Characterization and complexity results on jumping finite automata

Abstract

In a jumping finite automaton, the input head can jump to an arbitrary position within the remaining input after reading and consuming a symbol. We characterize the corresponding class of languages in terms of special shuffle expressions and survey other equivalent notions from the existing literature. Moreover, we present several results concerning computational hardness and algorithms for parsing and other basic tasks concerning jumping finite automata.Proposition 10. Let M 1 , M 2 , M 3 be arbitrary languages.

Problems on Finite Automata and the Exponential Time Hypothesis

“…Interestingly enough, also the problem of detecting emptiness of the intersection of only two JFA languages is NP-hard. This and a related study on ETH-based complexity can be found in [37]. For the intersection of k JFAs, the proof of Proposition 2 actually shows the analogous result also in that case.…”

“…Frankly speaking, we are only aware of the papers [29,37] on these topics. Returning to the survey of Holzer and Kutrib [7], it becomes clear that there are quite a many hard problems related to finite automata and regular expressions that have not yet been examined with respect to exact algorithms and ETH.…”

“…In other words, a JFA may first jump to an arbitrary position of the input and then apply the rule there. This model was introduced in [36] and further studied in [37]. It is relatively easy to see that the languages accepted by JFAs are just the inverses of the Parikh images of the regular languages, or, in other words, the commutative (or permutation) closure of the regular languages, or, yet in different terminology, the inverses of the Parikh images of semilinear sets.…”

“…Hence, UNIVERSALITY is hard for JFAs, as well. Classical complexity considerations on these formalisms are contained in [37][38][39][40][41]; observe that mostly the input is given in the form of Parikh vectors of numbers encoded in binary, while we will consider the input given in unary-encoded Parikh vectors below (namely, as words, i.e., as elements of the free monoid), since JFAs were introduced this way in [36]. Yet another way to formally look at how JFAs operate incorporates the use of the shuffle operation.…”

“…It can be therefore shown that the uniform word problem for JFAs is NP-hard. Analyzing the proof given in [37], Theorem 54, we can conclude: Notice that this problem can be solved in time O˚pn!q on arbitrary input alphabets, feeding all permutations of an input word of length n into the given automaton, interpreted as an NFA. There is also a dynamic programming algorithm solving UNIVERSAL MEMBERSHIP for q-state JFAs (that is not improvable by Theorem 8, assuming ETH).…”