Abstract
We study several classical decision problems on finite automata under the (Strong) Exponential Time Hypothesis. We focus on three types of problems: universality, equivalence, and emptiness of intersection. All these problems are known to be CoNP-hard for nondeterministic finite automata, even when restricted to unary input alphabets. A different type of problems on finite automata relates to aperiodicity and to synchronizing words. We also consider finite automata that work on commutative alphabets and those working on two-dimensional words.
Highlights
Many computer science students will get the impression, at least when taught the basics on the Chomsky hierarchy in their course on Formal Languages, that finite automata are fairly simple devices, and it is expected that typical decidability questions on finite automata are easy ones
For instance, the non-emptiness problem for finite automata is solvable in polynomial time, as well as the uniform word problem. (Even tighter descriptions of the complexities can be given within classical complexity theory, but this is not so important for our presentation here, as we mostly focus on polynomial versus exponential time.) This contrasts to the respective statements for higher levels of the Chomsky hierarchy
Finite automata can be viewed as edge-labeled directed graphs, and as many combinatorial problems are harder on directed graphs compared to undirected ones, it should not come as such a surprise that many interesting questions are NP-hard for finite automata
Summary
Many computer science students will get the impression, at least when taught the basics on the Chomsky hierarchy in their course on Formal Languages, that finite automata are fairly simple devices, and it is expected that typical decidability questions on finite automata are easy ones. (Even tighter descriptions of the complexities can be given within classical complexity theory, but this is not so important for our presentation here, as we mostly focus on polynomial versus exponential time.) This contrasts to the respective statements for higher levels of the Chomsky hierarchy. Notice that minimizing a DFA with s states takes time Ops log sq with Hopcroft’s algorithm, so that the running time of first converting a q-state NFA into an equivalent DFA (Op2q q) and minimizing a 2q -state DFA (in time Op2q 2log q q “ Op2q q) For (E MPTINESS OF) I NTERSECTION, our results are summarized, whose entries are to be read similar to those of Table 1 All these problems are already computationally hard for tally NFAs, i.e., NFAs on unary inputs. O(1), i.e., in P unary binary unbounded q unary opkq a opminpk log q, qqq k minpk log q, 1.5 ̈ qq Theorem 3 & Proposition 1 q binary opminpk, 2q qq minpk log q, 22q log qq Proposition 3 & Proposition 4 q unbounded opk log qq k log q
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