Difference between revisions of "Decidable predicate"
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| − | An $n$-place predicate $P$ given on a certain set of constructive objects (for example, natural numbers) $M$ for which there exists an algorithm which enables one to find the value (T or F) of $P$ on any $n$-tuple $a_1,\ldots,a_n$ of elements in $M$. In other words, a predicate is decidable if, regarded as an $n$-place function on $M$ with values in the set $\{\text T,\text F\}$, it is a [[ | + | An $n$-place predicate $P$ given on a certain set of constructive objects (for example, natural numbers) $M$ for which there exists an algorithm which enables one to find the value (T or F) of $P$ on any $n$-tuple $a_1,\ldots,a_n$ of elements in $M$. In other words, a predicate is decidable if, regarded as an $n$-place function on $M$ with values in the set $\{\text T,\text F\}$, it is a [[computable function]]. |
| − | When the concept of a [[ | + | When the concept of a [[recursive function]] or some equivalent concept is used as a mathematical refinement of the concept of computability, then the term "recursive predicate" is usually employed instead of "decidable predicate" . |
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| − | So, $P$ is a decidable predicate if the set of $n$-tuples on which $P$ takes the value T (true) is a [[ | + | So, $P$ is a decidable predicate if the set of $n$-tuples on which $P$ takes the value T (true) is a [[decidable set]]. |
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| + | [[Category:Logic and foundations]] | ||
Latest revision as of 18:17, 9 November 2014
An $n$-place predicate $P$ given on a certain set of constructive objects (for example, natural numbers) $M$ for which there exists an algorithm which enables one to find the value (T or F) of $P$ on any $n$-tuple $a_1,\ldots,a_n$ of elements in $M$. In other words, a predicate is decidable if, regarded as an $n$-place function on $M$ with values in the set $\{\text T,\text F\}$, it is a computable function.
When the concept of a recursive function or some equivalent concept is used as a mathematical refinement of the concept of computability, then the term "recursive predicate" is usually employed instead of "decidable predicate" .
Comments
So, $P$ is a decidable predicate if the set of $n$-tuples on which $P$ takes the value T (true) is a decidable set.
Decidable predicate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decidable_predicate&oldid=31759