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Set builder notation

  1. Jan 23, 2005 #1
    How would you describe this set in plain English:
    [tex]\text{POS}_\lambda(\alpha) = \alpha_\lambda^0 \cup \{\kappa - \{\lambda\}|\kappa \in \alpha_\lambda^+\}[/tex]

    where:
    [itex]\lambda[/itex] is a literal
    [itex]\kappa[/itex] is an [itex]\vee[/itex] clause with the set [itex]\kappa = \vee_{j\leq m}\;\lambda_j [/itex] represented as [itex] \kappa = \{\lambda_j|j \leq m \}[/itex]
    [itex]\{\lambda\}[/itex] is a unit clause consisting of just a single literal [itex]\lambda[/itex]
    a CNF formula [itex]\alpha[/itex] is a set [itex]\alpha = \wedge_{i\leq n}\;\kappa_i [/itex] represented as [itex] \alpha = \{\kappa_i|i \leq n \}[/itex]

    a clause [itex]\kappa[/itex] is called [itex]\lambda-positive[/itex] if [itex]\lambda \in \kappa [/itex]
    a clause [itex]\kappa[/itex] is called [itex]\lambda-negative[/itex] if [itex]\neg \lambda \in \kappa [/itex]
    a clause [itex]\kappa[/itex] is called [itex]\lambda-neutral[/itex] if [itex]\kappa [/itex] is neither [itex]\lambda-positive [/itex] nor [itex]\lambda-negative [/itex]
    if [itex]\alpha[/itex] is a set of clauses, [itex]\alpha_\lambda^+[/itex] is the set of [itex]\lambda-positive [/itex] clauses of [itex]\alpha[/itex], [itex]\alpha_\lambda^-[/itex] is the set of [itex]\lambda-negative [/itex] clauses of [itex]\alpha[/itex], and [itex]\alpha_\lambda^0[/itex] is the set of [itex]\lambda-neutral [/itex] clauses of [itex]\alpha[/itex].


    At first, I was reading [itex]\text{POS}_\lambda(\alpha)[/itex] as the union of the [itex]\lambda-neutral [/itex] clauses of [itex]\alpha[/itex] with the [itex]\lambda-positive [/itex] clauses of [itex]\alpha[/itex] but excluding the unit clause [itex]\{\lambda\}[/itex]. But apparently it is intended to be read as the union of the [itex]\lambda-neutral [/itex] clauses of [itex]\alpha[/itex] with the [itex]\lambda-positive [/itex] clauses of [itex]\alpha[/itex] but with ALL of the [itex]\lambda[/itex] terms EXCLUDED from those clauses.

    WHY is it read that way? Or, to put it another way, why was it not written as [itex]\text{POS}_\lambda(\alpha) = \alpha_\lambda^0 \cup \{\kappa - \lambda|\kappa \in \alpha_\lambda^+\}[/itex]. (Why did they put their [itex]\lambda[/itex] in brackets {}?) And how would you write set-builder notation for "the [itex]\lambda-positive [/itex] clauses of [itex]\alpha[/itex] but excluding just the unit clause [itex]\{\lambda\}[/itex]."

    This comes from Computability, Complexity and Languages by Davis, Sigal and Weyuker, section 12.4, discussion of the Davis-Putnam rules for algorithms to manipulate CNF formulas.
     
  2. jcsd
  3. Jan 23, 2005 #2

    Hurkyl

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    Bleh, one day I'll learn this terminology.

    The expression A - B is the set formed from everything in the set A that is not in the set B. Thus, to "remove" the element λ from the set κ, you have to subtract off the singleton set containing λ.
     
  4. Jan 23, 2005 #3
    Hmmm...

    It becomes surprisingly clear when you put it that way.

    Thanks.
     
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