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Characterizations of rough finite state automata

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Abstract

The purpose of this paper is to discuss the properties such as separatedness, connectedness and retrievability of a rough finite state automaton. The results for the relationship among the above properties are discussed and the concept of product of two rough finite state automata is introduced.

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Notes

  1. An example is given in “Appendix”.

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Acknowledgments

The authors are greatly indebted to the referees for their valuable observations and suggestions for improving the presentation of the paper.

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Correspondence to Shambhu Sharan.

Appendix A

Appendix A

Here, we construct a rough finite state machine (an RFSA with initial and final states), which accepts imprecise statements. We begin with the following.

Definition

A rough finite state machine (or RFSM) is 6-tuple \(M=(Q,R,X,\delta ,I,H)\), where \((Q,R,X,\delta )\) is an RFSA, \(I\) is a definable set in \((Q,R)\) called the initial configuration, and \(H\subseteq Q\) called the set of final states of \(M\).

The following is the concept of acceptability of a string by a RFSM.

Definition

Let \(M=(Q,R,X,\delta ,I,H)\) be a RFSM. Then the set of strings definitely accepted (resp. possibly accepted) by \(M\) is denoted by \(\underline{\beta }_{M}\) (resp. \(\overline{\beta }_{M}\)) , and defined by

$$\begin{aligned} \underline{\beta }_{M}&=\{x\in X: \underline{\delta (I,x)}\cap H\ne \phi \} \, \text { and}\\ \overline{\beta }_{M}&=\{x\in X: \overline{\delta (I,x)}\cap H\ne \phi \}. \end{aligned}$$

The following is an example of rough set concerning dengue diagnosis.

Example

During a certain season in New Delhi, doctors find that patients having blotched red skin and muscular pain articulations definitely suffer from dengue, those having no blotched red skin but muscular pain articulations or blotched red skin but no muscular pain articulations may or may not suffer from dengue, and patients having no blotched red skin and no muscular pain articulations do not suffer from dengue. Thus the disease dengue may be characterized by a rough set of symptoms whose lower approximation is {blotched red skin and muscular pain articulations} and upper approximation is {blotched red skin and muscular pain articulations, no blotched red skin but muscular pain articulations, blotched red skin but no muscular pain articulations}.

Let us write the above observations in a tabular form in Table 5.

Table 5 Decision table for detecting dengue

In the above table blotched red skin and muscular pain articulations are condition attributes, and Dengue is a decision attribute. We want to explain what causes the patient suffer from dengue (or not), i.e., to describe the set of patients \(\{P1, P2, P5\} (\{P3, P4, P6\})\) in terms of condition attributes: blotched red skin and muscular pain articulations.

The data set is inconsistent because the decisions for Patients 1, 3 and Patients 5, 6 are contradictory, therefore the problem cannot be solved exactly but only approximately. Let us observe what the data are telling us:

  • Patient 2 is certainly suffering from dengue;

  • Patient 4 is not suffering from dengue;

  • Patient 1, 2, 3, 5, and 6 can possibly may or may not be suffering from dengue.

Hence, approximation of the set of patients {P1, P2, P5} are:

lower approximation= {P2};

upper approximation={P1, P2, P3, P5, P6}.

Now, we construct an RFSM, which recognizes the disease dengue (described in above table). We denote the dengue symptoms blotched red skin or muscular pain articulations by \(1\) and no blotched red skin or no muscular pain articulations by \(0\).

Consider a RFSM \(M=(Q,R,X,\delta ,I,H)\), where \(Q=\{q_{0},q_{1},q_{2}, q_{3}, q_{4},\, q_{5},q_{6},q_{7}\}\), \(X=\{0,1\}\) \(Q/R=\{\{q_{0}\},\{q_{1},q_{2}\},\{q_{3},q_{5}\},\{q_{6}\},\{q_{4},q_{7}\}\}\), \(I=\{q_{0}\}\) and \(H=\{q_{1},q_{4},q_{6}\}\) and the rough transition map \(\delta\) is given in Table 6.

Table 6 State transition table

Now, \(\underline{\delta (q_0,0)}=\underline{\delta (q_0,01)}=\underline{\delta (q_0,10)}=\phi , \underline{\delta (q_0,1)}=\{q_3,q_5\}\), \(\underline{\delta (q_0,11)}=\{q_{6}\}\) and \(\overline{\delta (q_0,0)}=\{q_{1}, q_2\}, \overline{\delta (q_0,1)}=\{q_{3},q_{5}\}\cup \{q_{6}\}, \overline{\delta (q_0,01)}=\{q_{4},q_{7}\},\, \overline{\delta (q_0,10)}= \overline{\delta (q_0,11)}=\{q_{6}\}\).

Thus, \(\underline{\beta }_{M}=\{11\}\) and \(\overline{\beta }_{M}=\{0,1,01,10,11\}\), or that the set \(\{11\}\) is definitely accepted and the set \(\{0,1,01,10,11\}\) is possibly accepted by RFSM \(M\).

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Sharan, S., Srivastava, A.K. & Tiwari, S.P. Characterizations of rough finite state automata. Int. J. Mach. Learn. & Cyber. 8, 721–730 (2017). https://doi.org/10.1007/s13042-015-0372-3

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