Essential Information & explanations, latest texts & monographs on
Conjunctive_normal_form.
Conjunctive normal form
Conjunctive Normal Form or CNF is a method of standardizing and normalizing formulas in Boolean logic. It is similar to the Canonical sum of products form used in circuit theory. A logical formula is considered to be in CNF if and only if it is a single conjunction of disjunctions. Thus all simple conjunctions are in CNF, but also all simple disjunctions are degenerately in CNF. For example, all of the following formulas are in CNF:
A ∧ B
¬A ∧ (B ∨ C)
(A ∨ B) ∧ (¬B ∨ C ∨ ¬D) ∧ (D ∨ ¬E)
(¬B ∨ C)
But the following are not:
¬(B ∨ C)
(A ∧ B) ∨ C
A ∧ (B ∨ (D ∧ E))
The above three formulas are respectively equivalent to the following three formulas that are in conjunctive normal form:
¬B ∧ ¬C
(A ∨ C) ∧ (B ∨ C)
A ∧ (B ∨ D) ∧ (B ∨ E)
Note that all logical formulas can be converted into conjunctive normal form through repeated application of the distributive law of disjunction over conjunction, thus when making proofs on formulas or on the structure of formulas it is often convenient to assume that everything is in CNF. However, in some cases
conversion to CNF can lead to an exponential explosion of the formula. For example, in CNF form, logical formulas of the following form have 2^n terms:
(X1 ∧ Y1) ∨ (X2 ∧ Y2) ∨ ... ∨ (Xn ∧ Yn)
Conjunctive Normal Form can be taken further to yield the Clausal normal form of a logical formula, that is used to preform first order resolution.
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. 3-SAT is the problem of finding a satisfying assignment to a boolean formula expressed in CNF such that each disjunction contains at most 3 variables. This has been shown to be NP-complete. The corresponding 2-SAT problem however can be solved in
linear time.
See Also:
The above article is adapted from from Wikipedia All Wikipedia article text is available under the terms of the GNU Free Documentation License
Recent related patents from USPTO:
Bibliographic Resources
Updates and comments at Essential Facts blog
Are you interested in Feng Shui?
Price Theory Resources
World Class Photographers
Some philosophical movements
Top PDF and eBook Downloads
|
|
Interesting Links
Sports
Kitchen Knowledge
Hollywood Icons
Mythology
Philosophy
Politics
Retirement
Accounting & Finance
Automobiles
Marketing
Psychology
Academic Subjects
Ancient Greeks
Art & Design
Biology
Biology & Biologists
California
Cats & Dogs
Ethics
Logic
The Greats
Architectural Dates & Places
* Mathematics & Mathematicians
Medical Update d06
More
Chromosomes and Genomics
Enginering Systems 1
Investments
Transportation
Mathematics
Brilliant Mathematicians
Classic Authors
Fear No Exams
Nexus
Characters & countries
Computers + 2
Science Plus
Science & Computers
Quantum Theory
Home, Site-Map
|