Two-Way Automata Versus Logarithmic Space

We strengthen a previously known connection between the size complexity of two-way finite automata ([inline-graphic not available: see fulltext]) and the space complexity of Turing machines (tms). Specifically, we prove that every s-state [inline-graphic not available: see fulltext] has a poly(s)-state [inline-graphic not available: see fulltext] that agrees with it on all inputs of length ≤s if and only if NL⊆L/poly, andevery s-state [inline-graphic not available: see fulltext] has a poly(s)-state [inline-graphic not available: see fulltext] that agrees with it on all inputs of length ≤2s if and only if NLL⊆LL/polylog. Here, [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] are the deterministic and nondeterministic [inline-graphic not available: see fulltext], NL and L/poly are the standard classes of languages recognizable in logarithmic space by nondeterministic tms and by deterministic tms with access to polynomially long advice, and NLL and LL/polylog are the corresponding complexity classes for space O(loglogn) and advice length poly(logn). 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.