Prequisite: Regular Language, Regular Grammar, Finite Automaton Deterministic Finite Automata (DFA) vs Nondeterministic Finite Automata (NFA). FINITE AUTOMATON HOW TOHow to prove a Language is not Regular? (using DFA).This is one of the most useful theoretical computation model. However, that is way beyond the usual course of learning DFA.Deterministic Finite Automaton ( DFA) is the simplest version of Finite Automaton and is used to accept Regular Languages in Theory of Computation. To make the proof even more formal, we could use some formal proof system such as HOL Light, Isabelle, Coq, etc. if $s$ is accepted by $A'$, $s$ will be accepted by $A$ by the "new definition".if $s$ is accepted by $A$ by the "new definition", $s$ will be accepted by $A'$.To make the proof more formal, we should prove each of following separately, using detailed analysis of the transitions made by $A$ and $A'$ on an arbitrary input string $s$. The language accepted by $A'$ is exactly the language accepted by $A$ by the "new definition".
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