 Conjunctive normal form

In Boolean logic, a formula is in conjunctive normal form (CNF) if it is a conjunction of clauses, where a clause is a disjunction of literals. As a normal form, it is useful in automated theorem proving. It is similar to the product of sums form used in circuit theory.
All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of oneliteral clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form (DNF), the only propositional connectives a formula in CNF can contain are and, or, and not. The not operator can only be used as part of a literal, which means that it can only precede a propositional variable.
Contents
Examples and counterexamples
All of the following formulas are in CNF:
The last formula is in CNF because it can be seen as the conjunction of the two singleliteral clauses A and B. Incidentally, this formula is also in disjunctive normal form. The following formulae are not in CNF:
The above three formulae are respectively equivalent to the following three formulas that are in CNF:
Conversion into CNF
Every propositional formula can be converted into an equivalent formula that is in CNF. This transformation is based on rules about logical equivalences: the double negative law, De Morgan's laws, and the distributive law.
Since all logical formulae can be converted into an equivalent formula in conjunctive normal form, proofs are often based on the assumption that all formulae are CNF. However, in some cases this conversion to CNF can lead to an exponential explosion of the formula. For example, translating the following nonCNF formula into CNF produces a formula with 2^{n} clauses:
In particular, the generated formula is:
This formula contains 2^{n} clauses; each clause contains either X_{i} or Y_{i} for each i.
There exist transformations into CNF that avoid an exponential increase in size by preserving satisfiability rather than equivalence.^{[1]}^{[2]} These transformations are guaranteed to only linearly increase the size of the formula, but introduce new variables. For example, the above formula can be transformed into CNF by adding variables as follows:
An interpretation satisfies this formula only if at least one of the new variables is true. If this variable is Z_{i}, then both X_{i} and Y_{i} are true as well. This means that every model that satisfies this formula also satisfies the original one. On the other hand, only some of the models of the original formula satisfy this one: since the Z_{i} are not mentioned in the original formula, their values are irrelevant to satisfaction of it, which is not the case in the last formula. This means that the original formula and the result of the translation are equisatisfiable but not equivalent.
An alternative translation includes also the clauses . With these clauses, the formula implies ; this formula is often regarded to "define" Z_{i} to be a name for .
Firstorder logic
In first order logic, conjunctive normal form can be taken further to yield the clausal normal form of a logical formula, which can be then used to perform firstorder resolution.
Computational complexity
An important set of problems in computational complexity involves finding assignments to the variables of a boolean formula expressed in Conjunctive Normal Form, such that the formula is true. The kSAT problem is the problem of finding a satisfying assignment to a boolean formula expressed in CNF such that each disjunction contains at most k variables. 3SAT is NPcomplete (like any other kSAT problem with k>2) while 2SAT is known to have solutions in polynomial time.
Typical problems in this case involve formulas in "3CNF" form: conjunctive normal form with no more than 3 variable per conjunct. Examples of such formulas encountered in practice can be very large, for example with 100,000 variables and 1,000,000 conjuncts.
Converting from firstorder logic
To convert firstorder logic to CNF:^{[3]}
 Convert to negation normal form.
 Eliminate implications: convert to
 Move NOTs inwards by repeatedly applying DeMorgan's Law. Specifically, replace with ; replace with ; and replace with x.
 Standardize variables
 For sentences like (∀x P(x)) ∨ (∃x Q(x)) which use the same variable name twice, change the name of one of the variables. This avoids confusion later when we drop the quantifiers.
 From ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃y Loves(y, x)]. we obtain: ∀x [∃y Animal(y) ∧ ¬Loves(x, y)] ∨ [∃z Loves(z,x)].
 Skolemize the statement
 ∃x P(x) into P(A), where A is a new constant (consult linked article for more details)
 Drop universal quantifiers
 Distribute ORs over ANDs.
Notes
 ^ Tseitin (1968)
 ^ Jackson and Sheridan (2004)
 ^ Artificial Intelligence: A modern Approach [1995...] Russel and Norvig
See also
References
 Paul Jackson, Daniel Sheridan: Clause Form Conversions for Boolean Circuits. In: Holger H. Hoos, David G. Mitchell (Eds.): Theory and Applications of Satisfiability Testing, 7th International Conference, SAT 2004, Vancouver, BC, Canada, May 10–13, 2004, Revised Selected Papers. Lecture Notes in Computer Science 3542, Springer 2005, pp. 183–198
 G.S. Tseitin: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Structures in Constructive Mathematics and Mathematical Logic, Part II, Seminars in Mathematics (translated from Russian), pp. 115–125. Steklov Mathematical Institute (1968)
External links
Categories: Normal forms (logic)
 Convert to negation normal form.
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