Proving the Limitations of Recognizable Languages

✅ A recognizable language is defined as a language that can be recognized by a finite automaton.

❌ Not all possible languages are recognizable; there exist non-recognizable languages.

🔐 The pumping lemma is a tool used to prove that certain languages are not recognizable.

🔑 The pumping lemma is used to prove that certain languages are not recognizable by finite automata.

💡 The intuition behind the lemma is that a finite automaton needs to keep track of the number of 'a's and 'b's in a word and ensure that the number of 'b's matches the number of 'a's encountered so far.

⚠️ However, since the number of 'a's is unbounded, a finite automaton would require an infinite number of states, which is not possible.

🔑 There is a nine-state automaton that recognizes a certain language.

📍 If a word starts with 'n' and is followed by exactly nine 'b's, it should be accepted by the automaton.

🔄 By examining the path taken by the automaton, we can see that it visits a state twice.

📝 The Pumping Lemma for recognizable languages is explained.

🔄 An example is given to demonstrate the lemma's application.

❌ The automaton does not recognize certain words in the language.

🔍 The pumping lemma states that recognizable languages have a property where any long enough word can be divided into three parts.

✏️ These three parts consist of the initial part, the part that can be repeated, and the last part that brings the word to the final state.

✅ For recognizable languages, if a word is longer than a certain length, there must be a part that can be repeated, and it must still be accepted by the automaton.

🔑 The video discusses the pumping lemma for recognizable languages and its key observations.

🔍 One important observation is that the state 'y' is not an empty word and must be visited twice during the process.

📝 Another observation is that the combined lengths of 'x' and 'y' must be no larger than the number of states in the language.

🔑 The pumping lemma states that for a recognizable language, there exists a number p such that words of length at least p can be split into three parts and the middle part can be pumped or repeated multiple times while still being part of the language.

🔑 If a language does not satisfy the conditions of the pumping lemma, it is not recognizable.

🔑 The pumping lemma provides a method to determine if a language is recognizable and can be applied to various languages.