Use Corollary 7.5.2 to show that each of the following sets...

Use Corollary 7.5.2 to show that each of the following sets is not regular. a) The set of strings over $\{a, b\}$ with the same number of $a$ 's and $b$ 's. b) The set of palindromes of even length over $\{a, b\}$. c) The set of strings over $$\{()$,$\} in which the parentheses are paired. Examples include$$ $\lambda,(),()(),(())()$. d) The language $\left\{a^i(a b)^j(c a)^{2 i} \mid i, j>0\right\}$.

Use Corollary 7.5.2 to show that each of the following sets is not regular.
a) The set of strings over $\{a, b\}$ with the same number of $a$ 's and $b$ 's.
b) The set of palindromes of even length over $\{a, b\}$.
c) The set of strings over $$\{()$,$\} in which the parentheses are paired. Examples include$$ $\lambda,(),()(),(())()$.
d) The language $\left\{a^i(a b)^j(c a)^{2 i} \mid i, j>0\right\}$.

Key Concepts

Regular Languages

Regular languages are languages that can be recognized by finite automata, described by regular expressions, or generated by regular grammars. Their key attribute is that they have a finite amount of memory, which restricts the kinds of dependencies they can track in strings. This limitation is central when proving that certain languages are not regular because the structure of the language exceeds what any finite automaton can remember.

Equivalence Relations and Equivalence Classes in Automata Theory

Equivalence relations partition the set of all strings into classes where two strings are equivalent if appending any continuation to both strings either always produces a member of the language or never does. In the context of automata theory, a language is regular if and only if there are finitely many such equivalence classes. Proving that there are infinitely many distinguishable equivalence classes is a common method to demonstrate a language's non-regularity.

Myhill-Nerode Theorem and Its Corollaries

The Myhill-Nerode theorem provides a necessary and sufficient condition for a language to be regular via the finiteness of the number of equivalence classes defined by the language’s indistinguishability relation. Corollary 7.5.2 (or equivalent results) leverages this idea to show non-regularity by exhibiting an infinite set of strings that are pairwise distinguishable with respect to the language. This method is a powerful alternative to the pumping lemma for proving that some languages are inherently non-regular.

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