For log-space machines, why do we care about PRGs? Assumptions


I believe RL can be derandomized in polynomial time:

We need to store the following state:

The position of the head on the input.
The state of the TM

Both can be stored with O(log n) bits.
We use dynamic programming, building a table of poly(n) size encapsulating the state of the TM. Then, we march along the random bits, calculating the probability of each TM state after reading a given bit.



This should give us exact acceptance probability in $poly(n)$ time.

Observation

People care about prgs for log space machines.

Question:

Why? What can we do with a prg for RL that we can't do with an exact acceptance probability algo for RL ?

What is implied by a O(log n) seed length PRG for RL?

Thanks!
You show that RL$\subseteq$P. Better derandomization could show that RL=L.