44 lines
1.9 KiB
TeX
44 lines
1.9 KiB
TeX
\documentclass{article}
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\usepackage{mathpartir}
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\begin{document}
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\begin{figure}
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\begin{mathpar}
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\begin{array}{rclll}
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\delta_c \O & = & \O & & \\
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\delta_c \varepsilon & = & \O & & \\
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\delta_c a & = & \{\varepsilon\} & , & if \ a == c \\
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& | & \O & , & otherwise \\
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\delta_c (r_1;r_2) & = & \{r;r_2 \ | \ r \in \delta_c r_1\} \ \cup
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\ (\delta_c r_2)
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& , & if \ \varepsilon \in (\delta_c r_1) \\
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& | & \{r;r_2 \ | \ r \in \delta_c r_1\} & , & otherwise \\
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\delta_c (r_1 + r_2) & = & (\delta_c r_1) \ \cup \ (\delta_c r_2) & & \\
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\delta_c (r*) & = & \{r';r \! * \ | \ r' \in \delta_c r\} & & \\
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\end{array}
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\end{mathpar}
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\caption{Antimirov derivatives}
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\label{fig:antimirov}
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\end{figure}
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\begin{figure}[h]
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\begin{mathpar}
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\begin{array}{rcl}
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w \models \varepsilon & \iff & w = nil \\
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w \models b & \iff & (|w|=1) \land (w_0 \models b) \\
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w \models (r_1 ; r_2) & \iff & (w=w_1 \cdot w_2) \land
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(w_1 \models r_1) \land
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(w_2 \models r_2) \\
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w \models (r_1 + r_2) & \iff & (w \models r_1) \ \lor \
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(w \models r_2) \\
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w \models r* & \iff & (w \models \varepsilon) \ \lor \
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(w=w_1 \cdot w_2) \land
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(w_1 \models r) \land
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(w_2 \models r*) \\
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\end{array}
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\end{mathpar}
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\caption{Semantics of regular expression}
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\label{fig:re-semantics}
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\end{figure}
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\end{document}
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