Synchronizing series-parallel deterministic finite automata with loops and related problems

Bruchertseifer, Jens; Fernau, Henning

doi:10.1051/ita/2021005

Cited by 2 publications

(7 citation statements)

References 47 publications

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“…In the following, we will concentrate on the length-lexicographical ordering ≤ ll as the partial order, where v ≤ ll w means that either |v| < |w| or that |v| = |w| and v ≤ lex w, where v ≤ lex w refers to a lexicographical (total) order induced by a given total order on the alphabet. In and in Bruchertseifer and Fernau (2021), we are also discussing other (natural) partial orders. It should be noted that in some cases, the extension variants are solvable in polynomial time, while other cases lead to NP-or co-NP-hard problems.…”

Section: Extensions and Orderings Of Wordsmentioning

confidence: 99%

“…In Bruchertseifer and Fernau (2021), we also showed that EXT DFA-SW-≤ | is W[3]-hard, where | refers to the (scattered) subsequence ordering. But it is an open question if this extension problem belongs to W[3].…”

Section: Extensions and Orderings Of Wordsmentioning

confidence: 99%

“…Possible applications of this problem are explained in Kisielewicz et al (2015). The problem has also been considered from the viewpoint of approximation by Berlinkov (2014) and parameterized complexity by Bruchertseifer and Fernau (2021), Fernau et al (2015), Fernau and Krebs (2017), Montoya and Nolasco (2018).…”

Section: Introductionmentioning

confidence: 99%

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Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Fernau

Bruchertseifer²

2022

Math. Struct. Comp. Sci.

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The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $\textsf{W}[2]$ -hardness results, we show that these problems belong to $\textsf{A}[2]$ , $\textsf{W}[\textsf{P}],$ and $\textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $\textsf{W}[\textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $\textsf{W}[\textsf{Sync}]$ or are hard or complete for this new class.

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Section: Extensions and Orderings Of Wordsmentioning

confidence: 99%

Section: Extensions and Orderings Of Wordsmentioning

confidence: 99%

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Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Fernau

Bruchertseifer²

2022

Math. Struct. Comp. Sci.

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“…Since δ ′ agrees with δ on every word in Σ * it holds that δ ′ (Q, w) ⊆ S. The only state in Q ′ \Q is r and for r we have δ ′ (r, w) = r. Hence, δ ′ (Q ′ , w) ⊆ (S ∪{r}) and δ ′ (Q ′ , wc) = {t}. The relation R = (Q\S)× {r} demands that while reading wc on the set Q we reach a situation on which r is active but no state in Q\S is active 5 and does not become active again anymore. This is the case when reading the prefix w of wc since δ ′ (Q, w) ⊆ S and δ ′ (r, w) = r. Together with δ ′ (Q ′ , wc) = {t} ⊆ S we get R ⊆≺ 1 wc .…”

Section: • ∝mentioning

confidence: 99%

“…We will also observe W [1]-hardness results from the reductions given in [15]. So far, only little is known (see for example [10,24,5]) about the parameterized complexity of all the different synchronization variants considered in the literature.…”

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confidence: 94%

Synchronization under Dynamic Constraints

Wolf¹

2019

Preprint

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Imagine an assembly line where a box with a lid and liquid in it enters in some unknown orientation. The box should leave the line with the open lid facing upwards with the liquid still in it. To save costs there are no complex sensors or image recognition software available on the assembly line, so a reset sequence needs to be computed. But how can the dependencies of the deforming impact of a transformation of the box, such as "do not tilt the box over when the lid is open" or "open the lid again each time it gets closed" be modeled? We present three attempts to model constraints of these kinds on the order in which the states of an automaton are transitioned by a synchronizing word. The first two concepts relate the last visits of states and form constraints on which states still need to be reached, whereas the third concept concerns the first visits of states and forms constraints on which states might still be reached. We examine the computational complexity of different variants of the problem, whether an automaton can be synchronized with a word that respects the constraints defined in the respective concept, and obtain nearly a full classification. While most of the problems are PSPACE-complete we also observe NP-complete variants and variants solvable in polynomial time. One of them is the careful synchronization problem for partial weakly acyclic automata (which are partial automata, the states of which can be ordered such that no transition leads to a smaller state), which is shown to be solvable in time O(k 2 n 2 log(n)) where n is the size of the state set and k is the size of the alphabet. The algorithm even computes a synchronizing words as a witness. This is quite surprising as the careful synchronization problem uses to be a hard problem for most classes of automata. We will also observe a drop of the complexity if we track the orders of states on several paths simultaneously instead of tracking the set of active states. Further, we give upper bounds on the length of a synchronizing word depending on the size of the input relation and show that the Černý conjecture holds for partial weakly acyclic automata.

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Synchronizing series-parallel deterministic finite automata with loops and related problems

Cited by 2 publications

References 47 publications

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Synchronizing words and monoid factorization, yielding a new parameterized complexity class?

Synchronization under Dynamic Constraints

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