I am trying to understand the concept of nondeterminism in the context of computational complexity, particularly with reference to problems like PATH, which is in the class NL. I have read that PATH is in NL because we can "nondeterministically select the nodes of a path connecting $s$ to $t$." However, I am unclear on what exactly "nondeterministically" means in this context.
Specifically:
- Does "nondeterministically" imply that the machine effectively explores all possible choices simultaneously, as if performing a form of parallel computation?
- Why is PATH not in $L$ (deterministic logarithmic space)? Is it because deterministic machines must keep track of visited nodes and systematically brute-force through all possible paths, which could require more than logarithmic space?
- How should I conceptualise the difference between how deterministic and nondeterministic machines approach problems like PATH in terms of time and space?
Any clarification would be greatly appreciated!
