## Extremal Solutions of Inequations over Lattices with Applications to Supervisory Control (0)

Venue: | Theoretical Computer Science |

Citations: | 15 - 7 self |

### BibTeX

@ARTICLE{Kumar_extremalsolutions,

author = {Ratnesh Kumar and Vijay K. Garg},

title = {Extremal Solutions of Inequations over Lattices with Applications to Supervisory Control},

journal = {Theoretical Computer Science},

year = {},

volume = {148},

pages = {67--92}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study the existence and computation of extremal solutions of a system of inequations defined over lattices. Using the Knaster-Tarski fixed point theorem, we obtain sufficient conditions for the existence of supremal as well as infimal solution of a given system of inequations. Iterative techniques are presented for the computation of the extremal solutions whenever they exist, and conditions under which the termination occurs in a single iteration are provided. These results are then applied for obtaining extremal solutions of various inequations that arise in computation of maximally permissive supervisors in control of logical discrete event systems (DESs) first studied by Ramadge and Wonham. Thus our work presents a unifying approach for computation of supervisors in a variety of situations. Keywords: Fixed points, lattices, inequations, discrete event systems, supervisory control, language theory. 1 Introduction Given a set X and a function f : X ! X, x 2 X is called a fixed p...

### Citations

3829 |
Introduction to Automata Theory, Languages, and Computation
- Hopcroft, Ullman
- 1979
(Show Context)
Citation Context ...ma . If there exists m 2 N such that Km+1 = Km , then Km = sup C(K). Using the arguments similar to those given in [20] it can be shown that such an m exists whenever K and L(G) are regular languages =-=[8]-=-. This new technique for iteratively computing sup C(K) is a direct consequence of the theory developed in this paper. If we require that the extremal language be controllable as well as prefix closed... |

3398 | Communicating sequential processes
- Hoare
- 1985
(Show Context)
Citation Context ...ualitative changes, called events, in the system. At the logical level of abstraction [7], the behavior of a DES can be described using the set of all possible sequences of events that it can execute =-=[33, 8]-=-. Thus the space of logical behaviors of DESs is a certain power set, and so it can be studied from a lattice theoretic perspective. The framework of supervisory control was introduced by Ramadge and ... |

1064 |
Lattices and Order
- Davey, Priestley
- 2002
(Show Context)
Citation Context ...paper by LassezNguyen -Sonenberg [15] provides a nice historical account of these fixed point theorems. Several other fixed point results have since been discovered and are reported in papers such as =-=[4, 18, 23, 7, 1, 6]-=-. The notion of optimal fixed points and their properties are discussed in [17, 14]. In this paper we study the existence and computation of extremal solutions of a system of inequations ff i (x)sg i ... |

568 | The control of discrete event systems
- Ramadge, Wonham
- 1989
(Show Context)
Citation Context ...the space of logical behaviors of DESs is a certain power set, and so it can be studied from a lattice theoretic perspective. The framework of supervisory control was introduced by Ramadge and Wonham =-=[27, 28]-=- for developing the techniques for controlling the qualitative behavior of such systems. A supervisor in this setting is event driven, and dynamically disables some of the events from occurring so tha... |

529 |
A lattice-theorethic Fixpoint Theorem and its applications
- Tarski
- 1955
(Show Context)
Citation Context ...---the set of all subsets of a given set---together with the containment partial order. One of the initial results on extremal fixed points of functions defined over lattices is due to Knaster-Tarski =-=[24]-=-. It states that every monotone function possesses an infimal as well as a supremal fixed point. Another result provides methods of computing the infimal and supremal fixed points under stronger condi... |

460 |
SupeMsory control of a class of discrete event processes
- Ramadge, Wonham
- 1987
(Show Context)
Citation Context ...the space of logical behaviors of DESs is a certain power set, and so it can be studied from a lattice theoretic perspective. The framework of supervisory control was introduced by Ramadge and Wonham =-=[21]-=- for developing the techniques for controlling the qualitative behavior of such systems. A supervisor in this setting is event driven, and dynamically disables some of the events from occurring so tha... |

225 |
Predicate Calculus and program semantics
- Dijkstra, Scholten
- 1990
(Show Context)
Citation Context ...haviors of DESs. We first introduce the notions of dual, co-dual, inverse, and converse of a function and study their properties. Some of these terminology is taken from the work of Dijkstra-Scholten =-=[5]-=- who first investigated these concepts in the setting of predicates and predicate transform1 ers. Kumar-Garg-Marcus [12] applied some of this work to supervisory control of DESs represented as program... |

210 |
Data types as lattices
- SCOTT
- 1976
(Show Context)
Citation Context ...paper by LassezNguyen -Sonenberg [15] provides a nice historical account of these fixed point theorems. Several other fixed point results have since been discovered and are reported in papers such as =-=[4, 18, 23, 7, 1, 6]-=-. The notion of optimal fixed points and their properties are discussed in [17, 14]. In this paper we study the existence and computation of extremal solutions of a system of inequations ff i (x)sg i ... |

135 |
On the supremd controllable sublanguage of a given language
- Wonham, Ramadge
- 1987
(Show Context)
Citation Context ... composition of functions, it follows that f is disjunctive. Hence it follows from Corollary 2 that the supremal controllable sublanguage of a language K ` L(G), denoted sup C(K), exists, as expected =-=[20]-=-. Since f(\Delta) = pr(\Delta)\Sigma u , it follows from Figure 2 and Proposition 4 that f \Gamma1 (\Delta) = [(\Delta)=\Sigma u ]\Sigma . Hence the following iterative scheme of Corollary 2 can be us... |

118 | Notes on control of discrete-event systems
- Wonham
- 2002
(Show Context)
Citation Context ...rmore, if the supervisor's observation is filtered through a mask function M defined over the event set \Sigma, then the desired behavior can be achieved under control only when it is also observable =-=[16, 3]-=-. A language H ` L(G) is said to be observable if 8s; t 2 pr(H); oe 2 \Sigma : [M(s) = M(t); soe 2 pr(H); toe 2 L(G)] ) [toe 2 pr(H)]: (12) It was shown in [9, Theorem 2] that this is equivalent to su... |

98 |
Think globally, act locally: Decentralized supervisory control
- Rudie, Wonham
- 1992
(Show Context)
Citation Context ...ESs, represented as languages over a certain event set, under complete as well as partial observation. The results presented here can also be applied for computation of modular [34] and decentralized =-=[19, 29]-=- supervisors, and also for computing supervisors for controlling the non-terminating behaviors [25, 14, 32]. The work presented here thus presents a unifying approach for existence and computation of ... |

77 |
Modeling and Control of Logical Discrete Event Systems
- Kumar, Garg
- 1995
(Show Context)
Citation Context ...t masks all but the last event, i.e., f M(ffl) := ffl; 8s 2 \Sigma ; oe 2 \Sigma : f M(soe) := M(s)oe: In case the supervisor is local so that it is only able to control those events that it observes =-=[10], the-=-n the desired behavior must satisfy a condition stronger than observability, called normality. A language H ` L(G) is said to be normal if M(pr(H)) " L(G) ` pr(H); (14) where M : 2 \Sigma ! 2 \Si... |

60 |
Decentralized supervisory control of discrete-event systems
- Lin, Wonham
- 1988
(Show Context)
Citation Context ...bservable if 8s; t 2 pr(H); oe 2 \Sigma : [M(s) = M(t); soe 2 pr(H); toe 2 L(G)] ) [toe 2 pr(H)]: (12) In case the supervisor is local so that it is only able to control those events that it observes =-=[11, 19, 21], the-=-n the desired behavior must satisfy a condition stronger than observability, called normality. A language H ` L(G) is said to be normal if M(pr(H)) " L(G) ` pr(H); (13) where M : 2 \Sigma ! 2 \Si... |

60 |
Supervisory control of discrete-event systems
- Wonham
- 2007
(Show Context)
Citation Context ... logical behaviors of DESs, represented as languages over a certain event set, under complete as well as partial observation. The results presented here can also be applied for computation of modular =-=[34]-=- and decentralized [19, 29] supervisors, and also for computing supervisors for controlling the non-terminating behaviors [25, 14, 32]. The work presented here thus presents a unifying approach for ex... |

59 |
Modular control and coordination of discreteevent systems
- Wong, Wonham
- 1998
(Show Context)
Citation Context ...bservable if 8s; t 2 pr(H); oe 2 \Sigma : [M(s) = M(t); soe 2 pr(H); toe 2 L(G)] ) [toe 2 pr(H)]: (12) In case the supervisor is local so that it is only able to control those events that it observes =-=[11, 19, 21], the-=-n the desired behavior must satisfy a condition stronger than observability, called normality. A language H ` L(G) is said to be normal if M(pr(H)) " L(G) ` pr(H); (13) where M : 2 \Sigma ! 2 \Si... |

41 |
Fixed Point. Theorems and Semantics: A Folk Tale
- Lassez, Nguyen, et al.
- 1982
(Show Context)
Citation Context ...as well as a supremal fixed point. Another result provides methods of computing the infimal and supremal fixed points under stronger conditions than monotonicity. The paper by LassezNguyen -Sonenberg =-=[15]-=- provides a nice historical account of these fixed point theorems. Several other fixed point results have since been discovered and are reported in papers such as [4, 18, 23, 7, 1, 6]. The notion of o... |

30 | On controllability and normality of discrete event dynamical systems
- Kumar, Garg, et al.
- 1991
(Show Context)
Citation Context ...that for any language K, [K \Sigma u " L(G) ` K] , [K \Sigma u " L(G) ` K]: This equivalence was first demonstrated in [2] under the assumption that K is prefix closed, and without this assu=-=mption in [11]. Thu-=-s by setting K = pr(H), we obtain that the two inequations of (15) are equivalent to the following: pr(H )\Sigma u " L(G) ` pr(H); pr(H) ` H; (16) Using the fact that H ` L(G), we next show that ... |

27 | Formulas for calculating supremal controllable and normal sublanguages,” Syst
- Brandt, Garg, et al.
- 1990
(Show Context)
Citation Context ... f (\Delta) = (\Delta)\Sigma u . Since f is disjunctive, it follows from Lemma 6 that for any language K, [K \Sigma u " L(G) ` K] , [K \Sigma u " L(G) ` K]: This equivalence was first demons=-=trated in [2]-=- under the assumption that K is prefix closed, and without this assumption in [11]. Thus by setting K = pr(H), we obtain that the two inequations of (15) are equivalent to the following: pr(H )\Sigma ... |

26 |
Induction rules and termination proofs
- Hitchcock, Park
- 1972
(Show Context)
Citation Context ...paper by LassezNguyen -Sonenberg [15] provides a nice historical account of these fixed point theorems. Several other fixed point results have since been discovered and are reported in papers such as =-=[4, 18, 23, 7, 1, 6]-=-. The notion of optimal fixed points and their properties are discussed in [17, 14]. In this paper we study the existence and computation of extremal solutions of a system of inequations ff i (x)sg i ... |

20 |
Some tractable supervisory control problems for discrete-event systems modeled by Büchi automata
- Ramadge
- 1989
(Show Context)
Citation Context ...The results presented here can also be applied for computation of modular [34] and decentralized [19, 29] supervisors, and also for computing supervisors for controlling the non-terminating behaviors =-=[25, 14, 32]-=-. The work presented here thus presents a unifying approach for existence and computation of supervisory control policies in a variety of settings. A Remark on Inverse Operation In this appendix we pr... |

18 |
Supervisory control of discrete event processes with partial observations
- Cieslak, Desclaux, et al.
- 1988
(Show Context)
Citation Context ...rmore, if the supervisor's observation is filtered through a mask function M defined over the event set \Sigma, then the desired behavior can be achieved under control only when it is also observable =-=[16, 3]-=-. A language H ` L(G) is said to be observable if 8s; t 2 pr(H); oe 2 \Sigma : [M(s) = M(t); soe 2 pr(H); toe 2 L(G)] ) [toe 2 pr(H)]: (12) It was shown in [9, Theorem 2] that this is equivalent to su... |

18 | Predicates and predicate transformers for supervisory control of discrete event dynamical systems
- Kumar, Garg, et al.
- 1993
(Show Context)
Citation Context ...roperties. Some of these terminology is taken from the work of Dijkstra-Scholten [5] who first investigated these concepts in the setting of predicates and predicate transform1 ers. Kumar-Garg-Marcus =-=[12]-=- applied some of this work to supervisory control of DESs represented as programs consisting of a finite number of conditional assignment statements. In this paper, we further extend these work and ap... |

14 |
Wonham: Supervision of Infinite Behavior of Discrete-Event Systems
- Thistle, M
- 1994
(Show Context)
Citation Context ...The results presented here can also be applied for computation of modular [34] and decentralized [19, 29] supervisors, and also for computing supervisors for controlling the non-terminating behaviors =-=[25, 14, 32]-=-. The work presented here thus presents a unifying approach for existence and computation of supervisory control policies in a variety of settings. A Remark on Inverse Operation In this appendix we pr... |

12 |
Algorithm design: a recursion transformation framework
- Paull
- 1988
(Show Context)
Citation Context ...f fixed points of functions defined over lattices have been studied in computer science literature for applications such as theory of recursive functions, program termination, algorithm design, etc., =-=[19]-=-. Lattices are partially ordered sets with the property that least upper bound and greatest lower bound of any pair of lattice elements is defined. A commonly encountered example of lattices is a powe... |

8 |
A Lattice-Theoretical Fixedpoint Theorem and its Applications
- Tarski
- 1955
(Show Context)
Citation Context ...---the set of all subsets of a given set---together with the containment partial order. One of the initial results on extremal fixed points of functions defined over lattices is due to Knaster-Tarski =-=[31]-=-. It states that every monotone function possesses an infimal as well as a supremal fixed point. Another result provides methods of computing the infimal and supremal fixed points under stronger condi... |

6 | On supervisory control of sequential behaviors
- Kumar, Garg, et al.
- 1992
(Show Context)
Citation Context ...The results presented here can also be applied for computation of modular [34] and decentralized [19, 29] supervisors, and also for computing supervisors for controlling the non-terminating behaviors =-=[25, 14, 32]-=-. The work presented here thus presents a unifying approach for existence and computation of supervisory control policies in a variety of settings. A Remark on Inverse Operation In this appendix we pr... |

5 |
On the infimal closed and controllable superlanguage of a given language
- Lafortune, Chen
- 1990
(Show Context)
Citation Context ...r sup PC(K) was first reported in [2] under the assumption that K is prefix closed. Our derivation shows that we do not need to impose this assumption. The formula for inf PC(K) was first reported in =-=[13]-=-. We can also study the computation of extremal languages that are relative closed (rather than prefix closed) and controllable. In this case we must consider the extremal solutions of the following t... |

4 |
Formulas for observability of discrete event dynamical systems
- Kumar
- 1993
(Show Context)
Citation Context ...ollary 2 that the supremal relative closed sublanguage as well as infimal relative closed superlanguage of a given language K ` Lm (G), denoted sup R(K) and inf R(K), respectively, exist, as expected =-=[9]-=-. Furthermore it follows from the first part of Theorem 4 that sup R(K) = K \Gamma pr \Gamma1 (Lm (G) \Gamma K) = K \Gamma (Lm (G) \Gamma K )\Sigma : On the other hand, it follows from the second part... |

4 |
Optimal Fixed Points of Logic Programs
- Lassez, Maher
- 1985
(Show Context)
Citation Context ...heorems. Several other fixed point results have since been discovered and are reported in papers such as [4, 18, 23, 7, 1, 6]. The notion of optimal fixed points and their properties are discussed in =-=[17, 14]-=-. In this paper we study the existence and computation of extremal solutions of a system of inequations ff i (x)sg i (x)g in defined over lattices, where x 2 X is the variable of inequations and n 2 N... |

3 |
The theoretical aspects of the optimal fixedpoint
- Manna, Shamir
- 1976
(Show Context)
Citation Context ...heorems. Several other fixed point results have since been discovered and are reported in papers such as [4, 18, 23, 7, 1, 6]. The notion of optimal fixed points and their properties are discussed in =-=[17, 14]-=-. In this paper we study the existence and computation of extremal solutions of a system of inequations ff i (x)sg i (x)g in defined over lattices, where x 2 X is the variable of inequations and n 2 N... |

3 |
The convergence of functions to fixed points of recursive definitions
- Manna, Shamir
- 1978
(Show Context)
Citation Context |

3 |
The infimal prefix closed and observable superlanguage of a given language
- Rudie, Wonham
- 1990
(Show Context)
Citation Context ...; soe 2 pr(H); toe 2 L(G)] ) [toe 2 pr(H)]: (12) It was shown in [9, Theorem 2] that this is equivalent to sup P [ f M \Gamma1 f M(pr(H))] " L(G) ` pr(H); (13) where the map f M , first introduce=-=d in [22], is a &qu-=-ot;modified" observation mask that masks all but the last event, i.e., f M(ffl) := ffl; 8s 2 \Sigma ; oe 2 \Sigma : f M(soe) := M(s)oe: In case the supervisor is local so that it is only able to ... |

3 |
Scanning the Issue
- Ho
- 1989
(Show Context)
Citation Context ...stems that involve quantities that are discrete and which evolve according to the occurrence of certain discrete qualitative changes, called events, in the system. At the logical level of abstraction =-=[7]-=-, the behavior of a DES can be described using the set of all possible sequences of events that it can execute [33, 8]. Thus the space of logical behaviors of DESs is a certain power set, and so it ca... |

1 |
Continuously generated fixed points
- Bracho
- 1989
(Show Context)
Citation Context |

1 |
Fixed points of increasing functions
- Hoare
- 1990
(Show Context)
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1 |
Formulas for observability of discrete event systems
- Kumar
- 1883
(Show Context)
Citation Context ...bservable if 8s; t 2 pr(H); oe 2 \Sigma : [M(s) = M(t); soe 2 pr(H); toe 2 L(G)] ) [toe 2 pr(H)]: (12) In case the supervisor is local so that it is only able to control those events that it observes =-=[11, 19, 21], the-=-n the desired behavior must satisfy a condition stronger than observability, called normality. A language H ` L(G) is said to be normal if M(pr(H)) " L(G) ` pr(H); (13) where M : 2 \Sigma ! 2 \Si... |

1 |
iv--vii
- Preface
- 1988
(Show Context)
Citation Context ...ualitative changes, called events, in the system. At the logical level of abstraction [7], the behavior of a DES can be described using the set of all possible sequences of events that it can execute =-=[33, 8]-=-. Thus the space of logical behaviors of DESs is a certain power set, and so it can be studied from a lattice theoretic perspective. The framework of supervisory control was introduced by Ramadge and ... |