Showing papers by "Tomáš Masopust published in 2017"
TL;DR: In this paper, it was shown that the problem of deciding universality of partially ordered NFAs with unbounded alphabets is PSpace -complete. But the complexity of determining universality for poNFAs with fixed alphables was shown to be poNFA-complete.
Abstract: Partially ordered NFAs (poNFAs) are NFAs where cycles occur only in the form of self-loops. A poNFA is universal if it accepts all words over its alphabet. Deciding universality is PSpace -complete for poNFAs. We show that this remains true when restricting to fixed alphabets. This is nontrivial since standard encodings of symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP -complete complexity bound is obtained if all self-loops in the poNFA are deterministic. We find that such restricted poNFAs (rpoNFAs) characterize R -trivial languages, and establish the complexity of deciding if the language of an NFA is R -trivial. The limitation to fixed alphabets is essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSpace -complete. Consequently, we obtain the complexity results for inclusion and equivalence problems. Finally, we show that the languages of rpoNFAs are definable by deterministic (one-unambiguous) regular expressions.
20 citations
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TL;DR: It is shown that for modular systems the problems are EXPSPACE-complete, and hence there is neither a polynomial-time nor a poynomial-space algorithm solving them.
Abstract: We study the complexity of deciding whether a modular discrete event system is detectable (resp. opaque, A-diagnosable). Detectability arises in the state estimation of discrete event systems, opacity is related to the privacy and security analysis, and A-diagnosability appears in the fault diagnosis of stochastic discrete event systems. Previously, deciding weak detectability (opacity, A-diagnosability) for monolithic systems was shown to be PSPACE-complete. In this paper, we study the complexity of deciding weak detectability (opacity, A-diagnosability) for modular systems. We show that the complexities of these problems are significantly worse than in the monolithic case. Namely, we show that deciding modular weak detectability (opacity, A-diagnosability) is EXPSPACE-complete. We further discuss a special case where all unobservable events are private, and show that in this case the problems are PSPACE-complete. Consequently, if the systems are all fully observable, then deciding weak detectability (opacity) for modular systems is PSPACE-complete.
18 citations
TL;DR: This work identifies the states of the ∼ k -canonical DFA whose union forms the language L ( A ) and use them to construct the required Boolean combination, and studies the computational and descriptional complexity of related problems.
13 citations
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TL;DR: The aim of this paper is to bring several interesting results from complexity theory and to illustrate their relevance to supervisory control by proving new nontrivial results concerning nonblockingness in modularsupervisory control of discrete event systems modeled by finite automata.
Abstract: Complexity analysis becomes a common task in supervisory control. However, many results of interest are spread across different topics. The aim of this paper is to bring several interesting results from complexity theory and to illustrate their relevance to supervisory control by proving new nontrivial results concerning nonblockingness in modular supervisory control of discrete event systems modeled by finite automata.
8 citations
TL;DR: In this paper, a controllable and coobservable sublanguage of the specification by using additional communications between supervisors is constructed, which can be used for both prefix-closed and non-prefix-closed specifications.
Abstract: In decentralized supervisory control, several local supervisors cooperate to accomplish a common goal (specification). Controllability and coobservability are the key conditions to achieve a specification in the controlled system. We construct a controllable and coobservable sublanguage of the specification by using additional communications between supervisors. Namely, we extend observable events of local supervisors via communication and apply a fully decentralized computation of local supervisors. Coobservability is then guaranteed by construction. Sufficient conditions to achieve the centralized optimal solution are discussed. Our approach can be used for both prefix-closed and non-prefix-closed specifications.
8 citations
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TL;DR: It is shown that it is PTime-hard and that it remains P time-hard even for minimal DFAs, and the lower-bound complexity of separability of regular languages by piecewise testable languages is shown.
Abstract: Piecewise testable languages form the first level of the Straubing-Th\'erien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. The question has not yet been addressed for NFAs. We fill in this gap by showing that it is PSpace-complete. The main result is then the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separability by piecewise testable languages is known to be decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even for minimal DFAs.
3 citations
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TL;DR: This work solves the universality problem for confluent, self-loop deterministic poNFAs by showing that it is PSpace-complete if the alphabet may grow polynomially, and provides a lower-bound complexity for some other problems, including inclusion, equivalence, and $k$-piecewise testability.
Abstract: An automaton is partially ordered if the only cycles in its transition diagram are self-loops. The expressivity of partially ordered NFAs (poNFAs) can be characterized by the Straubing-Therien hierarchy. Level 3/2 is recognized by poNFAs, level 1 by confluent, self-loop deterministic poNFAs as well as by confluent poDFAs, and level 1/2 by saturated poNFAs. We study the universality problem for confluent, self-loop deterministic poNFAs. It asks whether an automaton accepts all words over its alphabet. Universality for both NFAs and poNFAs is a PSpace-complete problem. For confluent, self-loop deterministic poNFAs, the complexity drops to coNP-complete if the alphabet is fixed but is open if the alphabet may grow. We solve this problem by showing that it is PSpace-complete if the alphabet may grow polynomially. Consequently, our result provides a lower-bound complexity for some other problems, including inclusion, equivalence, and $k$-piecewise testability. Since universality for saturated poNFAs is easy, confluent, self-loop deterministic poNFAs are the simplest and natural kind of NFAs characterizing a well-known class of languages, for which deciding universality is as difficult as for general NFAs.
2 citations
TL;DR: In this paper, it was shown that separability of regular languages by piecewise testable languages is PSpace -complete and that it remains PTime -hard even if the input automata are minimal DFAs.
2 citations
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TL;DR: In this paper, it was shown that there is no polynomial-time algorithm for computing a DFA of the infimal prefix-closed, controllable and observable superlanguage.
Abstract: The infimal prefix-closed, controllable and observable superlanguage plays an essential role in the relationship between controllability, observability and co-observability -- the central notions of supervisory control theory. Existing algorithms for its computation are exponential and it is not known whether a polynomial algorithm exists. In this paper, we study the state complexity of this language. State complexity of a language is the number of states of the minimal DFA for the language. For a language of state complexity $n$, we show that the upper-bound state complexity on the infimal prefix-closed and observable superlanguage is $2^n + 1$ and that this bound is asymptotically tight. It proves that there is no algorithm computing a DFA of the infimal prefix-closed and observable superlanguage in polynomial time. Our construction further shows that such a DFA can be computed in time $O(2^n)$. The construction involves NFAs and a computation of the supremal prefix-closed sublanguage. We study the computation of the supremal prefix-closed sublanguage and show that there is no polynomial-time algorithm that computes an NFA of the supremal prefix-closed sublanguage of a language given as an NFA even if the language is unary.
1 citations
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TL;DR: In this paper, it is shown that for deterministic finite automata without non-trivial cycles, checking weak (periodic) detectability remains intractable, even for very simple DESs.
Abstract: Detectability of discrete event systems (DESs) is a question whether the current and subsequent states can be determined based on observations. Shu and Lin designed a polynomial-time algorithm to check strong (periodic) detectability and an exponential-time (polynomial-space) algorithm to check weak (periodic) detectability. Zhang showed that checking weak (periodic) detectability is PSpace-complete. This intractable complexity opens a question whether there are structurally simpler DESs for which the problem is tractable. In this paper, we show that it is not the case by considering DESs represented as deterministic finite automata without non-trivial cycles, which are structurally the simplest deadlock-free DESs. We show that even for such very simple DESs, checking weak (periodic) detectability remains intractable. On the contrary, we show that strong (periodic) detectability of DESs can be efficiently verified on a parallel computer.