# On Indexing and Compressing Finite Automata

An index for a finite automaton is a powerful data structure that supports locating paths labeled with a query pattern, thus solving pattern matching on the underlying regular language. In this paper, we solve the long-standing problem of indexing arbitrary finite automata. Our solution consists in finding a partial co-lexicographic order of the states and proving, as in the total order case, that states reached by a given string form one interval on the partial order, thus enabling indexing. We provide a lower bound stating that such an interval requires O(p) words to be represented, p being the order's width (i.e. the size of its largest antichain). Indeed, we show that p determines the complexity of several fundamental problems on finite automata: (i) Letting σ be the alphabet size, we provide an encoding for NFAs using ⌈logσ⌉ + 2⌈log p⌉ + 2 bits per transition and a smaller encoding for DFAs using ⌈logσ⌉ + ⌈log p⌉ + 2 bits per transition. This is achieved by generalizing the Burrows-Wheeler transform to arbitrary automata. (ii) We show that indexed pattern matching can be solved in Õ(m· p^2) query time on NFAs. (iii) We provide a polynomial-time algorithm to index DFAs, while matching the optimal value for p. On the other hand, we prove that the problem is NP-hard on NFAs. (iv) We show that, in the worst case, the classic powerset construction algorithm for NFA determinization generates an equivalent DFA of size 2^p(n-p+1)-1, where n is the number of NFA's states.

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