5.1 ��, 24/9/02: ��

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5.1 ��, 24/9/02: ��

�� ' �� . �� , �, ��, �� . �� ��  �� ��  (regular languages).

�� :

�� $\Sigma$ �� , �� �� � ��. �.�. $\Sigma = {\left\{{0, 1}\right\}}$ � $\Sigma$ = �� .
�� �� $\Sigma$ �� $\Sigma$ , �� (� �� $\epsilon$ ). �.�. �� $\Sigma = {\left\{{0, 1}\right\}}$ �� $\Sigma$ , �� ${\left\vert{w}\right\vert} = 4$ . � �� $\epsilon$ �� 0.
�� �� �� $\Sigma$ �� , �� , �� $\Sigma$ . �.�., �� $\Sigma = {\left\{{0, 1}\right\}}$ ��

$\displaystyle Q = {\left\{{w: \exists k\in {\mathbf N}: {\left\vert{w}\right\vert} = k^2}\right\}}$
�� . �� .
�� $\alpha = \alpha_1 \ldots \alpha_m$ �� $\beta = \beta_1 \ldots \beta_n$ �� $\Sigma$ �� , �� �� (concatenation) $\alpha\beta$ �� $\alpha_1\ldots\alpha_m\beta_1\ldots\beta_n$ , �� . �� $\alpha\epsilon = \epsilon\alpha = \alpha$ , �� $\alpha$ .
�� $\pi$ �� �� (prefix) �� $\alpha$ �� $\beta$ �.�. $\alpha = \pi\beta$ . �� $\pi$ �� �� (suffix) �� $\alpha$ �� $\beta$ �.�. $\alpha = \beta\pi$ . �� .
�� $\Sigma$ ��

$\displaystyle L_1L_2 = {\left\{{xy: x\in L_1, y\in L_2}\right\}}.$
�� $L^0 = {\left\{{\epsilon}\right\}}$ �� $n\ge 1$ $L^n = L L \cdots L$ (�� ). ��

$\displaystyle L^*=\bigcup_{k=0}^\infty L^k$
��

$\displaystyle L^+=\bigcup_{k=1}^\infty L^k.$
�� , �� $\Sigma$ �� $\Sigma$ �� $\Sigma$ �� , �� $\Sigma^*$ �� �� �� $\Sigma$ �� $\Sigma^+$ �� $\Sigma$ . ��, �� $\Sigma$ �� $L \subseteq \Sigma^*$ .
��

$\displaystyle {\left\{{+,-,\epsilon}\right\}}1{\left\{{0,1}\right\}}^*.$
�� , �� ${\left\{{+,-,\epsilon}\right\}}$ , �� (�� ${\left\{{1}\right\}}$ �� ) �� ${\left\{{0,1}\right\}}^*$ . ��, �� , �� , �� , �� $\Sigma = {\left\{{0,1,+,-}\right\}}$ .
�� ��  �� , �� . �� . �� , �� .

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