# AI-Portal Artificial Intelligence for Enthusiasts

## Prenex Normal Form

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In First Order Predicate Logic ( FOPL ), one can represent an expression using the operators

*For any X, Expression*

*Exists X such that Expression.*

Where Expression is a logic expression , either a predicate or a combination of logic operators and other expressions.

In order to represent an Expression in **Prenex Normal Form**, one must do some transformation. For an expression to be in Prenex Normal Form, all the "Any" and "Exists" Logical operators must precede a whole stand-alone expression. For example, expression

*(Any X in P(X) ) Or ( Exists Y in Q(Y))* it''s not in Prenex Normal Form.

Expression

*(Any X in ( P(X) and Q (Y) ) )* **is** in Prenex Normal Form .

Prenex Normal Form is useful in Knowledge representation for obvious reasons: memorizing in computer memory, specific attributes: We evaluate first the "Any"/"Exists" operators and the following expression can be substituted to simple expression.

To convert a normal expression to Prenex Normal Form, we consider the following transformation rules:

*( Any X in Expr) or (Another_Expr) *changes to *(Any X in ( Expr or Another_Expr )) *with mentioning the fact that X has no **free **appearances in Another_Expr ( if free they are renamed to another **free **variable )

*( Any X in Expr) and (Another_Expr)* changes to *(Any X in (Expr and Another_Expr ))* with same mentioning.

The same rules apply to *Exists *operator.