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lfa:2025:lab01 [2025/10/05 23:26] tpruteanu created |
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| === 2.1. Write DFAs for the following languages: === | === 2.1. Write DFAs for the following languages: === | ||
| - | **2.1.1.** $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of ones} \} $. | + | |
| + | **2.1.1.** The language of binary words which encode odd numbers (the last digit is least significative). | ||
| + | |||
| + | |||
| + | /*<hidden> | ||
| + | {{:lfa:2022:lfa2022_lab2_ex3.png?300|}} | ||
| + | * We pass through the word, marking the parity of the last character that we have read. | ||
| + | </hidden>*/ | ||
| + | |||
| + | **2.1.2.** $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of ones} \} $. | ||
| Line 27: | Line 36: | ||
| - | **2.1.2.** The language of binary words which contain **exactly** two ones. | + | **2.1.3.** The language of binary words which contain **exactly** two ones. |
| /*<hidden> | /*<hidden> | ||
| Line 39: | Line 47: | ||
| - | **2.1.3.** The language of binary words which encode odd numbers (the last digit is least significative). | + | **2.1.4.** (*) The language of words which encode binary numbers divisible by 3. |
| + | **2.1.5.** (* *) The language of words which encode binary numbers divisible by 3, represented in reverse order (the least significant digit is first). | ||
| - | /*<hidden> | + | **2.1.6.** (*) 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. |
| - | {{:lfa:2022:lfa2022_lab2_ex3.png?300|}} | + | |
| - | * We pass through the word, marking the parity of the last character that we have read. | + | |
| - | </hidden>*/ | + | |
| - | + | ||
| - | + | ||
| - | **2.1.4.** (*) The language of words which encode numbers divisible by 3. | + | |
| - | + | ||
| - | /* | + | |
| - | <hidden> | + | |
| - | {{:lfa:2022:lfa2022_lab2_ex4.png?400|}} | + | |
| - | * State 0: 3k [initial state and final state] | + | |
| - | * State 1: 3k + 1 | + | |
| - | * State 2: 3k + 2 | + | |
| - | * When we read a 0, the number we had so far is multiplied by 2: | + | |
| - | * $ 3k \overset{*2}{\longrightarrow} 3k $ | + | |
| - | * $ 3k + 1 \overset{*2}{\longrightarrow} 3k + 2 $ | + | |
| - | * $ 3k + 2 \overset{*2}{\longrightarrow} 3k + 1 $ | + | |
| - | * When we read a 1, the number we had so far is multiplied by 2 and we also add 1: | + | |
| - | * $ 3k \overset{*2}{\longrightarrow} 3k \overset{+1}{\longrightarrow} 3k + 1 $ | + | |
| - | * $ 3k + 1 \overset{*2}{\longrightarrow} 3k + 2 \overset{+1}{\longrightarrow} 3k $ | + | |
| - | * $ 3k + 2 \overset{*2}{\longrightarrow} 3k + 1 \overset{+1}{\longrightarrow} 3k + 2 $ | + | |
| - | </hidden> | + | |
| - | */ | + | |
| - | + | ||
| - | **2.1.5.** (*) 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} $. | + | |
| - | + | ||
| - | /* | + | |
| - | <hidden> | + | |
| - | + | ||
| - | {{:lfa:2022:lfa2022_lab2_ex6_v2.png?400|}} | + | |
| - | * injuraturile redirectati-le catre AU :)) (si MP) | + | |
| - | * All missing transitions lead to a sink state (where there is a transition that loops in that state for any character) | + | |
| - | </hidden> | + | |
| - | */ | + | |
| - | + | ||
| - | + | ||
| - | **2.1.6.** The set of all binary strings having the substring 00101. | + | |
| + | **2.1.7.** The language of all binary words having the substring 00101. | ||
| /*<hidden> | /*<hidden> | ||
| Line 87: | Line 60: | ||
| * Analyze how each incoming character changes the current available sequence. For example, if we are in state **E** and we read character **1** we reach a final state, but if we read **0** we go back to state **C** since the available seq will be **00** | * Analyze how each incoming character changes the current available sequence. For example, if we are in state **E** and we read character **1** we reach a final state, but if we read **0** we go back to state **C** since the available seq will be **00** | ||
| </hidden>*/ | </hidden>*/ | ||
| + | |||
| + | **2.1.8.** The language of binary words that start and end with different digits. | ||
| /*<hidden> | /*<hidden> | ||
| Line 105: | Line 80: | ||
| **2.2.1** | **2.2.1** | ||
| - | {{:lfa:2024:lab1-2_2_1.png?500|}} | + | {{:lfa:2024:lab1-2_2_2.png?300|}} |
| - | /* Cuvinte nevide în care după un grup de 0-uri consecutive urmează fie nimic, fie un număr impar de 1-uri consecutive, | + | /* Cel puțin un grup de: 0 urmat de un număr impar de 1-uri urmat de un număr par de 0-uri */ |
| - | și după un grup consecutiv de 1-uri urmează fie nimic, fie un număr par de 0-uri consecutive */ | + | |
| **2.2.2** | **2.2.2** | ||
| - | {{:lfa:2024:lab1-2_2_2.png?300|}} | + | {{:lfa:2024:lab1-2_2_1.png?500|}} |
| - | /* Cel puțin un grup de: 0 urmat de un număr impar de 1-uri urmat de un număr par de 0-uri */ | + | /* Cuvinte nevide în care după un grup de 0-uri consecutive urmează fie nimic, fie un număr impar de 1-uri consecutive, |
| + | și după un grup consecutiv de 1-uri urmează fie nimic, fie un număr par de 0-uri consecutive */ | ||
| **2.2.3** | **2.2.3** | ||
| Line 136: | Line 111: | ||
| iar dacă începe cu b. prima secvență este de al doilea tip */ | iar dacă începe cu b. prima secvență este de al doilea tip */ | ||
| + | |||
| ---- | ---- | ||
| + | **2.3.1** What happens if we switch all final states to non-final states and vice-versa in a DFA? | ||
| + | |||
| + | **2.3.2** 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. | ||
| + | |||
| + | **2.3.3** If no such cycle described above exist, L(M) is finite. | ||
| + | |||
| + | **2.3.4** 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 $. | ||