Tomáš Masopust

Jan Komenda, Tomáš Masopust, Jan H. van Schuppen

The requirement of a language to be conditionally decomposable is imposed on a specification language in the coordination supervisory control framework of discrete-event systems. In this paper, we present a polynomial-time algorithm for the verification whether a language is conditionally decomposable with respect to given alphabets. Moreover, we also present a polynomial-time algorithm to extend the common alphabet so that the language becomes conditionally decomposable. A relationship of conditional decomposability to nonblockingness of modular discrete-event systems is also discussed in this paper in the general settings. It is shown that conditional decomposability is a weaker condition than nonblockingness.

Liyong Lin, Tomáš Masopust, W. Murray Wonham et al.

A reduction of a source distribution is a collection of smaller sized distributions that are collectively equivalent to the source distribution with respect to the property of decomposability. That is, an arbitrary language is decomposable with respect to the source distribution if and only if it is decomposable with respect to each smaller sized distribution (in the reduction). The notion of reduction of distributions has previously been proposed to improve the complexity of decomposability verification. In this work, we address the problem of generating (optimal) reductions of distributions automatically. A (partial) solution to this problem is provided, which consists of 1) an incremental algorithm for the production of candidate reductions and 2) a reduction validation procedure. In the incremental production stage, backtracking is applied whenever a candidate reduction that cannot be validated is produced. A strengthened substitution-based proof technique is used for reduction validation, while a fixed template of candidate counter examples is used for reduction refutation; put together, they constitute our (partial) solution to the reduction verification problem. In addition, we show that a recursive approach for the generation of (small) reductions is easily supported.

Tomáš Masopust, Xiang Yin

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.

Tomáš Masopust

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.

Tomáš Masopust

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.

Tomáš Masopust

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.

Tomáš Masopust

One of the most interesting questions concerning hierarchical control of discrete-event systems with partial observations is a condition under which the language observability is preserved between the original and the abstracted plant. Recently, we have characterized two such sufficient conditions---observation consistency and local observation consistency. In this paper, we prove that the condition of observation consistency is undecidable for non-regular (linear, deterministic context-free) languages. The question whether the condition is decidable for regular languages is open.

Jiří Balun, Tomáš Masopust

Opacity is a property expressing whether a system may reveal its secret to a passive observer (an intruder) who knows the structure of the system but has a limited observation of its behavior. Several notions of opacity have been studied, including current-state opacity, K-step opacity, and infinite-step opacity. We study K-step opacity that generalizes both current-state opacity and infinite-step opacity, and asks whether the intruder cannot decide, at any time, whether or when the system was in a secret state during the last K observable steps. We design a new algorithm deciding K-step opacity the complexity of which is lower than that of existing algorithms and that does not depend on K. We then compare K-step opacity with other opacity notions and provide new transformations among the notions that do not use states that are neither secret nor non-secret (neutral states) and that are polynomial with respect to both the size of the system and the binary encoding of K.

Tomáš Masopust

Critical observability is a property of cyber-physical systems to detect whether the current state belongs to a set of critical states. In safety-critical applications, critical states model operations that may be unsafe or of a particular interest. De Santis et al. introduced critical observability for linear switching systems, and Pola et al. adapted it for discrete-event systems, focusing on algorithmic complexity. We study the computational complexity of deciding critical observability for systems modeled as (networks of) finite-state automata and Petri nets. We show that deciding critical observability is (i) NL-complete for finite automata, that is, it is efficiently verifiable on parallel computers, (ii) PSPACE-complete for networks of finite automata, that is, it is very unlikely solvable in polynomial time, and (iii) undecidable for labeled Petri nets, but becoming decidable if the set of critical states (markings) is finite or co-finite, in which case the problem is as hard as the non-reachability problem for Petri nets.