Error tolerance for the recognition of faulty strings in a regulated grammar using fuzzy sets

Abstract

To overcome the limitations of context-free and context-sensitive grammars, regulated grammars have been proposed. In this paper, an algorithm is proposed for the recognition of faulty strings in regulated grammar. Furthermore, depending on the errors and certainty, it is decided whether the string belongs to the language or not based on string membership value. The time complexity of the proposed algorithm is O(|G ² _R |·|w|), where |G_R| represents the number of production rules and |w| is the length of the input string, w. The reader is provided with numerical examples by applying the algorithm to regularly controlled and matrix grammar. Finally, the proposed algorithm is applied in the Hindi language for the recognition of faulty strings in regulated grammar as a real-life application.

Appendix 1: Numerical example

In this appendix, the proposed algorithm has been applied to a matrix grammar.

Example 4.2

Consider G_M = (G, r_m), where G = {{S, C, D}, {c, d}, S, {m₀, m₁, m₂, m₃, m₄}} be the matrix grammar of example 2.2 where

$$ \begin{aligned} m_{0} & = (p_{0} :S \to CD), \\ m_{1} & = (p_{1} :C \to cC,p_{2} :D \to cD), \\ m_{2} & = (p_{3} :C \to dC,p_{4} :D \to dD), \\ m_{3} & = (p_{5} :C \to c,p_{6} :D \to c), \\ m_{4} & = (p_{7} :C \to d,p_{8} :D \to d), \\ \end{aligned} $$

The control set is r_m = m₀(m₁, m₂) * (m₃, m₄).

Equivalent CNF for the grammar G_M:

$$ \begin{aligned} m_{0}^{'} & = (p_{0} :S \to CD), \\ m_{1}^{'} & = (p_{1}^{1} :C \to A_{1} C,p_{1}^{2} :A_{1} \to c,p_{2}^{1} :D \to A_{2} D,p_{2}^{2} :A_{2} \to c), \\ m_{2}^{'} & = (p_{3}^{1} :C \to A_{3} C,p_{3}^{2} :A_{3} \to d,p_{4}^{1} :D \to A_{4} D,p_{4}^{2} :A_{4} \to d), \\ m_{3}^{'} & = (p_{5} :C \to c,p_{6} :D \to c), \\ m_{4}^{'} & = (p_{7} :C \to d,p_{8} :D \to d) \\ \end{aligned} $$

The control set is r_m′ is m₀′(m₁′, m₂′) * (m₃′, m₄′).

Using fuzzy replace operator:

$$ \begin{array}{*{20}l} {pr_{9} :A_{1} \to \_} \hfill & {pr_{10} :A_{2} \to \_} \hfill \\ {pr_{11} :A_{3} \to \_} \hfill & {pr_{12} :A_{4} \to \_} \hfill \\ {pr_{13} :C \to \_} \hfill & {pr_{14} :D \to \_} \hfill \\ {pr_{15} :C \to \_} \hfill & {pr_{16} :D \to \_} \hfill \\ \end{array} $$

Using fuzzy add operator:

$$ \begin{array}{*{20}l} {pr_{17} :A_{1} \to \_c|c\_} \hfill & {pr_{18} :A_{2} \to \_c|c\_} \hfill \\ {pr_{19} :A_{3} \to \_d|d\_} \hfill & {pr_{20} :A_{4} \to \_d|d\_} \hfill \\ {pr_{21} :C \to \_c|c\_} \hfill & {pr_{22} :D \to \_c|c} \hfill \\ {pr_{23} :C \to \_d|d\_} \hfill & {pr_{24} :D \to \_d|d\_} \hfill \\ \end{array} $$

Using fuzzy remove operator:

$$ \begin{array}{*{20}l} {pr_{25} :A_{1} \to \lambda } \hfill & {pr_{26} :A_{2} \to \lambda } \hfill \\ {pr_{27} :A_{3} \to \lambda } \hfill & {pr_{28} :A_{4} \to \lambda } \hfill \\ {pr_{29} :C \to \lambda } \hfill & {pr_{30} :D \to \lambda } \hfill \\ {pr_{31} :C \to \lambda } \hfill & {pr_{32} :D \to \lambda } \hfill \\ \end{array} $$

On considering string w = ddcd ∉ L. Table A1 dep

Table A1 Various derivation for the string $w = cdcdcccd$.

icts few ways for deriving the string w = ddcd.

Table A2 represents the confidence level for the string wusing E-set.

Table A2 Confidence level for the string w using E-set.

Table A3 and A4 represent the confidence level for the string wusing T_f-set and final interpretations with fuzzy confidence for the string w = cdcdcccd respectively.

$$ \mu (w) = \hbox{max} (0,\;0,\;0.75,\;0,\;0,\;0,\;0.5,\;0,\;0,\;0) = 0.75 $$

By choosing a certain level of confidence λ_c, we say string is accepted if μ(w) ≥ λ_c.

Table A3 Confidence level for the string w using T_f-set.

Table A4 Final interpretations with fuzzy confidence for the string w = cdcdcccd.