{
  "translations": {
    "en": {
      "q3": {
        "type": "true/false",
        "description":"Prove that $L = \\{a^jb^k: k = j^2\\}$ is not CFL.<br/>Assume $L$ is a CFL.",
        "question": "[T|F] Since $L$ is CFL, the pumping lemma holds",
        "answer": "True",
        "choices": []
      },
      "winL": {
        "type": "select",
        "description":"Prove that $L = \\{a^jb^k: k = j^2\\}$ is not a CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose a long string $w \\in L$, such that $|w| \\ge m$.",
        "question": "Which of the following is a string $\\in L$ and $\\ge m$?",
        "answer": ["$w = a^mb^{m^2}$"],
        "choices": ["$w = a^mb^{2m}$", "$w = a^mb^{m^2}$", "$w = a^mb^mb^{m}$", "aabbbb"]
      },
      "cases": {
        "type": "select",
        "description": "Prove that $L = \\{a^jb^k: k = j^2\\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose $w = a^mb^{m^2}$ where $m$ is the constant in the pumping lemma. (Note that $w$ must be choosen such that $|w| \\ge m$.)<br/><br/>The pumping lemma requires that for any CFL, there is a partition $w$ into $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><br/>The condition $|vxy| \\le m$ means that we must consider every possible case for substrings of $w$ with $m$ or fewer symbols.",
        "question": "For $w = a^mb^{m^2}$, what are all the variations of substrings with length $\\le m$?",
        "answer": ["a's only", "some a's followed by some b's", "b's only"],
        "choices": ["a's only", "some a's followed by some b's", "b's only", "some b's followed by some a's", "c's only"]
      },
      "case1": {
        "type": "select",
        "description":"Now we need to consider each of the three cases in turn, and show that none of them permit a decomposition of $w$ that can be pumped.<br/></br>Let $w = a^mb^{m^2}$. The pumping lemma requires that for any CFL, there is a partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><b>Case 1:</b> $vxy$ is $a$'s only. Let $v = a^{t_1}$ and $y = a^t_2$ where $0 < t_1 + t_2 \\le m$.",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^{m+t_1+t_2}b^{m^2}$",
        "choices": ["$a^{m+t_1+t_2}b^{m^2}$", "$a^{m+t_1+t_2}b^m$", "$a^{m+t_1+t_2}b^{2m}$"]
      },
      "asnopump": {
        "type": "select",
        "description":"Now we need to consider each of the three cases in turn, and show that none of them permit a decomposition of $w$ that can be pumped.<br/></br>Let $w = a^mb^{m^2}$. The pumping lemma requires that for any CFL, there is a partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><b>Case 1:</b> $vxy$ is $a$'s only. Let $v = a^{t_1}$ and $y = a^t_2$ where $0 < t_1 + t_2 \\le m$.",
        "question": "Is $a^{m+t_1+t_2}b^{m^2}$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "case2": {
        "type": "select",
        "description":"Now we need to consider each of the three cases in turn, and show that none of them permit a decomposition of $w$ that can be pumped.<br/></br>Let $w = a^mb^{m^2}$. The pumping lemma requires that for any CFL, there is a partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><b>Case 2:</b> $vxy$ is some a's followed by some b's. Let $v = a^{t_1}$ and $y = b^{t_3}$.",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^{m+t_1}b^{m^2+t_3}$",
        "choices": ["$a^{m+t_1}b^{m^2+t_3}$", "$a^{m+t_1}b^{2m + t_3}$", "$a^{m+t_1}b^{m + t_3}$"]
      },
      "absnopump": {
        "type": "select",
        "description":"Now we need to consider each of the three cases in turn, and show that none of them permit a decomposition of $w$ that can be pumped.<br/></br>Let $w = a^mb^{m^2}$. The pumping lemma requires that for any CFL, there is a partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><b>Case 2:</b> $vxy$ is some a's followed by some b's. Let $v = a^{t_1}$ and $y = b^{t_3}$.",
        "question": "Is $a^{m+t_1}b^{m^2+t_3}$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"],
        "correctFeedback": ["We also need to consider another subcase of Case 2. What if either $v$ or $y$ is some a's followed by some b's? Hopefully it is clear that we can't pump those either, since we would then get alternating a's and b's, and such strings are not in $L$."]
      },
      "case3": {
        "type": "select",
        "description":"Now we need to consider each of the three cases in turn, and show that none of them permit a decomposition of $w$ that can be pumped.<br/></br>Let $w = a^mb^{m^2}$. The pumping lemma requires that for any CFL, there is a partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><b>Case 3:</b> $vxy$ is $b$'s only. Let $v = b^{t_1}$ and $y = b^{t_2}$ where $0 < t_1 + t_2 \\le m$.",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^mb^{m^2+t_1+t_2}$",
        "choices": ["$a^mb^{m^2+t_1+t_2}$", "$a^mb^{m+t_1+t_2}$", "$a^mb^{2m+t_1 + t_2}$"]
      },
      "csnopump": {
        "type": "select",
        "description":"Now we need to consider each of the three cases in turn, and show that none of them permit a decomposition of $w$ that can be pumped.<br/></br>Let $w = a^mb^{m^2}$. The pumping lemma requires that for any CFL, there is a partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$.<br/><b>Case 3:</b> $vxy$ is $b$'s only. Let $v = b^{t_1}$ and $y = b^{t_2}$ where $0 < t_1 + t_2 \\le m$.",
        "question": "Is $a^mb^{m^2+t_1+t_2}$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      }
    }
  }
}