Tight hierarchy of data-independent multi-head automata

Abstract We study the expressive power of 1-way data-independent finite automata with k heads (1 DiDFA (k)). The data-independence means that the trajectory of each head during a computation depends only on the length of the input word. It has long been known that “ k + 1 heads are better than k” for the (data-dependent) 1 DFA ( k ) , and (2-way) 2 DiDFA ( k ) . However, somewhat surprisingly, no such hierarchy has been known for the 1 DiDFA ( k ) . In this paper we show not only the tight head hierarchy for 1 DiDFA ( k ) but even stronger result as well, that for each k ≥ 1 , there are languages that can be recognized by a 1 DiDFA ( k + 1 ) , but cannot be recognized by k one-way heads even with a non-deterministic data dependent automaton. On the other hand, we show that if the data-independent automaton is allowed to be two-way, 3 heads are sufficient to simulate all 1 DiDFA ( k ) . Finally, we remark that the restriction of data-independence cannot be compensated by adding more heads.