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1

Tight Bounds for Cut-Operations on Deterministic Finite Automata

Drewes F., van der Merwe B., Holzer M., Jakobi S.

Fundamenta Informaticae

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2017

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Vol. 155, nr 1/2

89--110

EN

We investigate the state complexity of the cut and iterated cut operation for deterministic finite automata (DFAs), answering an open question stated in [M. BERGLUND, et al.: Cuts in regular expressions. In Proc. DLT, LNCS 7907, 2011]. These operations can be seen as an alternative to ordinary concatenation and Kleene star modelling leftmost maximal string matching. We show that the cut operation has a matching upper and lower bound of n states, if m = 1, and (n-1).m+n states, otherwise, on DFAs accepting the cut of two individual languages that are accepted by n-andm-state DFAs, respectively. In the unary case we obtain max(2n-1;m+n-2) states as a tight bound--notice that for m ≤n the bound for unary DFAs only depends on the former automaton and not on the latter. For accepting the iterated cut of a language accepted by an n-state DFA we find a matching bound of 1+(n+1). F( 1; n+2;-n+2; n+1 j-1 ) states on DFAs, if n ≥ 4 and where F refers to the generalized hypergeometric function. This bound is in the order of magnitude Θ((n - 1)!). Finally, the bound drops to 2n - 1 for unary DFAs accepting the iterated cut of an n-state DFA, if n ≥ 3, and thus is similar to the bound for the cut operation on unary DFAs.

2

On the Computational Complexity of Partial Word Automata Problems

Holzer M., Jakobi S., Wendlandt M.

Fundamenta Informaticae

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2016

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Vol. 148, nr 3/4

267--289

EN

We consider the computational complexity of problems related to partial word automata. Roughly speaking, a partial word is a word in which some positions are unspecified and a partial word automaton is a finite automaton that accepts a partial word language-here the unspecified positions in the word are represented by a "hole" symbol ◊ . A partial word language L' can be transformed into an ordinary language L by using a ◊-substitution. In particular, we investigate the complexity of the compression or minimization problem for partial word automata, which is known to be NP-hard. We improve on the previously known complexity on this problem, by showing PSPACE-completeness. In fact, it turns out that almost all problems related to partial word automata, such as, e.g., equivalence and universality, are already PSPACE- complete. Moreover, we also study these problems under the further restriction that the involved automata accept only finite languages. In this case, the complexities of the studied problems drop from PSPACE-completeness down to coNP-hardness and containment in ∑P2 depending on the problem investigated.

3

Minimization and Characterizations for Biautomata

Holzer M., Jakobi S.

Fundamenta Informaticae

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2015

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Vol. 136, nr 1/2

113--137

EN

We show how to minimize biautomata with adaptations of classical minimization algorithms for ordinary deterministic finite automata and moreover by a Brzozowski-like minimization algorithm by applying reversal and power-set construction twice to the biautomaton under consideration. Biautomata were recently introduced in [O. KL´I MA, L. POL´A K: On biautomata. RAIRO— Theor. Inf. Appl., 46(4), 2012] as a generalization of ordinary finite automata, reading the input from both sides. The correctness of the Brzozowski-like minimization algorithm needs a little bit more argumentation than for ordinary finite automata since for a biautomaton its dual or reverse automaton, built by reversing all transitions, does not necessarily accept the reversal of the original language. To this end we first use the recently introduced notion of nondeterminism for biautomata [M. HOLZER, S. JAKOBI: Nondeterministic Biautomata and Their Descriptional Complexity. In: 15th DCFS, Number 8031 of LNCS, 2013] and take structural properties of the forward- and backward-transitions of the automaton into account. This results in a variety of biautomata models, the accepting power of which is characterized. As a byproduct we give a simple structural characterization of cyclic regular and commutative regular languages in terms of deterministic biautomata.