# CNF (Conjunctive Normal Form)
- Conjunction of Disjunctions
- 'AND' of 'OR's
- $(0 \lor 1 \lor -2) \land (0 \lor -5)$
# DNF (Disjunctive Normal Form)
- Conjunction of Disjunctions
- 'AND' of 'OR's
- $(15 \land 1 \land -2) \lor (1 \land -5)$
- Easy to see if there is a SAT assignment
	- Is any Clause SAT?
	- Conversely In CNF finding an UNSAT assignment is easy

# Variables as Integers
- $x, y, z \mapsto 1, 2, 3$
- Efficient representation
- Can store assignments in an array
- Negation $-var$ has a literal meaning in code

# Encoding Problems to boolean formula
- For example $compile:$ `if x then y else z` $\mapsto (x \land y) \lor (\lnot x \land z)$ 
- What do you encode?
	- Equivalence of two programs?
	- $compile(A) \cancel\leftrightarrow compile(B)$ 
	- Encode this, if its SAT we have a counterexample


# NNF / Negation Normal Form
- Mixture of 'AND's and 'OR's
- But negation can only be in front of Variables
- Can be made easily using De Morgan
- Naive transformation from NNF to CNF
	- by merging via 'OR-distribution'
	- Merge on: $(a \lor b) \lor (c \land d) \mapsto (a \lor b \lor c) \land (a \lor b \lor d)$ 
- Exponential Fan out

# Tseitin Transformation
- Extract Formula nodes into new variables and create a constraint list 
- $((a \leftrightarrow b) \land (c \lor \overline{d}))$ 
	- $x_1 \leftrightarrow (a \leftrightarrow b)$
	- $x_2 \leftrightarrow \overline d$ 
	- $x_2 \leftrightarrow (c \lor x_2)$ 
	- $x_4 \leftrightarrow (x_1 \land x_3)$ 
- Then transform all of them to CNF and conjunct them
![[tseitin.png]]
- The resulting formula grows linearly in respect to the size of the original formula
- Sometimes taking dividing into binary operations is not optimal (xor optimum is 3)

## Plaisted–Greenbaum (PG)
- Only use implication instead of equivalence $\leftrightarrow$  
- Direction depends on polarity of the subformula's occurrence
	- Positive polarity: only need $x \rightarrow \varphi$
	- Negative polarity: only need $\varphi \rightarrow x$
	- Mixed polarity (e.g. under XOR/biconditional): need both, i.e. full Tseitin $x \leftrightarrow \varphi$
- Example $x \leftrightarrow (a \land b)$: positive occurrence needs $(\overline x \lor a) \land (\overline x \lor b)$; negative occurrence needs just $(\overline a \lor \overline b \lor x)$
	- Positive occurrence: $x$ appears unnegated in $F$, e.g. $F = x \lor c$ (as $a\land b$ becomes true, $F$ can only become 'more satisfied') $\Rightarrow$ only need $x \rightarrow (a \land b)$
	- Negative occurrence: $x$ appears negated in $F$, e.g. $F = \overline x \lor c$ (mirror case, $F$ becomes 'more satisfied' as $a \land b$ becomes false) $\Rightarrow$ only need $(a \land b) \rightarrow x$