{
  "translations": {
    "en": {
      "q2": {
        "type": "select",
        "description": "Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not a CFL",
        "question": "What is the first step?.",
        "answer": "Assume that $L$ is a CFL",
        "choices": ["Assume that $L$ is a CFL", "Assume that $L$ is not a CFL"]
      },
      "q3": {
        "type": "true/false",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL.",
        "question": "[T|F] Since $L$ is CFL, the pumping lemma holds",
        "answer": "True",
        "choices": []
      },
      "q4": {
        "type": "select",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose a long string $w \\in L$",
        "question": "Based on the Pumping Lemma, what is the main condition for the length of the string $w$?",
        "answer": "$|w| \\ge m$",
        "choices": ["$|w| \\ge m$", "$|w| \\le m$", "$|w| = m$"]
      },
      "q5": {
        "type": "true/false",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose a long string $w \\in L$",
        "question": "Based on the Pumping Lemma, $m$ is the number of states for any PDA that recognize $L$.",
        "answer": "True",
        "choices": []
      },
      "q6": {
        "type": "select",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose a long string $w \\in L$",
        "question": "Which of the following is a correct string $\\in L$?",
        "answer": "$w = a^{2m}b^{2m}c^md^m$",
        "choices": ["$w = a^mb^mc^md^m$", "$w = a^{2m}b^{2m}c^md^m$", "$w = a^{2n}b^{2m}c^nd^m$"]
      },
      "q7": {
        "type": "select",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose a long string $w \\in L$",
        "question": "What is the length of $w = a^{2m}b^{2m}c^md^m$?",
        "answer": "$6m$",
        "choices": ["$6m$", "$4m$", "$5m$", "$2m^2+2m$"]
      },
      "q8": {
        "type": "select",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose $w = a^{2m}b^{2m}c^md^m$ where $m$ is the constant in the pumping lemma. (Note that $w$ must be choosen such that $|w| \\ge m$.)",
        "question": "The next step is to decompose the string $w$ to...?",
        "answer": "$uvxyz$",
        "choices": ["$xyz$", "$uvxyz$"]
      },
      "q14": {
        "type": "select",
        "description":"Prove that $L = \\{ a^{2n}b^{2m}c^nd^m : n,m \\ge 0 \\}$ is not CFL.<br/>Assume $L$ is a CFL, therefore the pumping lemma holds.<br/>Choose $w = a^{2m}b^{2m}c^md^m$ where $m$ is the constant in the pumping lemma. (Note that $w$ must be choosen such that $|w| \\ge m$.)",
        "question": "Based on the Pumping Lemma, what are the main conditions to successfully decomposing $w$ to $uvxyz$?",
        "answer": ["$|vxy| \\le m$", "$|vy| \\geq 1$", "$uv^ixy^iz \\in L$ for all $i \\ge 0$"],
        "choices": ["$|vxy| \\le m$", "$|vy| \\geq 1$", "$uv^ixy^iz \\in L$ for all $i \\ge 0$"]
      },
      "q15": {
        "type": "select",
        "description":"The first condition that we need to think about is $|vxy| \\le m$. This condition means that we must test every possible $m$ letters substring in $w$",
        "question": "Think about $w = a^{2m}b^{2m}c^md^m$, select the distincted substrings with length $\\le m$",
        "answer": ["substring of $a$'s only", "substring of some $a$'s followed by some $b$'s", "substring of $b$'s only", "substring of some $b$'s followed by some $c$'s", "substring of $c$'s only", "substring of some $c$'s followed by some $d$'s", "substring of $d$'s only"],
        "choices": ["substring of $a$'s only", "substring of some $a$'s followed by some $b$'s", "substring of $b$'s only", "substring of some $b$'s followed by some $c$'s", "substring of $c$'s only", "substring of some $c$'s followed by some $d$'s", "substring of $d$'s only", "substring of some $a$'s followed by some $b$'s followed by some $c$'s", "substring of some $a$'s followed by some $b$'s followed by some $d$'s", "substring of some $b$'s followed by some $c$'s followed by some $d$'s", "substring of some $a$'s followed by some $b$'s followed by some $c$'s followed by some $d$'s"]
      },
      "q16": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 1:}$ $vxy$ 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": "$a^{2m+t_1+t_2}b^{2m}c^md^m$",
        "choices": ["$a^{2m+t_1+t_2}b^{2m}c^md^m$", "$a^{2m+t_1}b^{2m}c^md^m$", "$a^{2m+t_1-t_2}b^{2m}c^md^m$"]
      },
      "q17": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 1:}$ $vxy$ is $a$'s only, let $v = a^{t_1}$ and $y = a^t_2$ where $0 < t_1 + t_2 \\le m$",
        "question": "Is $a^{2m+t_1+t_2}b^{2m}c^md^m$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q18": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 2:}$ $vxy$ 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": "$a^{2m+t_1}b^{2m+t_3}c^md^m$",
        "choices": ["$a^{2m+t_1}b^{2m+t_3}c^md^m$", "$a^{2m+t_1}b^{2m - t_3}c^md^m$", "$a^{m-t_1}b^{m + t_3}$"]
      },
      "q19": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 2:}$ $vxy$ is some $a$'s followed by some $b$'s, let $v = a^{t_1}$ and $y = b^{t_3}$",
        "question": "Is $a^{2m+t_1}b^{2m+t_3}c^md^m$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q20": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 3:}$ $vxy$ 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": "$a^{2m}b^{2m+t_1+t_2}c^md^m$",
        "choices": ["$a^{2m}b^{2m+t_1+t_2}c^md^m$", "$a^{2m}b^{2m+t_1 - t_2}c^md^m$", "$a^{2m}b^{2m - t_1+t_2}c^md^m$"]
      },
      "q21": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 3:}$ $vxy$ is $b$'s only, let $v = b^{t_1}$ and $y = b^{t_2}$ where $0 < t_1 + t_2 \\le m$",
        "question": "Is $a^{2m}b^{2m+t_1+t_2}c^md^m$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q22": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 4:}$ $vxy$ is some $b$'s followed by some $c$'s, let $v = b^{t_1}$ and $y = c^{t_3}$ where $0 < t_1 + t_3 \\le m$",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^{2m}b^{2m+t_1}c^{m+t_3}d^m$",
        "choices": ["$a^{2m}b^{2m+t_1}c^{m+t_3}d^m$", "$a^{2m}b^{2m-t_1}c^{m+t_3}d^m$", "$a^{2m}b^{2m+t_1}c^{m-t_3}d^m$"]
      },
      "q23": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 4:}$ $vxy$ is some $b$'s followed by some $c$'s, let $v = b^{t_1}$ and $y = c^{t_3}$ where $0 < t_1 + t_3 \\le m$",
        "question": "Is $a^{2m}b^{2m+t_1}c^{m+t_3}d^m$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q24": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 5:}$ $vxy$ is $c$'s only, let $v = c^{t_1}$ and $y = c^{t_2}$ where $0 < t_1 + t_2 \\le m$",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^{2m}b^{2m}c^{m+t_1+t_2}d^m$",
        "choices": ["$a^{2m}b^{2m}c^{m+t_1+t_2}d^m$", "$a^{2m}b^{2m}c^{m+t_1}d^m$", "$a^{2m}b^{2m}c^{m+t_1-t_2}d^m$"]
      },
      "q25": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 5:}$ $vxy$ is $c$'s only, let $v = c^{t_1}$ and $y = c^{t_2}$ where $0 < t_1 + t_2 \\le m$",
        "question": "Is $a^{2m}b^{2m}c^{m+t_1+t_2}d^m$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q26": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 6:}$ $vxy$ is some $c$'s followed by some $d$'s, let $v = c^{t_1}$ and $y = d^{t_3}$ where $0 < t_1 + t_3 \\le m$",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^{2m}b^{2m}c^{m+t_1}d^{m+t_3}$",
        "choices": ["$a^{2m}b^{2m}c^{m+t_1}d^{m+t_3}$", "$a^{2m}b^{2m}c^{m-t_1}d^{m+t_3}$", "$a^{2m}b^{2m}c^{m+t_1}d^{m-t_3}$"]
      },
      "q27": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 6:}$ $vxy$ is some $c$'s followed by some $d$'s, let $v = c^{t_1}$ and $y = d^{t_3}$ where $0 < t_1 + t_3 \\le m$",
        "question": "Is $a^{2m}b^{2m}c^{m+t_1}d^{m+t_3}$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q28": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 7:}$ $vxy$ is $d$'s only, let $v = d^{t_1}$ and $y = d^{t_2}$ where $0 < t_1 + t_2 \\le m$",
        "question": "If $i = 2$ what is the value of $uv^2xy^2z$?",
        "answer": "$a^{2m}b^{2m}c^md^{m+t_1+t_2}$",
        "choices": ["$a^{2m}b^{2m}c^md^{m+t_1+t_2}$", "$a^{2m}b^{2m}c^md^{m+t_1-t_2}$", "$a^{2m}b^{2m}c^md^{m-t_1-t_2}$"]
      },
      "q29": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 7:}$ $vxy$ is $d$'s only, let $v = d^{t_1}$ and $y = d^{t_2}$ where $0 < t_1 + t_2 \\le m$",
        "question": "Is $a^{2m}b^{2m}c^md^{m+t_1+t_2}$ $\\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "q30": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 1:}$ $vxy$ is $a$'s only<br/>$\\textbf{Case 2:}$ $vxy$ is some $a$'s followed by some $b$'s<br/>$\\textbf{Case 3:}$ $vxy$ is $b$'s only<br/>$\\textbf{Case 4:}$ $vxy$ is some $b$'s followed by some $c$'s<br/>$\\textbf{Case 5:}$ $vxy$ is $c$'s only<br/>$\\textbf{Case 6:}$ $vxy$ is some $c$'s followed by some $d$'s<br/>$\\textbf{Case 7:}$ $vxy$ is $d$'s only",
        "question": "Are we done from all cases?",
        "answer": "No, there are some more cases",
        "choices": ["No, there are some more cases", "Yes"]
      },
      "q31": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$<br/>$\\textbf{Case 1:}$ $vxy$ is $a$'s only<br/>$\\textbf{Case 2:}$ $vxy$ is some $a$'s followed by some $b$'s<br/>$\\textbf{Case 3:}$ $vxy$ is $b$'s only<br/>$\\textbf{Case 4:}$ $vxy$ is some $b$'s followed by some $c$'s<br/>$\\textbf{Case 5:}$ $vxy$ is $c$'s only<br/>$\\textbf{Case 6:}$ $vxy$ is some $c$'s followed by some $d$'s<br/>$\\textbf{Case 7:}$ $vxy$ is $d$'s only",
        "question": "Yes, we have cases where $v$ or $y$ contain 2 or more distinct symbols. If $v$ or $u$ contains 2 distincit letters, then $uv^2xy^2z \\notin L$",
        "answer": "Yes",
        "choices": ["No", "Yes"]
      },
      "q32": {
        "type": "select",
        "description":"let $w = a^{2m}b^{2m}c^md^m$ we need to partition $w$ to $uvxyz$ such that $|vxy| \\le m$, $|vy| \\geq 1$, $uv^ixy^iz \\in L$ for all $i \\ge 0$",
        "question": "Now, is L CFL?",
        "answer": "No",
        "choices": ["No", "Yes"]
      }
    }
  }
}