# space complexity of DFA intersection problem

the DFA-intersection computation problem, given two DFAs specified on the input, compute the intersection DFA, or the decision problem to determine its emptiness, turns out to have wider/ deeper significance in computational complexity theory.[1][2] it is O(n2) time with no better lower bounds known, and tight bounds possibly having major implications on multiple complexity class separations. looking for insight/ refs.

what is the best known space complexity of determining DFA intersection (emptiness)? how does the best known algorithm work?

## 1 Answer

Solving intersection Non-Emptiness for 2 DFA's:

It essentially just becomes a reachability problem for the product DFA.

• Roughly, we can solve it deterministically in $O(n^2)$ time using $O(n^2)$ space.

• Or, we can solve it non-deterministically with $O(\log(n))$ space.

• By Savitch's Theorem, we can also solve it deterministically in $2^{O(\log^2(n))}$ time using $O(\log^2(n))$ space.

To the best of my knowledge, it is not known if we can solve it deterministically in polynomial time using $O(n^{2-\epsilon})$ space.

Feel free to share any thoughts or ideas in the comments. :)