Sum of Products form #
- A series of clauses
- Each with a series of variables or negated variables ANDed together.
- The clauses are ORed together.
- Any boolean expression can be expressed in SoP.
- An expression can be transformed into SoP algebraically.
Example: ab!c+def+g!hi+adf
Disjunctive Normal Form (DNF) #
- The same as SoP, except each clause must mention every variable.
- An expression in DNF corresponds to the true rows of a truth table, so it may be up to 2^n clauses.
- An expression can be transformed into DNF by evaluating all possible inputs.
Product of Sums Form #
- A series of clauses, each combining terms with OR.
- Those clauses are ANDed together.
- Also can represent any boolean expression.
Example (a+b+!c)(!b+c+d)(b+!c)
Canonical Conjunctive Normal Form (CNF) #
- PoS, where each clause mentions every variable.
- Corresponds to the false rows of a truth table.
Thinking about SAT #
- A formula in SoP form is always satisfiable (unless it has zero clauses). Each clause is a satisfying assignment.
- A formula in PoS form is satisfiable if one term in each clause can be made true simultaneously.
- This means that conversion to SoP (including DNF) will find all satisfying assignments.
- SAT instances are typically specified in PoS form, since SoP instances are easy.
Converting to DNF #
- Start with the HW starter code.
- We want an OR at the top, ANDs under that, and terms under that.
Overflow #
- Symbolic evaluation
- Partial evaluation
- Brute-force search
