Abstract Learning finite automata (termed as model learning) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity (two-way QFA are more powerful than classical finite automata in computation ability as well). As a different problem in quantum learning theory and quantum machine learning, in this paper, our purpose is to initiate the study of learning QFA with queries (naturally it may be termed as quantum model learning), and the main results are regarding learning two basic one-way QFA (1QFA): (1) we propose a learning algorithm for measure-once 1QFA (MO-1QFA) with query complexity of polynomial time and (2) we propose a learning algorithm for measure-many 1QFA (MM-1QFA) with query complexity of polynomial time, as well.

Opacity is an important security notion for discrete event systems (DESs) to measure the secret behaviors leaked to an intruder. The intruder is usually modeled as a malicious observer who always attempts to infer secret behaviors based on his dynamical observations of the system. In our earlier research, we proposed several opacity measures for DESs from the perspective of fuzzy logic, in which the intruder is assumed to only infer the current secret behaviors. Considering that modern computers usually have a powerful computation ability and rich memory resources, this article presents a new opacity measure of DESs, called as fuzzy infinite-step opacity (FIO) measure, where the intruder is assumed to have the ability of inferring all the secret behaviors that occurred in the past. In addition, the properties and the calculation of the FIO measure are also investigated. The FIO measure can be used to assess the secret-leakage situations of fuzzy DESs attacked by powerful intruders. For the purpose of illustration, we present a practical application of the FIO measure in evaluating the privacy protection mechanisms in location-based services.

The goal in the area of functions property testing is to determine whether a given black-box Boolean function has a particular given property or is ɛ-far from having that property. We investigate here several types of properties testing for Boolean functions (identity, correlations and balancedness) using the Deutsch-Jozsa algorithm (for the Deutsch-Jozsa (D-J) problem) and also the amplitude amplification technique. At first, we study here a particular testing problem: namely whether a given Boolean function f, of n variables, is identical with a given function g or is ɛ-far from g, where ɛ is the parameter. We present a one-sided error quantum algorithm to deal with this problem that has the query complexity [Formula: see text]. Moreover, we show that our quantum algorithm is optimal. Afterwards we show that the classical randomized query complexity of this problem is [Formula: see text]. Secondly, we consider the D-J problem from the perspective of functional correlations and let C( f, g) denote the correlation of f and g. We propose an exact quantum algorithm for making distinction between | C( f, g)| = ɛ and | C( f, g)| = 1 using six queries, while the classical deterministic query complexity for this problem is Θ(2 n ) queries. Finally, we propose a one-sided error quantum query algorithm for testing whether one Boolean function is balanced versus ɛ-far balanced using [Formula: see text] queries. We also prove here that our quantum algorithm for balancedness testing is optimal. At the same time, for this balancedness testing problem we present a classical randomized algorithm with query complexity of O(1/ ɛ 2 ). Also this randomized algorithm is optimal. Besides, we link the problems considered here together and generalize them to the general case.

Opacity, as an important security property, has been well investigated in crisp discrete event systems. However, in some practical systems, the general behaviors, secret and nonsecret behaviors obtained by the intruder need to be expressed as fuzzy sets rather than crisp sets. Under this situation, the intruder makes inferences by fuzzy logic rather than classic logic, thus the notions of the fuzzy opacity measures are desired to be established. First, by means of the basic laws of fuzzy logic, a general fuzzy opacity measure, as a template, is presented, in this article, for the given generalized characterizations of secret and nonsecret behaviors. Moreover, a specific measure, called as fuzzy initial-final opacity (FIFO) measure, is investigated under the assumption that the secrets and nonsecrets can be coded into the initial-final state-pairs. In addition, two variants of FIFO measure are also proposed under the assumptions that the secrets and nonsecrets are only related to the initial-states or final-states. With these fuzzy opacity measures, the system designers can evaluate the opacity degree of the systems to be designed. To show the applicability of the fuzzy opacity measures to practical systems, we finally present an illustrative example of evaluating the location privacy protection systems in location-based services by means of the fuzzy opacity measures.

In our earlier paper (i.e., Deng and Qiu, 2017), we have investigated the safe-diagnosis problem for discrete-event systems (DESs) with assumptions that events completely cannot be obtained and states can be measured by sensors under the so-called state-based decentralized architecture (SDA). In this paper, we continue to consider this problem under an economical decentralized architecture (EDA). The corresponding safe-diagnosability, as well as a polynomial-time verification algorithm is presented.

In thedistributed Deutsch–Jozsa promise problem, two parties are to determine whether their respective stringsx, y∈ {0,1}nare at theHamming distanceH(x, y) = 0 orH(x, y) = $\frac{n}{2}$ . Buhrmanet al.(STOC' 98) proved that the exactquantum communication complexityof this problem isO(logn) while thedeterministic communication complexityisΩ(n). This was the first impressive (exponential) gap between quantum and classical communication complexity. In this paper, we generalize the above distributed Deutsch–Jozsa promise problem to determine, for any fixed $\frac{n}{2}$ ⩽k⩽n, whetherH(x, y) = 0 orH(x, y) =k, and show that an exponential gap between exact quantum and deterministic communication complexity still holds ifkis an even such that $\frac{1}{2}$ n⩽k< (1 − λ)n, where 0 < λ < $\frac{1}{2}$ is given. We also deal with a promise version of the well-knowndisjointnessproblem and show also that for this promise problem there exists an exponential gap between quantum (and also probabilistic) communication complexity and deterministic communication complexity of the promise version of such a disjointness problem. Finally, some applications to quantum, probabilistic and deterministic finite automata of the results obtained are demonstrated.

Recently, we have introduced bi-fuzzy discrete event systems (BFDESs) and established their supervisory control theory under full observations and partial observations, respectively. Compared with the models of discrete event systems and fuzzy discrete event systems, BFDESs are more satisfactory models to characterize some high-uncertainty systems. In this paper, we study the fault diagnosis problems of BFDESs. First, the framework of state-based decentralized diagnosis is introduced. Then, the notion of state-based codiagnosability is defined and a necessary and sufficient condition for state-based codiagnosability is provided. In particular, a polynomial-time algorithm for state-based codiagnosability verification and an online decentralized diagnosis method are presented. In addition, a compact state-based decentralized diagnosis framework is introduced, and the notion of state-based jointly diagnosability and its verification algorithm are proposed. Finally, the implication relationships between the two diagnosabilities are investigated.

The supervisory control theory of fuzzy discrete-event systems (FDESs) for fuzzy language equivalence has been developed. However, in a way, language equivalence has limited expressiveness. Therefore, if the given specification can not be expressed by language equivalence, then the control for language equivalence does not work. In this paper, we further establish the supervisory control theory of FDESs for fuzzy simulation equivalence whose expressiveness is stronger than that of fuzzy language equivalence. First, we formalize the notions of fuzzy simulation and fuzzy simulation equivalence between two FDESs. Then, we present a method for deciding whether there is a fuzzy simulation or not. In addition, we also show several basic properties of fuzzy simulation relations. Afterward, we put forward the notion of fuzzy simulation-based controllability and, in particular, show that it serves as a necessary and sufficient condition for the existence of the fuzzy supervisors of FDESs. Moreover, we study the “range” control problem of FDESs. Some examples are given to illustrate the main results obtained.

It is well known that type-1 fuzzy sets (T1 FSs) have limited capabilities to handle some data uncertainties directly, and type-2 fuzzy sets (T2 FSs) can cover the shortcoming of T1 FSs to a certain extent. Fuzzy discrete event systems (FDESs) were proposed based on T1 FSs theory. Hence, FDES may not be a satisfactory model to characterize some high-uncertainty systems. In this paper, we propose a new model, called as bifuzzy discrete event systems (BFDESs), by combining classical DESs theory and T2 FSs theory. Then, we consider the supervisory control problem of BFDESs. The bifuzzy controllability theorem and nonblocking bifuzzy controllability theorem are demonstrated. In addition, an algorithm for checking the bifuzzy controllability condition is presented. In addition, two controllable approximations to an uncontrollable language are investigated in detail. An illustrative example is provided to show the applicability and the advantages of the BFDES model.

This paper investigates the minimization problem of fuzzy automata, aiming to obtain a procedure for finding a minimal state fuzzy automaton equivalent to a given one. The decision version of the minimization problem is as follows: Given a fuzzy automaton A and a natural number k, i.e., a pair (A, k), is there a k-state fuzzy automaton equivalent to A? We prove that the above problem is decidable for fuzzy automata over totally ordered lattices and then obtain a procedure for minimizing a given fuzzy automaton. To this end, we introduce the concept of systems of fuzzy polynomial equations, present a procedure for finding solutions of these systems and, finally, reduce the above decision problem to finding a solution of a system of fuzzy polynomial equations. It is worth pointing out that although some algorithms in the literature were claimed to be minimization algorithms, the term “minimization” there did not mean state minimization in our sense, since these algorithms did not aim at a minimal fuzzy automaton but found “reasonably” small fuzzy automata.

In order to more effectively cope with the real-world problems of vagueness, fuzzy discrete-event systems (FDESs) were proposed by Lin and Ying recently. Then we and Cao and Ying investigated the supervisory control of FDESs independently. In this paper, we are concerned with another important issue of FDESs, the failure diagnosis. More specifically: (1) we propose a ldquofuzzy diagnosabilityrdquo approach by introducing a fuzzy diagnosability function to characterize the diagnosability degree, which takes values in the interval [0,1] rather than { 0,1}; (2) based on the observability of events, we formalize the construction of the diagnosers that are used to perform fuzzy diagnosis; (3) a number of basic properties of the diagnosers are investigated. In particular, we present a necessary and sufficient condition for failure diagnosis of FDESs. Our results generalize the important consequences of the diagnosability for crisp discrete-event systems (DESs) introduced by Sampathetal. The newly proposed approach allows us to deal with the problem of diagnosability for both crisp DESs and FDESs; (4) in addition, a method for checking the fuzzy diagnosability for FDESs is proposed. Also, some examples are provided to illustrate the application of the diagnosability of FDESs.

In order to more effectively cope with the real-world problems of vagueness, impreciseness, and subjectivity, fuzzy discrete-event systems (FDESs) were proposed recently. Notably, FDESs have been applied to biomedical control for human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) treatment planning and sensory information processing for robotic control. Qiuindependently developed supervisory control theory of FDESs. We note that the controllability of events in Qiu's work is fuzzy, but the observability of events is crisp, and the observability of events in Cao and Ying's work is also crisp, although the controllability is not completely crisp since the controllable events can be disabled with any degrees. Motivated by the necessity to consider the situation that the events may be observed or controlled with some membership degrees, in this paper, we establish the supervisory control theory of FDESs with partial observations in which both the observability and controllability of events are fuzzy instead. We formalize the notions of fuzzy controllability condition and fuzzy observability condition. In addition, controllability and observability theorem of FDESs is set up in a more generic framework. In particular, we present a detailed computing flow to verify whether the controllability and observability conditions hold. Thus, this result can decide the existence of supervisors. Also, we use this computing method to check the existence of supervisors in the controllability and observability theorem of classical discrete-event systems (DESs), which is a new method and different from classical case. A number of examples are elaborated on to illustrate the presented results.

Fuzzy discrete event systems (DESs) were proposed recently by Lin and Ying [19], which may better cope with the real-world problems of fuzziness, impreciseness, and subjectivity such as those in biomedicine. As a continuation of [19], in this paper, we further develop fuzzy DESs by dealing with supervisory control of fuzzy DESs. More specifically: 1) we reformulate the parallel composition of crisp DESs, and then define the parallel composition of fuzzy DESs that is equivalent to that in [19]. Max-product and max-min automata for modeling fuzzy DESs are considered, 2) we deal with a number of fundamental problems regarding supervisory control of fuzzy DESs, particularly demonstrate controllability theorem and nonblocking controllability theorem of fuzzy DESs, and thus, present the conditions for the existence of supervisors in fuzzy DESs; 3) we analyze the complexity for presenting a uniform criterion to test the fuzzy controllability condition of fuzzy DESs modeled by max-product automata; in particular, we present in detail a general computing method for checking whether or not the fuzzy controllability condition holds, if max-min automata are used to model fuzzy DESs, and by means of this method we can search for all possible fuzzy states reachable from initial fuzzy state in max-min automata. Also, we introduce the fuzzy n-controllability condition for some practical problems, and 4) a number of examples serving to illustrate the applications of the derived results and methods are described; some basic properties related to supervisory control of fuzzy DESs are investigated. To conclude, some related issues are raised for further consideration.

Computing with words (CW) as a methodology, means computing and reasoning by the use of words in place of numbers or symbols, which may conform more to humans' perception when describing real-world problems. In this paper, as a continuation of a previous paper, we aim to develop and deepen a formal aspect of CW. According to the previous paper, the basic point of departure is that CW treats certain formal modes of computation with strings of fuzzy subsets instead of symbols as their inputs. Specifically, 1) we elaborate on CW via Turing machine (TM) models, showing the time complexity is at least exponential if the inputs are strings of words; 2) a negative result of (6) not holding is verified which indicates that the extension principle for CW via TMs needs to be re-examined; 3) we discuss CW via context- free grammars and regular grammars and the extension principles for CW via these formal grammars are set up; 4) some equivalences between fuzzy pushdown automata (respectively, fuzzy finite-state automata) fuzzy context-free grammars (respectively, fuzzy regular grammars) are demonstrated in the sense that the inputs are instead strings of words; 5) some instances are described in detail. Summarily formal aspect of CW is more systematically established more deeply dealt with while some new problems also emerge.

It reveals some equivalences between automata based on complete residuated lattice-valued logic (called l valued automata) and the truth-value lattice of the underlying logic (i.e. residuated lattice). In particular, it demonstrates several basic equivalent characterizations on the retrievability of l valued automata. Finally, the connections of the homomorphisms between two l valued automata to continuous mappings and open mappings are clarified. So this paper establishes further the more profound fuzzy automata theory.