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Empirical studies in the size of diagnosers and verifiers for diagnosability analysis

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Abstract

Diagnosability is an intrinsic property of the language generated by discrete event systems (DES) and the computational procedure to determine whether a language possesses or not this property is called diagnosability verification. For regular languages, diagnosability verification is carried out by building either diagnoser or verifier automata; the former is known to have worst-case exponential complexity whereas the latter has polynomial complexity in the size of state space of the automaton that generates the language. A question that has been asked for some time now is whether, in average, the state size of diagnosers is no longer exponential. This claim has been supported by the size of diagnoser automata usually obtained in practical and classroom examples, having, in some cases, state space size much smaller than that of verifiers. In an effort to clarify this matter, in this paper we carry out an experimental study on the average state size of diagnosers and verifiers by means of two experiments: (i) an exhaustive experiment, in which ten sets of automata with moderate cardinality were generated and for these sets of automata, diagnosers and verifiers were built, being the exact average state size for these specific instances calculated; (ii) an experiment with sampling, which considers 1660 sets of different instance sizes and, for each one, sample sets of 10,000 automata are randomly generated with uniform distribution and we compute sets of diagnosers and verifiers for each set of randomly generated automata, which have been used to estimate an asymptotic model for the average state sizes of diagnosers and verifiers.

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Notes

  1. A set A possesses a total order relation, denoted as \(\preccurlyeq \), if for any a,b,c ∈ A the following conditions are satisfied: (i) \(a \preccurlyeq a\) for any a (reflexivity); (ii) if \(a \preccurlyeq b\) and \(b \preccurlyeq a\), then a = b (antisymmetry); (iii) if \(a \preccurlyeq b\) and \(b \preccurlyeq c\), then \(a \preccurlyeq c\) (transitivity); (iv) either \(a \preccurlyeq b\) or \(b \preccurlyeq a\) (totality).

  2. The notation ⌊x⌋denotes the integer part of x, which is defined as\(\lfloor x \rfloor =\max \{ m \in \mathbb {N} : m\leq x\}\).

  3. The reader may find useful to follow the explanation with the help of Example 1.

  4. = (X ′,Σ,f ′,Γ′,𝜖) is an automaton isomorphic to G through ψ ′ : X → X ′ where ψ ′ : X → X ′ is the mapping X ′ = {ψ(x) : x ∈ X}, where ψ(x) is defined in Eq. 11.

  5. Base two has been chosen since the worst case computational complexity for diagnoser is \(\mathcal {O}(2^{n})\).

  6. This scale has been chosen since the worst case computational complexity of verifiers proposed by Moreira et al. (2011) and Moreira et al. (2016) is \(\mathcal {O}(n^{2})\)

  7. A box plot is a graphical display that provides a visual representation of the five-number summary of a data set: first quartile Q1, median, third quartile Q3 and the maximum value. The box of the plot contains the central 50% of the distribution, from the first (Q 1) to the third (Q 3) quartile. A line inside the box marks the median. The lines extending from the box are called whiskers, and encompass the rest of the data, except for potential outliers (data that lie more than 1.5I Q R = 1.5(Q 3 − Q 1) below the Q 1 and above Q 3), which are shown separately (Hair et al. 2007).

  8. The calculation was carried out using the open source statistic package R (The R foundation 2016).

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Acknowledgments

The authors are in debit with Prof. Stéphane Lafortune, University of Michigan, Ann Arbor, for the encouragement to submit this paper for publication and for suggesting its title. They also would like to thank the Brazilian Research Council (CNPq) for financial support.

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Authors and Affiliations

  1. Departamento de Ingeniería Eléctrica y Electrónica, Universidad Nacional de Colombia, Cra. 30 #45-03, Edif. 453, Of. 202, Bogotá D.C., Colombia

    Leonardo Bermeo Clavijo

  2. Departamento de Engenharia Elétrica, Universidade Federal do Rio de Janeiro, 21949-900, Rio de Janeiro, RJ, Brazil

    João C. Basilio

Authors
  1. Leonardo Bermeo Clavijo
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  2. João C. Basilio
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Correspondence to João C. Basilio.

Additional information

This work was carried out while L. B. Clavijo was a D.Sc. student at the Electrical Engineering Post-graduation Program of the Federal University of Rio de Janeiro.

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Clavijo, L.B., Basilio, J.C. Empirical studies in the size of diagnosers and verifiers for diagnosability analysis. Discrete Event Dyn Syst 27, 701–739 (2017). https://doi.org/10.1007/s10626-017-0260-y

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