Added new section on automated theorem proving

content/research/atp/definitional-cnf.md

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+---
+title: "Definitional CNF"
+math: true
+---
+
+Satisfiability algorithms have been optimized to accept formulas in Conjunctive Normal Form (CNF).
+One issue is that depending on the algorithm used to convert the formula, it can take either exponential
+time or space. To get around this we will use an algorithm to produce an equisatisfiable algorithm albeit
+non-equivalent. This will create a linear increase in the size of the formula, however, it also only
+takes linear time to compute.
+
+
+Procedure:
+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.
+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.
+
+Consider the expression $b + ac$. Then,
+
+$$
+\begin{align*}
+\text{Define } x \iff ac \\\\
+(b + x) * (x \iff ac) \\\\
+\text{Define } y \iff b + x \\\\
+y * (x \iff ac) * (y \iff b + x)
+\end{align*}
+$$
+
+3. Finally, convert the bi-implifications using the following tautologies:
+
+$$
+\begin{align*}
+x \iff (y * z) &\equiv (-x + y) (-x + z)(x+-y+-z) \\\\
+x \iff (y + z) &\equiv (-x + y + z)(x + -y)(x + -z)
+\end{align*}
+$$
+
+Extending the example,
+$$
+y(-x + a)(-x + c)(x + -a + -c)(y \iff b + x) \\\\
+y(-x + a)(-x + c)(x + -a + -c)(-y + b + x)(y + -b)(y + -x)
+$$