2
$\begingroup$

When reading about descriptive complexity theory, I have read about a "commutative transitive closure operator". I understand transitive closure operators, but what is a commutative transitive closure operator?

$\endgroup$
1
  • 3
    $\begingroup$ Where in "From descriptive complexity" did you see the term used? The source probably provides more context. $\endgroup$
    – Vijay D
    Commented Feb 25, 2013 at 10:42

1 Answer 1

12
$\begingroup$

Take a relation $R \subseteq X \times X$. Now, let $R^\dagger$ be the converse relation to $R$, and let $\Delta_X$ be the identity relation on $X$.

The commutative transitive closure $R^*$ is the smallest relation $S $ such that $\Delta_X \cup R \cup R^\dagger \cup S \circ S \subseteq S$.

Intuitively, think of the relation $R$ as being the edge relation on a directed graph. The commutative transitive closure tells you whether or not there is a path between two nodes, if you let yourself ignore the directedness.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.