Neste artigo, apresentamos duas novas formas normais para as gramáticas lineares fuzzy. Além disso, introduzimos duas novas classes de máquinas fuzzy. A primeira é a classe dos autômatos lineares fuzzy (FLA). Mostraramos que este tipo de autômato é equivalente as gramáticas lineares fuzzy, i.e, provamos que a classe das linguagens lineares fuzzy é reconhecida pelos autômatos lineares fuzzy. Por fim, introduzimos a classe dos autômatos não-determinísticos de 2-fitas fuzzy (FNTA) e mostramos a equivalência entre estes e os autômatos lineares fuzzy.

Palavras-chave:
Gramáticas Fuzzy; Gramáticas Lineares Fuzzy; Linguagens Fuzzy; Autômatos Fuzzy

Autoria

VALDIGLEIS S. COSTA ^**Corresponding author: Valdigleis S. Costa - E-mail: valdigleis@ppgsc.ufrn.br.

Doutorando em Ciência da Computação, Programa de Pós-graduação em Sistemas e Computação, DIMAp, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, CEP 59078-970, Caixa postal 1524, Natal/RN - Brasil.Universidade Federal do Rio Grande do NorteBrazilNatal, RN, BrazilDoutorando em Ciência da Computação, Programa de Pós-graduação em Sistemas e Computação, DIMAp, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, CEP 59078-970, Caixa postal 1524, Natal/RN - Brasil.

BENJAMÍN BEDREGAL

Departamento de Informática e Matemática Aplicada, CCET, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, CEP 59078-970, Caixa postal 1524, Natal/RN - Brasil. E-mail: bedregal@dimap.ufrn.brUniversidade Federal do Rio Grande do NorteBrazilNatal, RN, BrazilDepartamento de Informática e Matemática Aplicada, CCET, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, CEP 59078-970, Caixa postal 1524, Natal/RN - Brasil. E-mail: bedregal@dimap.ufrn.br

*Corresponding author: Valdigleis S. Costa - E-mail: valdigleis@ppgsc.ufrn.br.

SCIMAGO INSTITUTIONS RANKINGS

Fórmulas

Fórmulas (90)

(2.1)

(2.2)

(2.3)

(4.1)

(4.2)

(4.3)

(4.4)

R_{l} (w) = \{\begin{cases} x_{1}, & i f |w| > 0 \\ λ, & e l s e \end{cases} M_{l} (w) = \{\begin{cases} x_{2} \dots x_{n - 1} x_{n}, & i f |w| > 1 \\ λ, & e l s e \end{cases} R_{r} (w) = \{\begin{cases} x_{n}, & i f |w| > 0 M_{r} (w) = \{\begin{cases} x_{1} x_{2} \dots x_{n - 1}, & i f |w| > 1 \\ λ, & e l s e \end{cases} \\ λ, & e l s e \end{cases}

A = {(x, μ_{A} (x)) ∣ x \in U}

\otimes_{i = 1}^{n} x_{i} = \otimes (\otimes_{i = 1}^{n - 1} x_{i}, x_{n}) = \otimes (x_{1}, x_{2}, \dots, x_{n})

\oplus_{i = 1}^{n} x_{i} = \oplus (\oplus_{i = 1}^{n - 1} x_{i}, x_{n}) = \oplus (x_{1}, x_{2}, \dots, x_{n})

Γ_{w, w'} = {σ | (\exists x, y, α, β \in Σ *), w = x α y, w' = x β y a n d α \overset{σ}{\to} β \in \hat{P}}

(2.1)

d_{G} (w, w') = \lor Γ_{w, w'}

(2.2)

{\hat{d}}_{G} (w, w') = \{\begin{matrix} 1, & i f w = w' \\ \underset{w'' \in (V \cup Σ)^{+}}{\oplus} (\otimes ({\hat{d}}_{G} (w, w'') d_{G} (w'', w'))), & e l s e \end{matrix}

\hat{d} (X, a) = m a x (m i n (\hat{d} (X, A), d (A, a)), m i n (\hat{d} (X, B), d (B, a))) = m a x (m i n (m a x (m i n (1, 0.8), m i n (\hat{d} (X, B), 0.4)), 0.5), m i n (\hat{d} (X, B), 0.2)) = m a x (m i n (m a x (0.8, m i n (0.8, 0.4)), 0.5), m i n (0.8, 0.2)) = m a x (m i n (0.8, 0.5), 0.2) = 0.5

(2.3)

μ_{G} (w) = \underset{_{α \in V}}{\oplus} (\otimes (\hat{S} (α), \hat{d} (α, w)))

α \overset{σ}{\to} β \in \hat{P}

α \overset{σ}{\to} β \in \hat{P}

α \overset{σ}{\to} β \in \hat{P}

\hat{P}

\hat{P}

A \overset{σ}{\to} u B v

\hat{P}

A \overset{σ}{\to} u B v

\hat{P}

A \overset{1}{\to} u C

C \overset{σ}{\to} B v

\hat{P}

A \overset{σ}{\to} u B v

\hat{P}

{\hat{d}}_{G} (A, u B v) = \oplus (\otimes ({\hat{d}}_{G} (A, A), d_{G} (A, u B v)) = \oplus (\otimes (1, σ) = σ = d_{G} (A, u B v)

{\hat{d}}_{G'} (A, u B v) = \oplus (\otimes ({\hat{d}}_{G'} (A, u C), d_{G'} (u C, u B v))) = \oplus (\otimes (\oplus (\otimes ({\hat{d}}_{G'} (A, A), d_{G'} (A, u C))), d_{G'} (u C, u B v))) = \oplus (\otimes (\oplus (\otimes (1, 1)), σ)) = \oplus (\otimes (1, σ)) = σ = d_{G} (A, u B v) = {\hat{d}}_{G} (A, u B v) .

\hat{P}

\hat{P}

V_{L}^{G} \cap V_{R}^{G}

A \overset{σ'}{\to} B u

\hat{P}

A \overset{1}{\to} C

C \overset{σ}{\to} B v

\hat{P}

A \overset{σ}{\to} B v

\hat{P}

{\hat{d}}_{G} (A, B v) = \oplus (\otimes ({\hat{d}}_{G} (A, A), d_{G} (A, B v)) = \oplus (\otimes (1, σ)) = σ .

{\hat{d}}_{G'} (A, B v) = \oplus (\otimes ({\hat{d}}_{G'} (A, C), d_{G'} (C, B v))) = \oplus (\otimes (\oplus (\otimes ({\hat{d}}_{G'} (A, A), d_{G'} (A, C))), d (C, B v))) = \oplus (\otimes (\oplus (\otimes (1, 1)), σ)) = \oplus (\otimes (1, σ)) = σ = {\hat{d}}_{G} (A, B v) .

\hat{P}

\hat{P}

A \overset{σ}{\to} y a

\hat{P}

A \overset{σ}{\to} y a

\hat{P}

A \overset{1}{\to} C a

C \overset{σ}{\to} y

A \overset{σ}{\to} y a

\hat{P}

A \overset{σ}{\to} a y

\hat{P}

A \overset{σ}{\to} a y

\hat{P}

A \overset{1}{\to} C a

C \overset{σ}{\to} y

A \overset{σ}{\to} a y

\hat{P}

{\hat{d}}_{G} (A, y a) = \oplus (\otimes ({\hat{d}}_{G} (A, A), d_{G} (A, y a))) = \oplus (\otimes (1, σ)) = σ .

{\hat{d}}_{G'} (A, a y) = \oplus (\otimes ({\hat{d}}_{G'} (A, C a), d_{G'} (C a, y a))) = \oplus (\otimes (1, σ)) = σ = {\hat{d}}_{G} (A, y a) .

λ_{s u c}^{(q, w)}

L_{s u c}^{(q, w)}

R_{s u c}^{(q, w)}

λ_{s u c}^{(q, w)} \cup L_{s u c}^{(q, w)} \cup R_{s u c}^{(q, w)}

(4.1)

D_{M} (q, w, q', w') = \{\begin{cases} (q, q')_{λ}, & i f (q', w') \in λ_{s u c}^{(q, w)} \\ (q, q')_{R_{l} (w)}, & i f q \in Q_{L} and (q', w') \in L_{s u c}^{(q, w)} \\ (q, q')_{R_{r} (w)}, & i f q \in Q_{R} and (q', w') \in R_{s u c}^{(q, w)} \\ 0, & any other case. \end{cases}

D_{M} (q_{0}, a b a a, p_{0}, b a a) = (q_{0}, p_{0})_{R_{l} (a b a a)} = (q_{0}, p_{0})_{a} = 0.9

{\bar{D}}_{M} (q, w, q'', w'') = \{\begin{cases} D (q, w, q'', w'') & i f (q'', w'') \in S U C^{(q, w)} \\ \oplus_{(q', w') \in S U C^{(q, w)}} (\otimes (D_{M} (q, w, q', w'), {\bar{D}}_{M} (q', w', q'', w''))) & i f S U C^{(q, w)} \neq 0 \\ 0, & any other case. \end{cases}

{\bar{D}}_{M} (q_{0}, a b c, p_{0}, λ) = m a x (m i n (D_{M} (q_{0}, a b c, p_{0}, b c), {\bar{D}}_{M} (p_{0}, b c, p_{0}, λ))) = m a x (m i n (0.9, m a x (m i n (0.7, 1)))) = m a x (m i n (0.9, 0.7)) = 0.7

(4.2)

d e g_{M} (w) = \underset{q \in (Q_{L} \cup Q_{R}), q' \in F}{\oplus} (\otimes (\hat{I} (q), {\bar{D}}_{M} (q, w, q', λ)))

\hat{P} = {q' \overset{σ}{\to} a q'' ∣ (q', q'')_{a} = σ, q' \in Q_{L}, q'' \in (Q_{L} \cup Q_{R}), a \in Σ \cup {λ}} \cup {q' \overset{σ'}{\to} q'' a ∣ (q', q'')_{a} = σ', q' \in Q_{R}, q'' \in (Q_{L} \cup Q_{R}), a \in Σ \cup {λ}} \cup {q' \overset{1}{\to} λ ∣ q' \in F} .

(q_{0}, q_{1})_{a_{1}} = σ_{1}, (q_{1}, q_{2})_{a_{2}} = σ_{2}, \dots, (q_{n - 2}, q_{n - 1})_{a_{n - 1}} = σ_{n - 1}, (q_{n - 1}, q_{n})_{a_{n}} = σ_{n}

(q_{0}, q_{1})_{a_{1}} = σ_{1}, (q_{1}, q_{2})_{a_{2}} = σ_{2}, \dots, (q_{n - 2}, q_{n - 1})_{a_{n - 1}} = σ_{n - 1}, (q_{n - 1}, q_{n})_{a_{n}} = σ_{n}

{\hat{d}}_{G} (q_{0}, w) = \oplus (\otimes ({\hat{d}}_{G} (q_{0}, w'), d_{G} (w', w))) = \oplus (\otimes ({\hat{d}}_{G} (q_{0}, y_{1} q_{n} y_{2}), d_{G} (y_{1} q_{n} y_{2}, y_{1} y_{2}))) = \oplus (\otimes ({\hat{d}}_{G} (q_{0}, y_{1} q_{n} y_{2}), 1)) = \oplus ({\hat{d}}_{G} (q_{0}, w'))

μ_{G} (w) = \oplus (\otimes (\hat{I} (q_{0}), {\bar{D}}_{M} (q_{0}, w', q_{n}, λ))) .

(A, B)_{a} = σ, if A \overset{σ}{\to} a B or A \overset{σ}{\to} B a \in \hat{P} (A, C)_{a} = σ, if A \overset{σ}{\to} a \in \hat{P} .

(A_{1}, A_{2})_{a_{1}} = σ_{1}, (A_{2}, A_{3})_{a_{2}} = σ_{2}, \dots, (A_{n - 1}, A_{n})_{a_{n - 1}} = σ_{n - 1}, (A_{n}, C)_{a_{n}} = σ_{n}

d e g_{M} (w) = \oplus (\otimes (\hat{S} (A), {\bar{D}}_{M} (A, w, C, λ))) .

d e g_{M} (w) = \oplus (\otimes (\hat{S} (A), {\hat{d}}_{G} (A, w)))

ϕ' (q, a, q') = \{\begin{matrix} ϕ (q, a, q'), & if q, q' \in (Q_{L} \cup Q_{R}) \\ 1, & if q \in F, q' = q_{f} \\ 0 & else \end{matrix}

d e g_{M} (w) = σ \Leftrightarrow \oplus (\otimes (\hat{I} (q), {\bar{D}}_{M} (q, w, q', λ))) = σ .

{\bar{D}}_{M'} (q, w, q_{f}, λ) = \oplus (\otimes ({\bar{D}}_{M} (q, w, q', λ), 1)) = \oplus ({\bar{D}}_{M} (q, w, q', λ)) = {\bar{D}}_{M} (q, w, q', λ) .

d e g_{M'} (w) = \oplus (\otimes (\hat{I} (q), {\bar{D}}_{M'} (q, w, q_{f}, λ))) = \oplus (\otimes (\hat{I} (q), {\bar{D}}_{M} (q, w, q', λ))) = d e g_{M} (w) .

d e g_{M} (w) = \underset{q \in (Q_{L} \cup Q_{R})}{\oplus} (\otimes (\hat{I} (q), {\bar{D}}_{M} (q, w, q_{f}, λ)))

i. λ_{d e s}^{(q', w_{1}, w_{2})} = {(q', w_{1}, w_{2}) ∣ (q, q')^{λ} \in υ} . ii. L_{d e s}^{(q', w_{1}, w_{2})} = {(q', w'_{1}, w_{2}) ∣ w'_{1} = M_{l} (w_{1}) and (q, q')^{R_{l} (w_{1})} \in υ} . iii. R_{d e s}^{(q', w_{1}, w_{2})} = {(q', w_{1}, w'_{2}) ∣ w'_{2} = M_{l} (w_{2}) and (q, q')^{R_{l} (w_{2})} \in υ} . iv. Ω (q', w_{1}, w_{2}) = λ_{d e s}^{(q', w_{1}, w_{2})} \cup L_{d e s}^{(q', w_{1}, w_{2})} \cup R_{d e s}^{(q', w_{1}, w_{2})} .

(4.3)

f (q, w_{1}, w_{2}, q', w'_{1}, w'_{2}) = \{\begin{matrix} (q, q')^{λ}, & if (q', w'_{1}, w'_{2}) \in λ_{d e s}^{(q, w_{1}, w_{2})} \\ (q, q')^{R_{l} (w_{1})}, & if T (q) = 1 and (q', w'_{1}, w'_{2}) \in L_{d e s}^{(q', w_{1}, w_{2})} \\ (q, q')^{R_{l} (w_{2})}, & if T (q) = 2 and (q', w'_{1}, w'_{2}) \in R_{d e s}^{(q', w_{1}, w_{2})} \\ 0, & any other case \end{matrix}

\hat{f} (q, x, y, q', x', y') = \{\begin{cases} f (q, x, y, q', x', y'), & if (q, w_{1}^{'}, w_{2}^{'}) \in Ω^{(q, w_{1}, w_{2})} \\ \oplus_{(q_{i}, x_{1}, y_{1}) \in Ω^{(q, w_{1}, w_{2})}} & (\otimes (f (q, x, y, q_{i}, x_{1}, y_{2}), \hat{f} (q_{i}, x_{1}, y_{2}, q', x', y'))), if Ω^{(q, w_{1}, w_{2})} \neq 0 \\ 0, & any other case \end{cases}

\hat{f} (q_{0}, a c, b, q_{a c c}, λ, λ) = m a x (m i n (f (q_{0}, a c, b, q_{1}, b, c), \hat{f} (q_{1}, c, b, q_{a c c}, λ, λ))) = m a x (m i n (0.8, m a x (\otimes (1, 0.7)))) = m a x (m i n (0.8, 0.7)) = 0.7

(4.4)

A c c_{m} (w) = \underset{_{(x, y) \in \nabla_{w}, q \in Q}}{\oplus} (\otimes (O (q), \hat{f} (q, x, y^{R}, q_{a c c}, λ, λ)))

A c c_{m} (w) = σ \Leftrightarrow \underset{_{(x, y) \in \nabla_{w}, q \in Q}}{\oplus} (\otimes (O (q), \hat{f} (q, x, y^{R}, q_{a c c}, λ, λ))) = σ

d e g_{M} (w) = \underset{_{(x, y) \in \nabla_{w}, q \in Q}}{\oplus} (\otimes (O (q), {\bar{D}}_{M} (q, x y, q_{a c c}, λ))) = \underset{(x, y) \in \nabla_{w}, q \in Q}{\oplus} (\otimes (O (q), \hat{f} (q', x, y^{R}, q_{a c c}, λ, λ)))) = A c c_{m} (w) .

T (q) = \{\begin{cases} 1, & if q \in Q_{L} \\ 2, & else \end{cases}

d e g_{M} (w) = σ \Leftrightarrow \underset{q \in (Q_{L} \cup Q_{R})}{\oplus} (\otimes (\hat{I} (q), {\bar{D}}_{M} (q, w, q_{f}, λ))) = σ

A c c_{m} (w) = \underset{(x, y) \in \nabla_{w}, q \in Q}{\oplus} (\otimes (O (q), \hat{f} (q', x, y^{R}, q_{a c c}, λ, λ)))) = \underset{(x, y) \in \nabla_{w}, q \in Q}{\oplus} (\otimes (O (q), {\bar{D}}_{M} (q, x y, q_{a c c}, λ))) = d e g_{M} (w) .

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[1] *Corresponding author: Valdigleis S. Costa - E-mail: valdigleis@ppgsc.ufrn.br.