A representation of strongly connected automata and its applications
TL;DR: Some new results about automorphism groups of strongly connected automata are obtained by making use of a new type of representation of stronglyconnected automata introduced.
About: This article is published in Journal of Computer and System Sciences.The article was published on 1978-08-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Automata theory & Quantum finite automata.
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01 Aug 2019
TL;DR: This paper proposes properties to identify easy supervisory control problems and shows that the plant models and the requirement models together are a controllable, nonblocking, and maximally permissive supervisor, i.e., no synthesis is needed to calculate a supervisor.
Abstract: Supervisory control theory provides means to synthesize supervisors for cyber-physical systems based on models of the uncontrolled plant and models of the control requirements. In general, it has been shown that supervisory control synthesis is NP-hard, which is not beneficial for the applicability to industrial-sized systems. However, supervisory control synthesis seems to be easy for several industrial-sized systems compared to the theoretical worst-case complexity. In this paper, we propose properties to identify easy supervisory control problems. When a system satisfies these properties, we show that the plant models and the requirement models together are a controllable, nonblocking, and maximally permissive supervisor, i.e., no synthesis is needed to calculate a supervisor. Furthermore, these properties allow for local verification of each plant and requirement model separately.
7 citations
Additional excerpts
..., given a pair of states q1, q2 ∈ Q there exists a string s ∈ Σ∗ such that δ(q1, s) = q2 [14]....
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TL;DR: Algorithms for computing the least upper bound and the greatest lower bound of given two elements in each class which forms a lattice of automata are given.
Abstract: A class of Σautomata is considered as a partially ordered set by homomorphism relation. We show first that some classes of automata. e.g., quasiperfect automata, perfect automata and strongly cofinal automata, are lattices, and other classes, e.g., strongly connected automata, cyclic automata and cofinal automata, are not lattices. At the same time, we give algorithms for computing the least upper bound and the greatest lower bound of given two elements in each class which forms a lattice.
6 citations
TL;DR: This paper proposes a format for the requirements and a method to ensure that the crucial properties of controllability and nonblockingness directly hold, thus avoiding the most computationally expensive parts of synthesis.
5 citations
Additional excerpts
..., for any pair of states q1, q2 ∈ Q there exists a string s ∈ Σ∗ such that δ(q1, s) = q2, see [27]....
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22 Nov 2019
TL;DR: This thesis is to show that Supervisory Control Theory (SCT) as initiated by Ramadge-Wonham is very suitable for designing control systems for these large infrastructural systems, and proposes a method to transform a given set of plant and requirement models into a multilevel system needed v as input for multileVEL synthesis.
Abstract: In the coming decades, numerous locks and bridges in the Netherlands have to be renovated or replaced, as they reach their end of life cycle or have capacity problems. In the past, these infrastructural systems have been engineered, built, and maintained on a project basis, resulting in a large variety of solutions to the same engineering problem. Furthermore, more and more functionality of these infrastructural systems is being automated. Therefore, Rijkswaterstaat, part of the Dutch Ministry of Infrastructure and Water Management, is seeking methods for modularization and standardization to increase quality, increase evolvability, and decrease life-cycle costs of future infrastructural systems. The control system of these infrastructural objects is identified as one of the subsystems having a significant impact on the availability and reliability of the object. The aim of this thesis is to show that Supervisory Control Theory (SCT) as initiated by Ramadge-Wonham is very suitable for designing control systems for these large infrastructural systems. SCT is a research area focused on developing modelbased techniques for systems for which the uncontrolled behavior of the components is given and the task of the engineer is to restrict this behavior such that the system only exhibits desired behavior. SCT allows the engineer to specify what the system should do, not how the system should achieve this. Therefore, this synthesis-based engineering method may contribute to the aim of Rijkswaterstaat of increasing the quality and evolvability of control systems. In the past, applying supervisory control synthesis to industrial-size systems was often limited by the computational power, both in time and in memory. Recent developments from the literature as well as the characteristics of infrastructural systems are utilized in this thesis to synthesize supervisors for these systems. The thesis has the following four main contributions. First, monolithic synthesis of a basic lock, namely Lock III located at Tilburg, the Netherlands, showed not only that supervisory control synthesis is possible for these systems, but also that the synthesized supervisor does not impose any additional restrictions on the system besides the formulated requirements, i.e., the requirements together with the plant models form a maximally permissive, controllable, and nonblocking supervisor. Sufficient model properties are proposed to verify whether a given set of plant and requirement models can already act as a supervisor. If so, synthesis can be skipped, saving computational resources. Second, in case synthesis still needs to be performed, several divide-and-conquer synthesis techniques exist, such as modular, decentralized, and multilevel synthesis. The latter one closely resembles decomposition of a system, what is often done by engineers to cope with the size of the system. This thesis proposes a method to transform a given set of plant and requirement models into a multilevel system needed v as input for multilevel synthesis. The method uses Dependency Structure Matrices and clustering algorithms to identify such a multilevel structure. This allows for easy application of multilevel synthesis to infrastructural systems, and therefore opens the possibility to synthesize supervisors for large infrastructural systems. Third, several divide-and-conquer methods like multilevel synthesis ease the computation of local supervisors, but no longer guarantee that the complete system is nonblocking. A coordinator needs to be synthesized to resolve any conflict between supervisors. A compositional coordinator synthesis method is developed to synthesize such a coordinator efficiently. The method deploys known abstractions, like for example bisimulation semantics, synthesis equivalence, and variable unfolding, to simplify the models until a single one is obtained. Based on this abstracted model, a coordinator is synthesized with monolithic synthesis. Finally, this coordinator is refined back to the original model to resolve the conflicts between the synthesized supervisors. Fourth, several case studies with infrastructural systems have shown that the formulation of models has impact on the applicability and computational gain of the previously mentioned contributions. As modeling is an art, obtaining the ‘right’ model formulation of the system together with its requirements is nontrivial. Several modeling guidelines are proposed in this thesis, based on experience. Proposing these guidelines contributes to bridging the gap between theoretical research and industrial application of supervisory control synthesis. In 2017, 30 years of supervisory control theory of Ramadge-Wonham have been celebrated. Research presented in this thesis shows that the maturity level of this theory has reached a critical point where it is ready to be fully embraced by industry, as supervisors can be synthesized for large-scale infrastructural systems and not just for academic toy problems. A challenge lies ahead to support the industry in adapting synthesis-based engineering methods for their supervisory control systems.
4 citations
TL;DR: A method is provided to determine the structures of canonical S@?-automata (canonical C-Automata) whose endomorphism monoids are isomorphic to a given finite meet semilattice with the greatest element (Clifford monoid, respectively).
4 citations
References
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TL;DR: For a certain class of automata a necessary and sufficient condition, in terms of the group of the automaton, is given for insuring that an automaton can be represented as a direct product.
Abstract: This paper persues a discussion of certain algebraic properties of automata and their relationship to the structure (i.e., properties of the next state function) of automata. The device which is used for this study is the association of a group with each automaton. We introduce functions on automata and study the group of an automaton, a representation for the group elements and the direct product of automata. Finally, for a certain class of automata a necessary and sufficient condition, in terms of the group of the automaton, is given for insuring that an automaton can be represented as a direct product.
58 citations
TL;DR: The main result shows the group of operation-preserving transformations of a strongly connected au tomaton onto itself is isomorphic to a group of subsets of input sequences under a certain operation.
Abstract: This paper is mot iva ted by Fleck's s tudy [1] on certain classes of structurepreserving, nontrivial t ransformations of au tomata . In tha t paper the class of those transformations which preserve \"strongly-connectedness\" is completely characterized. An interesting subclass, the class of operation-preserving functions (which are essentially homomorphisms) is introduced there. Fleck showed tha t the set of all operation-preserving functions of an au tomaton A onto itself constitutes a group G(A). In [2] some of the properties of G(A) when A is strongly connected were studied. I t was shown in the lat ter paper tha t corresponding to every finite group G of regular permutat ions there is a strongly connected automaton A for which G = G(A). Since, in fact, the group G(A) determines the structure of A, it would appear tha t the structure of G(A) and of A should be related. The present paper investigates tha t relationship. The main result shows tha t the group of operation-preserving transformations of a strongly connected au tomaton onto itself is isomorphic to a group of subsets of input sequences under a certain operation.
37 citations
TL;DR: Arbitrary finite automata are decomposed into their major substructures, the primaries, and various characterizations of these transition-preserving functions on singly generated Automata are presented and are used as a basis for the reduction.
Abstract: Arbitrary finite automata are decomposed into their major substructures, the primaries. Several characterizations of homomorphisms, endomorphisms, isomorphisms, and automorphisms of arbitrary finite automata are presented via reduction to the primaries of the automata. Various characterizations of these transition-preserving functions on singly generated automata are presented and are used as a basis for the reduction. Estimates on the number of functions of each type are given.
36 citations
TL;DR: The algebraic properties of automata are investigated and the automorphism group of an automaton and a certain associated semigroup are the devices used in the study.
Abstract: The algebraic properties of automata are investigated. The automorphism group of an automaton and a certain associated semigroup are the devices used ir~ the study. Some relationships among various structures of the automaton, its group and semigroup are noted.
31 citations
TL;DR: The present paper considers somewhat of the reverse problem if G and H are groups of regular permutations on the finite sets S ~rnd 7' respec~ ~ivcly, there are nonunique strongly connected automata A and B whose automorphism groups are G andH respecLively.
Abstract: The direct product A X B of two automata A and B has been defined by Rabit~ a~d Scott [1] while the automorphism group of A X B has been investig,~tcd by Fleck [3I The latter showed tha~ the strongly connected automaton A with a transitive abelian auto-m0rphism group G(A) is the direct product of automata if and only if G(A) is isomorphic to the direct product of two groups. The present paper considers somewhat of the reverse problem. If G and H are groups of regular permutations on the finite sets S ~rnd 7' respec~ ~ivcly, there are nonunique strongly connected automata A and B whose automorphism groups are G and H respecLively. To what extent the automorphism group of A)< H is determined by G and H is studied. A sufficient conditiotl that G X H be the group of A X H is produced and it is shown that if G and H are cyclic, there are always automata A ~md B for which G X H is the automorphism group of A X B.
17 citations