speed up
  finding the solution.

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Backtracking Algorithm
contradiction
Input Problem
Check for Singles, update
Return No Solution
no contradiction
all variables set
Return solution
no more values
Guess Next Value, update
next value exists
contradiction
all variables set
Check for Singles, update
no contradiction
solve - no solution
solve - found solution
Recur on Sub-Problem
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Algorithm with Cases
contradiction
Input Problem
Check for Singles, update
Return No Solution
match - no solution
no contradiction
all variables set
Check Case-Base
Return solution
match - solution
no match
no more values
Guess Next Value, update
Record in Case-Base
next value exists
contradiction
all variables set
Check for Singles, update
Record in Case-Base
no contradiction
match - no solution
match - found solution
Check Case-Base
no match
solve - no solution
solve - found solution
Recur on Sub-Problem
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Similarity and Adaptation

• Similarity search in case-base depends on
  adaptation rules.
• Substitution adaptation make zero, one, or more
  substitutions of a variable name in target
  expression and search base for match.
• Case expressions in case-base stored in
  lexicographic order on conjuncts conjuncts
  ordered lexicographically on disjuncts.

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Other Possible Adaptations

• Substitution adaptation maintains the same number
  of variables and conjuncts.
• Possible to take two cases and form their
  conjunction to generate a new case. (But I do not
  know how to search efficiently for it.)

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Design Details

• Put all the problem details into a class that is
  manipulated by the algorithm. Can replace class
  with more efficient implementation later.
• Problem description consists of linked list of
  disjuncts and associative array of variable
  bindings.
• Cases stored as sorted list.

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Thank you!