. All-q-?-q-i-?-q-s and . For-all-?-?-l(g, there exist q ? Q i \ Q s and ? ? L(G, q ) such that P(? ) = P(?), that is, P(L(G, Q i ? Q s )) ? P(L(G, Q i \ Q s )), or equivalently

. P(l(g,-q-i-\-q-s,

A. , General algorithms for testing the ambiguity of finite automata and the double-tape ambiguity of finite-state transducers, International Journal of Foundations of Computer Science, vol.22, issue.04, p.40, 2011.

A. , D. Alur, and D. L. Dill, A theory of timed automata, Theoretical computer science, vol.126, issue.2, pp.183-235, 1994.

E. Athanasopoulou, N. Christoforos, and . Hadjicostis, Probabilistic approaches to fault detection in networked discrete event systems, IEEE transactions on neural networks, vol.16, issue.5, p.22, 2005.

[. Baccelli, Synchronization and linearity: an algebra for discrete event systems, vol.18, p.35, 1992.

. Badouel, Eric Badouel, Marek Bednarczyk, Andrzej Borzyszkowski, Benoît Caillaud, and Philippe Darondeau. Concurrent secrets. Discrete Event Dynamic Systems, vol.17, p.25, 2007.

[. Basile, An efficient approach for online diagnosis of discrete event systems, IEEE Transactions on Automatic Control, vol.54, issue.4, p.57, 2009.

[. Basile, On kdiagnosability of Petri nets via integer linear programming, Automatica, vol.48, issue.9, p.22, 2012.

[. Basile, Marking estimation of time Petri nets with unobservable transitions, Proceedings of 18th IEEE Conference on Emerging Technologies & Factory Automation, pp.1-7, 2013.

. References-[-basile, State estimation and fault diagnosis of labeled time petri net systems with unobservable transitions, IEEE Transactions on Automatic Control, vol.60, issue.4, p.59, 2015.

[. Béal, Unambiguous automata. Mathematics in Computer Science, vol.1, issue.4, p.39, 2008.

[. Benveniste, Diagnosis of asynchronous discrete-event systems: a net unfolding approach, IEEE Transactions on Automatic Control, vol.48, issue.5, pp.714-727, 2003.
URL : https://hal.archives-ouvertes.fr/inria-00072442

[. Bérard, Probabilistic opacity for markov decision processes, Information Processing Letters, vol.115, issue.1, p.25, 2015.

P. Bonhomme and ;. P. Bonhomme, State observer synthesis of real-time systems modeled by P-time petri nets, Proceedings of 18th IEEE Conference on Emerging Technologies & Factory Automation, vol.45, p.18, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01025860

[. Boukra, New representations for (max,+) automata with applications to performance evaluation and control of discrete event systems, International Conference on Foundations of Software Science and Computation Structures, vol.25, p.58, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00967973

. Bryans, Modelling opacity using Petri nets, Electronic Notes in Theoretical Computer Science, vol.121, p.24, 2005.

. Bryans, Opacity generalised to transition systems, International Journal of Information Security, vol.7, issue.6, p.24, 2008.

, Bisimulation relations for weighted automata, Theoretical Computer Science, vol.393, issue.1-3, p.45, 2008.

[. Cabasino, Fault detection for discrete event systems using Petri nets with unobservable transitions, 2010.

, Automatica, vol.46, issue.9, p.59, 2010.

[. Cabasino, Discrete event diagnosis using labeled Petri nets. An application to manufacturing systems, Control Engineering Practice, vol.19, p.51, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01479645

[. Cabasino, Diagnosability of discrete-event systems using labeled Petri nets, IEEE Transactions on Automation Science and Engineering, vol.11, issue.1, pp.144-153, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01479636

W. Caines, E. Peter, S. Caines, and . Wang, Classical and logic based regulator design and its complexity for partially observed automata, Proceedings of the 28th IEEE Conference on Decision and Control, p.17, 1989.

L. Cassandras, S. Cassandras, and . Lafortune, Introduction to discrete event systems, vol.29, p.30, 2009.

, Franck Cassez. The dark side of timed opacity, Advances in Information Security and Assurance, p.24, 2009.

, Franck Cassez. A note on fault diagnosis algorithms, Proceedings of 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference, pp.6941-6946, 2009.

. Cassez and . Cassez, Dynamic observers for fault diagnosis of timed systems, Proceedings of 49th IEEE Conference on Decision and Control, pp.4359-4364, 2010.

Y. Chen and G. Provan, Modeling and diagnosis of timed discrete event systems-A factory automation example, Proceedings of American Control Conference, vol.1, p.21, 1997.

[. Cohen, Max-plus algebra and system theory: where we are and where to go now, Annual reviews in control, vol.23, p.18, 1999.

[. Cong, Decentralized diagnosis by Petri nets and integer linear programming, IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol.48, issue.10, pp.1689-1700, 2017.

. References-[-cong, On-line verification of current-state opacity by petri nets and integer linear programming, Automatica, vol.94, issue.23, pp.205-213, 2018.

[. Corona, Marking estimation of Petri nets with silent transitions, IEEE Transactions on Automatic Control, vol.52, p.18, 2007.

B. Declerck, P. Declerck, and . Bonhomme, State estimation of timed labeled Petri nets with unobservable transitions, IEEE Transactions on Automation Science and Engineering, vol.11, p.18, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00879609

[. Dotoli, On-line fault detection in discrete event systems by petri nets and integer linear programming, Automatica, vol.45, issue.11, p.57, 2009.

G. Droste, P. Droste, and . Gastin, Weighted automata and weighted logics, Theoretical Computer Science, vol.380, issue.1-2, p.45, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00772642

[. Droste, Handbook of weighted automata, vol.35, p.45, 2009.

J. Fabre, On the construction of probabilistic diagnosers, IFAC Proceedings Volumes, vol.43, p.22, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01699592

S. Gaubert and J. Mairesse, Gaubert and Plus, 1997 ] Stephane Gaubert and Max Plus. Methods and applications of (max,+) linear algebra, Annual symposium on theoretical aspects of computer science, vol.44, p.35, 1997.

S. Gaubert, Performance evaluation of (max,+) automata, IEEE transactions on automatic Control, vol.40, p.45, 1995.

[. Ghazel, State observer for DES under partial observation with time Petri nets, Discrete Event Dynamic Systems, vol.19, issue.2, p.20, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00801148

A. Giua and C. Seatzu, Observability of place/transition nets, IEEE Transactions on Automatic Control, vol.47, p.20, 2002.

A. Giua and C. Seatzu, A survey on state estimation using Petri nets, Proceedings of 2014 European Control Conference, p.18, 2014.

[. Giua, State estimation of ?-free labeled Petri nets with contact-free nondeterministic transitions, Discrete Event Dynamic Systems, vol.15, p.18, 2005.

A. Giua, State estimation and fault detection using Petri nets, Proceedings of 32nd International Conference on Application and Theory of Petri Nets, p.17, 2011.

M. Gondran, M. Gondran, and . Minoux, Graphs, dioids and semirings: new models and algorithms, vol.41, p.35, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01304880

. Hadj-alouane, On the verification of intransitive noninterference in mulitlevel security, IEEE Transactions on Systems, Man, and Cybernetics, vol.35, issue.5, p.25, 2005.

. Hadj-alouane, Characterizing intransitive noninterference for 3-domain security policies with observability, IEEE transactions on automatic control, vol.50, issue.6, p.25, 2005.

. Hardouin, Observer design for (max,+) linear systems, IEEE Transactions on Automatic Control, vol.55, p.18, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00857841

. Hardouin, Observerbased controllers for max-plus linear systems, IEEE Transactions on Automatic Control, vol.62, pp.2153-2165, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01519026

. Hardouin, Control and state estimation for max-plus linear systems, Foundations and Trends® in Systems and Control, vol.6, issue.1, pp.1-116, 2018.

[. Heidergott, Max Plus at work: modeling and analysis of synchronized systems: a course on Max-Plus algebra and its applications, vol.48, p.35, 2014.

. Hernandez-flores, Diagnosability analysis of partially observable deadlock-free Petri nets, 2011.

, Proceedings of 3rd International Workshop on Dependable Control of Discrete Systems, pp.174-179, 2011.

. References-[-jacob, Overview of discrete event systems opacity: Models, validation, and quantification, Annual Reviews in Control, vol.41, p.25, 2016.

K. Jiang, R. Jiang, ;. Kumar, and . Jiang, Shengbing Jiang, Zhongdong Huang, Vigyan Chandra, and Ratnesh Kumar. A polynomial algorithm for testing diagnosability of discrete-event systems, IEEE Transactions on Automation Science and Engineering, vol.3, issue.1, p.57, 2001.

. Jiroveanu, G. Boel, . Jiroveanu, K. René, and . Boel, The diagnosability of Petri net models using minimal explanations, IEEE Transactions on Automatic Control, vol.55, issue.7, p.23, 2010.

B. Donald and . Johnson, Finding all the elementary circuits of a directed graph, SIAM Journal on Computing, vol.4, issue.1, p.80, 1975.

, Christoforos Keroglou and Christoforos N Hadjicostis. Initial state opacity in stochastic DES, Proc. ETFA, p.24, 2013.

C. Keroglou and C. N. Hadjicostis, Detectability in stochastic discrete event systems, Systems & Control Letters, vol.84, p.20, 2015.

C. Keroglou and C. N. Hadjicostis, Verification of detectability in probabilistic finite automata, Automatica, vol.86, p.20, 2017.

K. , L. Kirsten, and S. Lombardy, Deciding unambiguity and sequentiality of polynomially ambiguous min-plus automata, LIPIcs-Leibniz International Proceedings in Informatics, vol.3, p.39, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00360205

[. Klimann, Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton, Theoretical Computer Science, vol.327, issue.3, p.40, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00174171

[. Komenda, Max-plus algebra and discrete event systems, Proceedings of 20th IFAC World Congress, vol.50, p.18, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02481790

[. Komenda, Maxplus algebra in the history of discrete event systems, Annual Reviews in Control, vol.45, p.18, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01790864

[. Lahaye, Compositions of (max,+) automata. Discrete Event Dynamic Systems, vol.25, pp.323-344, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00846353

[. Lai, State estimation of max-plus automata with unobservable events, Automatica, vol.105, p.68, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02481619

[. Lai, A two-step approach for fault diagnosis of max-plus automata, proceedings 6th International Conference on Control, Decision and Information Technologies (CoDIT), p.68, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02481667

[. Lai, Verification of detectability for unambiguous weighted automata, IEEE Transactions on Automatic Control, 2019.

[. Lai, Initial-state detectability and initial-state opacity of unambiguous weighted automata. Automatica (submitted), 2019.

D. Lefebvre, C. Lefebvre, and . Delherm, Diagnosis of DES with Petri net models, IEEE Transactions on Automation Science and Engineering, vol.4, issue.1, pp.114-118, 2007.

[. Li, H. Li, and C. Hadjicostis, Minimum initial marking estimation in labeled Petri nets, IEEE Transactions on Automatic Control, vol.58, issue.1, pp.198-203, 2013.

F. Lin, Diagnosability of discrete event systems and its applications, Discrete Event Dynamic Systems, vol.4, p.22, 1994.

[. Lunze, J. Schröder, J. Lunze, and . Schröder, State observation and diagnosis of discrete-event systems described by stochastic automata, Discrete Event Dynamic Systems, vol.11, issue.4, p.22, 2001.

[. Ma, Basis marking representation of Petri net reachability spaces and its application to the reachability problem, IEEE Transactions on Automatic Control, vol.62, issue.3, p.20, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01786021

, Complexity of detectability, opacity and a-diagnosability for modular discrete event systems, Automatica, vol.101, p.20, 2019.

, Deciding detectability for labeled petri nets, Tomá? Masopust and Xiang Yin, p.20, 2019.

. References-[-masopust, Tomá? Masopust. Complexity of deciding detectability in discrete event systems, Automatica, vol.93, p.20, 2018.

L. Mazaré, Using unification for opacity properties, Proceedings of the 4th IFIP WG1, vol.7, pp.165-176, 2004.

W. M. Ozveren, A. S. Ozveren, and . Willsky, Observability of discrete event dynamic systems, IEEE Transactions on Automatic Control, vol.35, p.17, 1990.

P. J. Ramadge, Observability of discrete event systems, Proceedings of 25th IEEE Conference on Decision and Control, p.17, 1986.

. Ramírez-treviño, Observability of discrete event systems modeled by interpreted Petri nets, IEEE Transactions on Robotics and Automation, vol.19, issue.4, p.20, 2003.

. Ramírez-treviño, Online fault diagnosis of discrete event systems. a Petri net-based approach, IEEE Transactions on Automation Science and Engineering, vol.4, issue.1, pp.31-39, 2007.

Y. Ru, N. Christoforos, and . Hadjicostis, Sensor selection for structural observability in discrete event systems modeled by Petri nets, IEEE Transactions on Automatic Control, vol.55, issue.8, pp.1751-1764, 2010.

, Anooshiravan Saboori and Christoforos N Hadjicostis. Notions of security and opacity in discrete event systems. In Decision and Control, 46th IEEE Conference on, pp.5056-5061, 2007.

, Anooshiravan Saboori and Christoforos N Hadjicostis. Verification of initial-state opacity in security applications of des, 9th International Workshop on Discrete Event Systems, p.25, 2008.

, Anooshiravan Saboori and Christoforos N Hadjicostis. Verification of k-step opacity and analysis of its complexity, IEEE Transactions on Automation Science and Engineering, vol.8, issue.3, p.25, 2011.

, Anooshiravan Saboori and Christoforos N Hadjicostis. Current-state opacity formulations in probabilistic finite automata, IEEE Transactions on automatic control, vol.59, issue.1, pp.120-133, 2013.

, Anooshiravan Saboori and Christoforos N Hadjicostis. Verification of initial-state opacity in security applications of discrete event systems, Information Sciences, vol.246, issue.23, p.97, 2013.

[. Sakarovitch, Elements of automata theory, vol.6, p.35, 2009.

[. Sampath, Diagnosability of discrete-event systems, IEEE Transactions on Automatic Control, vol.40, p.59, 1995.

[. Sasi, L. Sasi, and F. Lin, Detectability of networked discrete event systems, Discrete Event Dynamic Systems, vol.28, issue.3, p.20, 2018.

M. P. Schützenberger, On the definition of a family of automata, Information and control, vol.4, issue.2-3, p.36, 1961.

L. Shu, F. Shu, and . Lin, Generalized detectability for discrete event systems, Systems & control letters, vol.60, issue.5, p.92, 2011.

L. Shu, Shaolong Shu and Feng Lin. I-detectability of discrete-event systems, IEEE Transactions on Automation Science and Engineering, vol.10, issue.1, p.97, 1920.

[. Shu, Detectability of discrete event systems, IEEE Transactions on Automatic Control, vol.52, p.26, 2007.

[. Shu, State estimation and detectability of probabilistic discrete event systems, Automatica, vol.44, pp.3054-3060, 2008.

, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol.10, issue.05, p.25, 2002.

K. Takai, R. Takai, and . Kumar, Verification and synthesis for secrecy in discrete-event systems, American Control Conference, p.25, 2009.

[. Thistle, G. Wonham-;-john, . Thistle, and . Murray-wonham, Control of infinite behavior of finite automata, SIAM Journal on Control and Optimization, vol.32, issue.4, p.46, 1994.

[. Thorsley, Diagnosability of stochastic discrete-event systems under unreliable observations, Proceedings of American Control Conference, p.22, 2008.

[. Tong, Verification of state-based opacity using Petri nets, IEEE Transactions on Automatic Control, vol.62, issue.6, p.24, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01786028

S. Tripakis, Fault diagnosis for timed automata, International symposium on formal techniques in real-time and fault-tolerant systems, vol.21, p.58, 2002.

[. Wang, On state estimation of timed choice-free Petri nets, Proceedings of 18th IFAC World Congress, vol.44, p.18, 2011.

[. Wang, The opacity of real-time automata, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol.37, issue.11, p.107, 2018.

[. Weber, A. Seidl, H. Weber, and . Seidl, On the degree of ambiguity of finite automata, Theoretical Computer Science, vol.88, issue.2, p.40, 1991.

[. Wen, A polynomial algorithm for checking diagnosability of Petri nets, Proceedings of IEEE International Conference on Systems, Man and Cybernetics, vol.3, p.17, 1976.

Y. Wu, N. Christoforos, and . Hadjicostis, Algebraic approaches for fault identification in discrete-event systems, IEEE Transactions on Automatic Control, vol.50, issue.12, p.22, 2005.

L. Wu, S. Wu, and . Lafortune, Comparative analysis of related notions of opacity in centralized and coordinated architectures, Discrete Event Dynamic Systems, vol.23, issue.3, p.24, 2013.

L. Yin, Codiagnosability and coobservability under dynamic observations: Transformation and verification, Automatica, vol.61, pp.241-252, 2015.

X. Lafortune, S. Yin, and . Lafortune, A new approach for the verification of infinite-step and k-step opacity using two-way observers, Automatica, vol.80, p.25, 2017.

L. Yin, X. Yin, and S. Lafortune, On the decidability and complexity of diagnosability for labeled Petri nets, IEEE Transactions on Automatic Control, vol.62, issue.11, pp.5931-5938, 2017.

L. Yin, Xiang Yin and Stéphane Lafortune. Verification complexity of a class of observational properties for modular discrete events systems, Automatica, vol.83, 1920.

[. Yin, Infinite-step opacity and k-step opacity of stochastic discrete-event systems, Automatica, vol.99, p.25, 2019.

X. Yin, Initial-state detectability of stochastic discrete-event systems with probabilistic sensor failures, Automatica, vol.80, p.20, 2017.

, Xiang Yin. Estimation and verification of partially-observed discrete-event systems, vol.20, p.25, 2019.

[. Yoo, L. Yoo, and S. Lafortune, Polynomial-time verification of diagnosability of partially observed discrete-event systems, IEEE Transactions on automatic control, vol.47, issue.9, p.57, 2002.

[. Zad, Fault diagnosis in discreteevent systems: Framework and model reduction, IEEE Transactions on Automatic Control, vol.48, p.61, 2003.

[. Zad, Fault diagnosis in discrete-event systems: Incorporating timing information, IEEE Transactions on Automatic Control, vol.50, issue.7, p.22, 2005.

L. Zaytoon, Overview of fault diagnosis methods for discrete event systems, Annual Reviews in Control, vol.37, issue.2, pp.308-320, 2013.

K. Zhang, The problem of determining the weak (periodic) detectability of discrete event systems is pspace-complete, Automatica, vol.81, p.20, 2017.

[. Zhao, Detectability measure for state estimation of discrete event systems, IEEE Transactions on Automatic Control, vol.64, issue.1, p.20, 2019.