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Formulas with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulas as the simplest. A theory has quantifier elimination if for every formula α {\displaystyle \alpha } , there exists another formula α Q F {\displaystyle \alpha _{QF}} without quantifiers that is equivalent to it ( modulo this ...
For example, there is a definition of primality using only bounded quantifiers: a number n is prime if and only if there are not two numbers strictly less than n whose product is n. There is no quantifier-free definition of primality in the language ,, +,, <, = , however. The fact that there is a bounded quantifier formula defining primality ...
Existential quantifiers. A formula () is true according to M and if there exists an evaluation ′ of the variables that differs from at most regarding the evaluation of x and such that φ is true according to the interpretation M and the variable assignment ′.
Some of the details can be found in the article Lindström quantifier. Conditional quantifiers are meant to capture certain properties concerning conditional reasoning at an abstract level. Generally, it is intended to clarify the role of conditionals in a first-order language as they relate to other connectives, such as conjunction or ...
For MSO formulas that have free variables, when the input data is a tree or has bounded treewidth, there are efficient enumeration algorithms to produce the set of all solutions, [6] ensuring that the input data is preprocessed in linear time and that each solution is then produced in a delay linear in the size of each solution, i.e., constant ...
Given a closed formula of first-order logic, first do the following: Attach a numerical subscript to every predicate letter, stating its degree; Translate all universal quantifiers into existential quantifiers and negation; Restate all atomic formulas of the form x=y as Ixy. Now apply the following algorithm to the preceding result:
of quantifiers for Q ∈ {∀,∃}. It is a special case of generalized quantifier. In classical logic, quantifier prefixes are linearly ordered such that the value of a variable y m bound by a quantifier Q m depends on the value of the variables y 1, ..., y m−1. bound by quantifiers Qy 1, ..., Qy m−1. preceding Q m. In a logic with (finite ...
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier (" ∃x" or "∃(x)" or ...