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sav08:normal_forms_for_propositional_logic [2008/03/09 12:45] thibaud Filled-in negation-normal paragraph |
sav08:normal_forms_for_propositional_logic [2008/03/09 13:33] thibaud Filled-in CNF |
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=== Disjunctive Normal Form === | === Disjunctive Normal Form === | ||
+ | Formulas in Disjunctive-normal form look like this: | ||
+ | $(x_1 \land x_2 \land \lnot x_3) \lor (\lnot x_1 \land x_3 \land x_4) \lor ...$ \\ | ||
+ | More formally $F = \bigvee^{n}_{i=1} D_i$ where $n \geq 0$. \\ | ||
+ | Each $D_i$ is a clause and is defined as $D_i = \bigwedge_{j=1}^{n_i} L_{ij}$. \\ | ||
+ | Each $L_{ij}$ is a literal. It's either an elementary proposition or its negation. | ||
- | Complete disjunctive normal form and truth tables. | + | Solving the SAT problem for DNF formulas is in P, but transforming an arbitrary propositional formula to DNF causes an exponential blow-up. |
- | * generating DNF from truth table | + | |
- | * generating DNF by transformations | + | |
- | === Conjunctive Normal Form === | + | DNF formulas can be easily generated from truth tables. Each row of the truth table that makes the formula true can be written as a clause. For a formula over $n$ variables, there are $2^{n}$ rows in the truth table. Over $n$ variables, there are $2^{2^{n}}$ different (i.e. non-equivalent) formulas. |
- | CNF | + | === Conjunctive Normal Form === |
- | Literal. Clause. | + | Formulas in Conjunctive-normal form look like this: |
+ | $(x_1 \lor x_2 \lor \lor x_3) \land (\lnot x_1 \lor x_3 \lor x_4) \land ...$ \\ | ||
+ | It's defined as $F = \bigwedge^{n}_{i=1} D_i$ and $D_i = \bigvee_{j=1}^{n_i} L_{ij}$ \\ | ||
+ | Like for DNF, $L_{ij}$ are elementary propositions or their negation. The terminology of clauses and literals also applies to CNF. | ||
- | No polynomial-time equivalence preserving transformation to CNF or to DNF. | + | There is no polynomial-time equivalence preserving transformation to CNF or to DNF. |
=== Complete Sets of Connectives === | === Complete Sets of Connectives === |