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lfa:2026:lab01 [2026/10/05 11:26] tpruteanu |
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| - | **1.1.2.** $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of ones} \} $. | + | **1.1.2.** The language of binary words that contains only the words 010 and 1010. |
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| + | **1.1.3.** $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of ones} \} $. | ||
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| - | **1.1.3.** The language of binary words which contain **exactly** two ones. | + | **1.1.4.** The language of binary words which contain **exactly** two ones. |
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| - | **1.1.4.** The language of all binary words having the substring 00101. | + | **1.1.5.** The language of all binary words having the substring 00101. |
| - | **1.1.5.** The language of binary words that start and end with different digits. | + | **1.1.6.** The language of binary words that start and end with different digits. |
| - | **1.1.6.** (*) The language of words which encode binary numbers divisible by 3. | + | **1.1.7.** (*) The language of words which encode binary numbers divisible by 3. |
| - | **1.1.7.** (* *) The language of words which encode binary numbers divisible by 3, represented in reverse order (the least significant digit is first). | + | **1.1.8.** (* *) The language of words which encode binary numbers divisible by 3, represented in reverse order (the least significant digit is first). |
| - | **1.1.8.** (*) The language of quaternary words (base 4), that follow the rule that every zero is immediately followed by a sequence of at least 2 consecutive threes and every one is immediately followed by a sequence of at most 2 consecutive twos. | + | **1.1.9.** (*) The language of quaternary words (base 4), that follow the rule that every zero is immediately followed by a sequence of at least 2 consecutive threes and every one is immediately followed by a sequence of at most 2 consecutive twos. |
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| + | **1.1.10.** (*) The langauge of binary words that have an equal number of **01** sequences as **10** sequences. | ||
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| **1.2.1** What happens if we switch all final states to non-final states and vice-versa in a DFA? | **1.2.1** What happens if we switch all final states to non-final states and vice-versa in a DFA? | ||
| - | **1.2.2** Prove that if L(M) is infinite, there has to be some cycle in the states graph, such that there is a path from the initial state to the cycle, and from the cycle to a final state. | + | **1.2.2** Prove that if L(M) is infinite, there has to be some cycle in the states graph, such that there is a path from the initial state to the cycle, and from the cycle to a final state. If no such cycle exists, L(M) has to be finite. |
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| - | **1.2.3** If no such cycle described above exist, L(M) is finite. | + | |
| - | **1.2.4** Show that if you can construct a DFA to accept L, than you can also construct a DFA to accept $ L \cup \{a\}, \forall a \in \Sigma $. | + | **1.2.3** Show that if you can construct a DFA to accept L, than you can also construct a DFA to accept $ L \cup \{a\}, \forall a \in \Sigma $. |