The Complexity of Class L

A major complexity classes are $L$, $POLYLOGTIME$ and $\oplus L$. A logarithmic Turing machine has a read-only input tape, a write-only output tape, and some read/write work tapes. The work tapes may contain $O(\log n)$ symbols. $L$ is the complexity class containing those decision problems that can be decided by a deterministic logarithmic Turing machine. We define the complexity class $POLYLOGTIME$ as the problems which can be decided by a random access machine in poly-logarithmic time. We prove there is problem in the complexity class $L$ which cannot be solved by a random access machine in poly-logarithmic time. Hence, we show $L \nsubseteq POLYLOGTIME$. The complexity class $\oplus L$ has the same relation to $L$ as $\oplus P$ does to $P$. We demonstrate there is a complete problem for $\oplus L$ that can be logarithmic space reduced to a problem in $L$. Consequently, we show $L = \oplus L$.