In the search for a logic capturing polynomial time most promising candidates are Choiceless Polynomial Time (CPT) and rank logic. Rank extends fixed-point with counting by operator over prime fields. We show that isomorphism problem CFI graphs ℤ 2 i cannot be defined in logic, even if base graph is totally ordered. However, CPT can define this problem. thereby separate from particular time.

In this article, we discuss measure theoretic characterizations for weighted composition operators in some operator classes on $L^{2}(Sigma)$ such as, $n$-power normal, $n$-power quasi-normal, $k$-quasi-paranormal and quasi-class$A$. Then we show that weighted composition operators can separate these classes.