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42 lines
1.4 KiB
Markdown
42 lines
1.4 KiB
Markdown
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---
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title: "Definitional CNF"
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math: true
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---
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Satisfiability algorithms have been optimized to accept formulas in Conjunctive Normal Form (CNF).
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One issue is that depending on the algorithm used to convert the formula, it can take either exponential
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time or space. To get around this we will use an algorithm to produce an equisatisfiable algorithm albeit
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non-equivalent. This will create a linear increase in the size of the formula, however, it also only
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takes linear time to compute.
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Procedure:
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1. First convert the formula to Negation Normal Form (NNF). This is done by making it so that negations are only applied to propositions and there is no implication or bi-implication symbols in the formula.
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2. Starting from the inner-most part of the expression to the outermost, define a new variable that describes the connective and add that definition to the expression.
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Consider the expression $b + ac$. Then,
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$$
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\begin{align*}
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\text{Define } x \iff ac \\\\
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(b + x) * (x \iff ac) \\\\
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\text{Define } y \iff b + x \\\\
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y * (x \iff ac) * (y \iff b + x)
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\end{align*}
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$$
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3. Finally, convert the bi-implifications using the following tautologies:
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$$
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\begin{align*}
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x \iff (y * z) &\equiv (-x + y) (-x + z)(x+-y+-z) \\\\
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x \iff (y + z) &\equiv (-x + y + z)(x + -y)(x + -z)
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\end{align*}
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$$
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Extending the example,
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$$
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y(-x + a)(-x + c)(x + -a + -c)(y \iff b + x) \\\\
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y(-x + a)(-x + c)(x + -a + -c)(-y + b + x)(y + -b)(y + -x)
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$$
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