# 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.
• @user13175: If we had log(n) seed length PRGS in L, then RL=L because we just try all the different seeds and take the majority answer. But I'm not sure about the converse (nonexistence of $\log n$ seeds in L $\implies$ RL $\neq$ L).