We explore bounds of {em time-space tradeoffs} in language recognition on {em two-way finite automata} for some special languages. We prove: (1) a time-space tradeoff upper bound for recognition of the languages $L_{EQ}(n)$ on {em two-way probabilistic finite automata} (2PFA): $TS={bf O}(nlog n)$, whereas a time-space tradeoff lower bound on {em two-way deterministic finite automata} is ${bf Omega}(n^2)$, (2) a time-space tradeoff upper bound for recognition of the languages $L_{INT}(n)$ on {em two-way finite automata with quantum and classical states} (2QCFA): $TS={bf O}(n^{3/2}log n)$, whereas a lower bound on 2PFA is $TS={bf Omega}(n^2)$, (3) a time-space tradeoff upper bound for recognition of the languages $L_{NE}(n)$ on exact 2QCFA: $TS={bf O}(n^{1.87} log n)$, whereas a lower bound on 2PFA is $TS={bf Omega}(n^2)$. It has been proved (Klauck, STOC00) that the exact one-way quantum finite automata have no advantage comparing to classical finite automata in recognizing languages. However, the result (3) shows that the exact 2QCFA do have an advantage in comparison with their classical counterparts, which has been the first example showing that the exact quantum computing have advantage in time-space tradeoff comparing to classical computing. Usually, two communicating parties, Alice and Bob, are supposed to have an access to arbitrary computational power in {em communication complexity} model that is used. Instead of that we will consider communication complexity in such a setting that two parties are using only finite automata and we prove in this setting that quantum automata are better than classical automata and also probabilistic automata are better than deterministic automata for some well known tasks.
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