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A formula of the predicate calculus is in prenex [1] normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. [2]
A fully quantified Boolean formula can be assumed to have a very specific form, called prenex normal form.It has two basic parts: a portion containing only quantifiers and a portion containing an unquantified Boolean formula usually denoted as .
In classical logic, every formula is logically equivalent to a formula in prenex normal form, that is, a string of quantifiers and bound variables followed by a quantifier-free formula. Quantifier elimination
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 ...
In mathematical logic, a formula of first-order logic is in Skolem normal form if it is in prenex normal form with only universal first-order quantifiers. Every first-order formula may be converted into Skolem normal form while not changing its satisfiability via a process called Skolemization (sometimes spelled Skolemnization ).
In mathematical logic, the rules of passage govern how quantifiers distribute over the basic logical connectives of first-order logic.The rules of passage govern the "passage" (translation) from any formula of first-order logic to the equivalent formula in prenex normal form, and vice versa.
This rule, which is used to put formulas into prenex normal form, is sound in nonempty domains, but unsound if the empty domain is permitted. The definition of truth in an interpretation that uses a variable assignment function cannot work with empty domains, because there are no variable assignment functions whose range is empty.
A formula in the language of second-order arithmetic is defined to be + if it is logically equivalent to a formula of the form where is . A formula is defined to be Π n + 1 1 {\displaystyle \Pi _{n+1}^{1}} if it is logically equivalent to a formula of the form ∀ X 1 ⋯ ∀ X k ψ {\displaystyle \forall X_{1}\cdots \forall X_{k}\psi } where ...