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In [[computability theory]] and [[mathematical logic]] the '''Tarski–Kuratowski algorithm''' is a [[non-deterministic algorithm]] which provides an upper bound for the complexity of formulas in the [[arithmetical hierarchy]] and [[analytical hierarchy]]. | |||
The algorithm is named after [[Alfred Tarski]] and [[Kazimierz Kuratowski]]. | |||
==Algorithm== | |||
The Tarski–Kuratowski algorithm for the arithmetical hierarchy: | |||
# Convert the formula to [[prenex normal form]]. | |||
# If the formula is quantifier-free, it is in <math>\Sigma^0_0</math> and <math>\Pi^0_0</math>. | |||
# Otherwise, count the number of alternations of quantifiers; call this ''k''. | |||
# If the first quantifier is [[Existential quantification|∃]], the formula is in <math>\Sigma^0_{k+1}</math>. | |||
# If the first quantifier is [[Universal quantification|∀]], the formula is in <math>\Pi^0_{k+1}</math>. | |||
==References== | |||
*Rogers, H. ''The Theory of Recursive Functions and Effective Computability'', MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1 | |||
{{DEFAULTSORT:Tarski-Kuratowski algorithm}} | |||
[[Category:Mathematical logic hierarchies]] | |||
[[Category:Computability theory]] | |||
[[Category:Theory of computation]] | |||
{{mathlogic-stub}} | |||
Revision as of 19:46, 10 August 2013
In computability theory and mathematical logic the Tarski–Kuratowski algorithm is a non-deterministic algorithm which provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy.
The algorithm is named after Alfred Tarski and Kazimierz Kuratowski.
Algorithm
The Tarski–Kuratowski algorithm for the arithmetical hierarchy:
- Convert the formula to prenex normal form.
- If the formula is quantifier-free, it is in and .
- Otherwise, count the number of alternations of quantifiers; call this k.
- If the first quantifier is ∃, the formula is in .
- If the first quantifier is ∀, the formula is in .
References
- Rogers, H. The Theory of Recursive Functions and Effective Computability, MIT Press. ISBN 0-262-68052-1; ISBN 0-07-053522-1