Abstract
We strengthen previously known connections between the size complexity of two-way finite automata (2FAS) and the space complexity of Turing machines. We prove that - every s-state 2NFA can be simulated on all poly(s)-long inputs by some poly(s)-state 2DFA if and only if NL ⊆ L/poly and - every s-state 2NFA can be simulated on all 2poly(s)-long inputs by some poly(s)-state 2DFA if and only if NLL ⊆ LL/polylog. Here, 2DFAS and 2NFAS are the deterministic and nondeterministic 2FAS, NL and L/poly are the standard space complexity classes, and NLL and LL/polylog are their counterparts for O(log log n) space and poly(log n) bits of advice. Our arguments strengthen and extend an old theorem by Berman and Lingas and can be used to obtain variants of the above statements for other modes of computation or other combinations of bounds for the input length, the space usage, and the length of advice.
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