uni_notes/10 Courses/02 - SoSe 2026/AT/29593975 - Regular Expressions.md

created, course, topic, related, type, status, tags

created	course	topic	related	type	status	tags
2026-04-08 10:55	29593850 - Automationtheory	languages	29593940 - Formal Languages#Finite representation of languages	lecture	🟢	university

📌 Summary

[!abstract]

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📝 Content

A regular expression r over an alphabet Sigma is defined recursively:

• emptyset, epsilon and each a in Sigma are regular expression, which represent the Languages L(emptyset) = emptyset, L(epsilon) = {epsilon} and L(a) = {a}
• If r and s are regular expressions then these are also regular expressions:
  • (r + s) with L(r + s) = L(r) union L(s)
  • (r s) with L(r s) = L(r)L(s)
  • r^* with L(r^*) = L(r)^*

[!EXAMPLE] The language L over Sigma = {a, b} containing the substring a b is regular, since it can be expressed using the regular expression r = (a +b)^* a b (a + b)^*

Equivalence of regular expressions

Two regular expressions r and s are equivalent (r eq.triple s or r hat(eq) s) if they generate the same language (L(r) eq L(s)).