# Conversion to cnf of

In Boolean logica formula is in conjunctive normal form CNF or clausal normal form if it is a conjunction of one or more clauseswhere a clause is a disjunction of literals ; otherwise put, it is a product of sums or an AND of ORs.

As a canonical normal formit is useful in automated theorem proving and circuit theory. All conjunctions of literals and all disjunctions of literals are in CNF, as they can be seen as conjunctions of one-literal clauses and conjunctions of a single clause, respectively. As in the disjunctive normal form DNFthe only propositional connectives a formula in CNF can contain are andorand not.

The not operator can only be used as part of a literal, which means that it can only precede a propositional variable or a predicate symbol. In automated theorem proving, the notion " clausal normal form " is often used in a narrower sense, meaning a particular representation of a CNF formula as a set of sets of literals.

Incidentally, the last two formulas are also in disjunctive normal form. Every formula can be equivalently written as a formula in conjunctive normal form. In particular this is the case for the three non-examples just mentioned; they are respectively equivalent to the following three formulas, which are in conjunctive normal form:. This transformation is based on rules about logical equivalences : double negation eliminationDe Morgan's lawsand the distributive law.

Since all propositional formulas 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. There exist transformations into CNF that avoid an exponential increase in size by preserving satisfiability rather than equivalence. An interpretation satisfies this formula only if at least one of the new variables is true.

This means that every model that satisfies this formula also satisfies the original one. This means that the original formula and the result of the translation are equisatisfiable but not equivalent. 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 first-order resolution.

In resolution-based automated theorem-proving, a CNF formula. 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 k -SAT problem is the problem of finding a satisfying assignment to a boolean formula expressed in CNF in which each disjunction contains at most k variables. Typical problems in this case involve formulas in "3CNF": conjunctive normal form with no more than three variables per conjunct.

Examples of such formulas encountered in practice can be very large, for example withvariables and 1, conjuncts. To convert first-order logic to CNF: [1]. From Wikipedia, the free encyclopedia. Categories : Normal forms logic.

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Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file. Download as PDF Printable version.Try running the cnf for the following. We use optional third-party analytics cookies to understand how you use GitHub. Learn more. You can always update your selection by clicking Cookie Preferences at the bottom of the page.

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You signed out in another tab or window. Accept Reject. Essential cookies We use essential cookies to perform essential website functions, e. Analytics cookies We use analytics cookies to understand how you use our websites so we can make them better, e. Save preferences. Not of proposition.This algorithm corresponds exactly to the one you saw on the lecture slidesbut this presentation gives a somewhat different perspective along with some further discussion.

In Homework 1you'll get to convert some formulas to CNF by hand. Or if you want, you could implement the algorithm below and implement an automatic converter! In lecturewe allowed 7 kinds of propositional formulas in our little language. It might be helpful to think about how you would implement formulas in a language like Java. You'd want a recursive datatype similar to trees, roughly like this:. If so, we can look at its two conjuncts, P and Qwhich are themselves instances of Formula.

A CNF formula is a restricted special case. It is a conjunction of "clauses," each of which is a disjunction of "literals," each of which is either a variable or a negated variable.

See why this recursive algorithm works? It must return a correct answer, assuming it doesn't recurse forever. You could prove that it won't recurse forever. Unfortunately, the case of P v Q above is very bad. Moreover, m and n may already be large, since P and Q may themselves contain v. So even though this alternative technique does result in a formula with additional variables, it keeps the total formula length far smaller. Note: We still have problems with P Q and P xor Qsince our pseudocode replaces these with expressions that are more than twice as long.

That can lead to exponential blowup if P and Q themselves contain and xor. You can avoid this problem by using the Tseitin transformation see below. Here is another way of thinking about this whole thing. Every intermediate step returns a formula that has already been converted to CNF.

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So the user never sees a non-CNF formula at all, except for the non-CNF formula that is implicit in the nested constructor calls above.

The CNFFormula constructors have to handle the same 7 cases as are in the pseudocode above, and they do it using exactly the same methods.

### Converting Context Free Grammar to Chomsky Normal Form

Now, as remarked in lecture, it's easy to write a SAT solver as. That's the "Satisfiability Hypothesis," or SH for short. In the CNF case, we were able to find a smarter version that keeps this formula compact by using switching variables.Prerequisite — Simplifying Context Free Grammars.

Step 1. Eliminate start symbol from RHS. Step 2. Eliminate null, unit and useless productions. If CFG contains null, unit or useless production rules, eliminate them. You can refer the this article to eliminate these types of production rules. Step 3. Eliminate terminals from RHS if they exist with other terminals or non-terminals.

Step 4.

## Chomsky Normal Form

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It only takes a minute to sign up. My main problem is that I do not know how to simplify the formula in the end, so even though I apply the rules in a correct way and reach the end of the question, being unable to simplify absorb etc.

However, the easiest technique I know of to do the conversion is to use Karnaugh maps. I let you do the CNF case on your own. There is an easy way of doing this. Draw a truth table for the given expression.

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## Conjunctive normal form

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