Prefix-closed Languages Research Articles

Comparing the notions of opacity for discrete-event systems

Opacity is an information flow property characterizing whether a system reveals its secret to a passive observer. Several notions of opacity have been introduced in the literature. We study the notions of language-based opacity, current-state opacity, initial-state opacity, initial-and-final-state opacity, K-step opacity, and infinite-step opacity. Comparing the notions is a natural question that has been investigated and summarized by Wu and Lafortune, who provided transformations among current-state opacity, initial-and-final-state opacity, and language-based opacity, and, for prefix-closed languages, also between language-based opacity and initial-state opacity. We extend these results by showing that all the discussed notions of opacity are transformable to each other. Besides a deeper insight into the differences among the notions, the transformations have applications in complexity results. In particular, the transformations are computable in polynomial time and preserve the number of observable events and determinism, and hence the computational complexities of the verification of the notions coincide. We provide a complete and improved complexity picture of the verification of the discussed notions of opacity, and improve the algorithmic complexity of deciding language-based opacity, infinite-step opacity, and K-step opacity.

P.3.e.009 - Epigenetic signature of early trauma: differences in the FKBP5 DNA methylation levels among psychotic patients, their healthy siblings and controls

We present here the notion of signature of trees and of languages, and its relationships with the theory of numeration systems. The signature of an ordered infinite tree (of bounded degree) is an infinite (bounded) sequence of integers, the sequence of the degrees of the nodes taken in the visit order of the canonical breadth-first traversal of the tree. A prefix-closed language defines such a tree augmented with labels on arcs, hence is associated with a signature. This way of ‘traversing’ a language is related to the notion of abstract numeration system, due to Lecomte and Rigo.After having set in detail the framework of signature, we study and characterise the signatures of rational languages. Using a known construction from numeration system theory, we show that these signatures form a special subclass of morphic words. We then use this framework to give an alternative definition to morphic numeration systems (also called Dumont–Thomas numeration systems). We finally highlight that the classes of morphic numeration systems and of (prefix-closed) rational abstract numeration systems are essentially the same.

Syntactic Complexity of Regular Ideals

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1, nn−1 + n − 1, and nn−2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.

The signature of rational languages

We present here the notion of signature of trees and of languages, and its relationships with the theory of numeration systems. The signature of an ordered infinite tree (of bounded degree) is an infinite (bounded) sequence of integers, the sequence of the degrees of the nodes taken in the visit order of the canonical breadth-first traversal of the tree. A prefix-closed language defines such a tree augmented with labels on arcs, hence is associated with a signature. This way of ‘traversing’ a language is related to the notion of abstract numeration system, due to Lecomte and Rigo.After having set in detail the framework of signature, we study and characterise the signatures of rational languages. Using a known construction from numeration system theory, we show that these signatures form a special subclass of morphic words. We then use this framework to give an alternative definition to morphic numeration systems (also called Dumont–Thomas numeration systems). We finally highlight that the classes of morphic numeration systems and of (prefix-closed) rational abstract numeration systems are essentially the same.

Distributed computation of supremal conditionally controllable sublanguages

ABSTRACTIn this paper, we further develop the coordination control framework for discrete-event systems with both complete and partial observations. First, a weaker sufficient condition for the computation of the supremal conditionally controllable sublanguage and conditionally normal sublanguage is presented. Then we show that this condition can be imposed by synthesising a-posteriori supervisors. The paper further generalises the previous study by considering general, non-prefix-closed languages. Moreover, we prove that for prefix-closed languages the supremal conditionally controllable sublanguage and conditionally normal sublanguage can always be computed in the distributed way without any restrictive conditions we have used in the past.

Operations on Automata with All States Final

We study the complexity of basic regular operations on languages represented by incomplete deterministic or nondeterministic automata, in which all states are final. Such languages are known to be prefix-closed. We get tight bounds on both incomplete and nondeterministic state complexity of complement, intersection, union, concatenation, star, and reversal on prefix-closed languages.

Coordination control of discrete-event systems revisited

In this paper, we revise and further investigate the coordination control approach proposed for supervisory control of distributed discrete-event systems with synchronous communication based on the Ramadge-Wonham automata framework. The notions of conditional decomposability, conditional controllability, and conditional closedness ensuring the existence of a solution are carefully revised and simplified. The paper is generalized to non-prefix-closed languages, that is, supremal conditionally controllable sublanguages of not necessary prefix-closed languages are discussed. Non-prefix-closed languages introduce the blocking issue into coordination control, hence a procedure to compute a coordinator for nonblockingness is included. The optimization problem concerning the size of a coordinator is under investigation. We prove that to find the minimal extension of the coordinator event set for which a given specification language is conditionally decomposable is NP-hard. In other words, unless P=NP, it is not possible to find a polynomial algorithm to compute the minimal coordinator with respect to the number of events.

Diagnosability analysis of hybrid systems cast in a discrete-event framework

This paper addresses the problem of assessing the diagnosability of hybrid systems modeled by a hybrid automaton coupling methods from the continuous and the discrete event model-based diagnosis fields. The discrete states of the hybrid automaton represent the modes of operation of the system for which the continuous dynamics are specified. The diagnosability of the continuously-valued part of the model is first analyzed and the new concept of mode signature is shown to characterize mode diagnosability from continuous measurements. Continuous dynamics are then abstracted by defining a set of signature-events associated to mode signature changes, preserving this way mode diagnosability. The behavior of the abstract hybrid system is then modeled by a prefix-closed language over the original event alphabet enriched by these additional events. Based on this language, diagnosability analysis of the hybrid system is cast into a discrete-event framework and hybrid diagnosability conditions are provided. A case study based on the Attitude and Orbit Control System of a spacecraft illustrates the method.

On the Computation of Supremal Sublanguages Relevant to Supervisory Control

Given a specification language, this paper discusses an iterative procedure for the computation of the supremal sublanguage, that possesses a conjunction of certain closed-loop properties, including controllability, normality and completeness. The iteration is stated in terms of (i) supremal sublanguage operators for each individual property, (ii) prefix-closures, and, (iii) language intersections. Within the iteration, the individual supremal sublanguage operators are only applied on prefix-closed languages, while the overall specification is not required to be prefix-closed. Our main result establishes finite convergence, provided that all parameters are regular.

Efficient robust supervisors for discrete event systems

This paper is a sequel to a previous publication by the same authors, in which an efficient modular solution to a robust supervisory control problem for discrete event systems modeled by finite automata with prefix-closed specification languages has been presented. This solution is based on a general recursive robust control scheme, which has been successfully applied to a number of problems. The additional contributions of the present paper are: (a) a slight generalization of the problem assumptions; (b) an alternative derivation of some of the results and an alternative formulation of the controller; (c) a detailed description of a very efficient on-line implementation algorithm; and (d) an illustrative practical example.

Modular Implementation of Robust Supervisory Controllers for Discrete Event Systems

In this paper, the problem of designing robust supervisory controllers for uncertain discrete event systems is studied. It is assumed that both the plant behavior and the design specifications can be represented by regular languages. It is also assumed that the unknown model of the uncertain plant belongs to a finite set of possibilities. To solve the problem, a modular approach based on a general recursive robust control scheme is developed. The contribution of the present paper consists in deriving a maximally permissive modular robust supervisory controller for the case of prefix-closed languages, having the advantage of linear complexity in the number of models as opposed to the exponential complexity of the worst-case scenario in the direct implementation of standard supervisors for augmented systems.

Decentralized Supervisory Control of Discrete-Event Systems Over Communication Networks

In this paper, we investigate the problem of designing embedded decentralized discrete-event controllers over communication networks. It is assumed that there is a path between every pair of processes in the network. The control objective is specified by a prefix-closed language that is controllable and observable, but not coobservable. The paper is focused on communication among processes necessary to meet the control objective. As such, process models are left unspecified; it is only required that disabling any of the controllable events do not block communication among processes. Our findings support the idea that in the presence of ideal communication channels, the protocol design for noncoobservable specifications can be reduced to the synthesis of communicating decentralized supervisors, and we propose solutions for a restricted class of problems. The paper is concluded with a positive result for the case where channels are unreliable.

Polynomial synthesis of supervisor for partially observed discrete-event systems by allowing nondeterminism in control

We study the supervisory control of discrete-event systems (DESs) under partial observation using nondeterministic supervisors. We formally define a nondeterministic control policy and also a control & observation compatible nondeterministic state machine and prove their equivalence. The control action of a nondeterministic supervisor is chosen online, nondeterministically from among a set of choices determined offline. Also, the control action can be changed online nondeterministically (prior to any new observation) in accordance with choices determined offline. The online choices, once made, can be used to affect the set of control action choices in future. We show that when control is exercised using a nondeterministic supervisor, the specification language is required to satisfy a weaker notion of observability, which we define in terms of recognizability and achievability. Achievability serves as necessary and sufficient condition for the existence of a nondeterministic supervisor, and it is weaker than controllability and observability combined. When all events are controllable, achievability reduces to recognizability. We show that both existence, and synthesis of nondeterministic supervisors can be determined polynomially. (For deterministic supervisors, only existence can be determined polynomially.) Both achievability and recognizability are preserved under union, and also under intersection (when restricted over prefix-closed languages). Using the intersection closure property we derive a necessary and sufficient condition for the range control problem for the prefix-closed case. Unlike the deterministic supervisory setting where the complexity of existence is exponential, our existence condition is polynomially verifiable, and also a supervisor can be polynomially synthesized.

Insights to Angluin's Learning

Among other domains, learning finite-state machines is important for obtaining a model of a system under development, so that powerful formal methods such as model checking can be applied.A prominent algorithm for learning such devices was developed by Angluin. We have implemented this algorithm in a straightforward way to gain further insights to practical applicability. Furthermore, we have analyzed its performance on randomly generated as well as real-world examples. Our experiments focus on the impact of the alphabet size and the number of states on the needed number of membership queries. Additionally, we have implemented and analyzed an optimized version for learning prefix-closed regular languages. Memory consumption is one major obstacle when we attempted to learn large examples.We see that prefix-closed languages are relatively hard to learn compared to arbitrary regular languages. The optimization, however, shows positive results.

Maximizing Robustness of Supervisors for Partially Observed Discrete Event Systems

We consider a robust supervisory control problem to synthesize a supervisor for the nominal plant model which maximizes robustness. Cury and Krogh have first addressed this problem by unnecessarily restricting the upper bound of the legal behavior. In our previous work, we have solved the problem without restricting the upper bound of the legal behavior when the specification is described by prefix-closed languages. In this paper, we extend our previous work to the partial (event) observation case. First, we synthesize a supervisor that maximizes robustness. This result shows that robustness can be optimized under partial observation while permissiveness cannot be optimized in general. We next consider a special case where all controllable events are observable. In this special case, we synthesize a maximally permissive supervisor for the nominal plant model which maximizes not only robustness but also permissiveness for the maximal set of admissible plant variations.

On A Generalization of Robust Supervisory Control of Discrete Event Systems

In this paper we consider a generalization of the robust supervisory control problem introduced by Lin (Lin, 1993), improved by Takai (Takai, 2000; Takai, 2002), Park and Lim (Park and Lim, 2000; Park and Lim, 2002) and Cury and Krogh (Cury and Krogh, 1999). Following the formalism in (Lin, 1993), we suppose the plant language L(G) ⊆ Σ* belongs to a finite collection of non-empty, prefix-closed languages L(G) Є {L1, L2,...,Ln}, where Li(≠0) ⊆ Σ*, i Є {1, 2,..., n}. The event-set Σ is partitioned into controllable (Σc) and uncontrollable (Σu) subsets respectively. We assume all events are observable, and the supervisor has no prior knowledge as to the value of L(G) Є {L1, L2,..., Ln}. For each Li ⊆ Σ* we suppose there exists a prefix-closed language Ki ⊆ Li. We present three conditions that are necessary and sufficient for the existence of a supervisor that enforces Ki if the plant language L(G) = Li. It is possible that for a given choice of the sets {L1, L2,..., Ln} and {K1, K2,..., Kn}, the conditions identified in this paper are not satisfied. This calls for finding a {K1,K2,...Kn}, such that ∀iЄ{1,2,...,n},Ki⊆Ki that meets the required conditions, and each K⌢ satisfies some property that we might be interested in. The search for a satisfactory {K1,K2,...,Kn} using the notion of monotone properties is also presented.

Maximizing robustness of supervisors for partially observed discrete event systems

We consider a robust supervisory control problem to synthesize a supervisor for the nominal plant model which maximizes robustness. Cury and Krogh have first addressed this problem by unnecessarily restricting the upper bound of the legal behavior. In our previous work, we have solved the problem without restricting the upper bound of the legal behavior when the specification is described by prefix-closed languages. In this paper, we extend our previous work to the partial (event) observation case. First, we synthesize a supervisor that maximizes robustness. This result shows that robustness can be optimized under partial observation, while permissiveness cannot be optimized in general. We next consider a special case where all the controllable events are observable. In this special case, we synthesize a maximally permissive supervisor for the nominal plant model which maximizes not only the robustness but also permissiveness for the maximal set of admissible plant variations.

Synthesis of maximally permissive and robust supervisors for prefix-closed language specifications

J.E.R. Cury and B.H. Krogh (1999) have formulated a robust supervisory control problem to synthesize a supervisor for the nominal plant which maximizes robustness. They have solved the problem by unnecessarily restricting the upper bound of the legal behavior. We show that the problem can be solved without restricting the upper bound of the legal behavior when the specification is described by prefix-closed languages. We synthesize a maximally permissive supervisor for the nominal plant which maximizes not only robustness, but also permissiveness for the maximal set of admissible plant variations.

Thue Specifications and Their Monadic Second-order Properties

This paper introduces oriented Thue specifications and their models which are simple directed edge-labeled graphs. The monadic second-order theory of these models is investigated and shown decidable for the class of suffix-bounded rational oriented Thue specifications. In addition, a characterization of prefix-closed context-free languages is given in terms of oriented Thue specifications.

Nonblocking supervisory control of nondeterministic systems via prioritized synchronization

Shayman and Kumar (1995) showed that supervisory control of nondeterministic discrete-event systems, in the presence of driven events, can be achieved using prioritized synchronous composition as a mechanism of control, and trajectory models as a modeling formalism, first introduced by Heymann (1990). The specifications considered in this earlier work were given by prefix-closed languages. In this paper, we extend this work to include markings so that nonclosed specifications and issues such as blocking can be addressed. It is shown that the usual notion of nonblocking, called language model nonblocking, may not be adequate in the setting of nondeterministic systems, and a stronger notion, called trajectory model nonblocking, is introduced. Necessary and sufficient conditions for the existence of language model nonblocking as well as trajectory model nonblocking supervisors are obtained for nondeterministic systems in the presence of driven events in terms of extended controllability and relative-closure conditions and a new condition called the trajectory-closure condition.