{
  "translations": {
    "en": {
      "gamma": {
        "type": "select",
        "description": "<b>Definition:</b> Let $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, F)$ be a NPDA.<br/>Then $L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),\\ p \\in F,\\ u \\in \\Gamma^*\\}$.",
        "question": "What is $\\Gamma$?",
        "answer": "Set of stack symbols",
        "choices": ["Start state", "Final state", "Set of alphabets", "Set of stack symbols", "Set of states", "Stack empty symbol"]
      },
      "Z": {
        "type": "select",
        "description": "<b>Definition:</b> Let $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, F)$ be a NPDA.<br/>Then $L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),\\ p \\in F,\\ u \\in \\Gamma^*\\}$.",
        "question": "What is $Z$?",
        "answer": "Stack empty symbol",
        "choices": ["Start state", "Final state", "Set of alphabets", "Set of stack symbols", "Set of states", "Stack empty symbol"]
      },
      "p": {
        "type": "select",
        "description": "<b>Definition:</b> Let $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, F)$ be a NPDA.<br/>Then $L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),\\ p \\in F,\\ u \\in \\Gamma^*\\}$.",
        "question": "To properly interpret $(q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),$<br/>$p \\in F, u \\in \\Gamma^*$, what does it mean that $q_0$ changed to $p$?",
        "answer": "The machine moved from start state $q_0$ to state $p$.",
        "choices": ["The machine moved from start state $q_0$ to state $p$.", "The string is consumed by the machine.", "The stack content is changed based on the transitions."]
      },
      "lambda": {
        "type": "select",
        "description": "<b>Definition:</b> Let $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, F)$ be a NPDA.<br/>Then $L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),\\ p \\in F,\\ u \\in \\Gamma^*\\}$.",
        "question": "To properly interpret $(q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),$<br/>$p \\in F, u \\in \\Gamma^*$, what does it mean that $w$ changed to $\\lambda$?",
        "answer": "The string is consumed by the machine.",
        "choices": ["The machine moved from start state $q_0$ to state $p$.", "The string is consumed by the machine.", "The stack content is changed based on the transitions."]
      },
      "stack": {
        "type": "select",
        "description": "<b>Definition:</b> Let $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, F)$ be a NPDA.<br/>Then $L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),\\ p \\in F,\\ u \\in \\Gamma^*\\}$.",
        "question": "To properly interpret $(q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),$<br/>$p \\in F, u \\in \\Gamma^*$, what does it mean that $Z$ changed to $u$?",
        "answer": "The stack content is changed based on the transitions.",
        "choices": ["The machine moved from start state $q_0$ to state $p$.", "The string is consumed by the machine.", "The stack content is changed based on the transitions."]
      },
      "irrelevant": {
        "type": "select",
        "description": "<b>Definition:</b> Let $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, F)$ be a NPDA.<br/>Then $L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, u),\\ p \\in F,\\ u \\in \\Gamma^*\\}$.<br/>The NPDA accepts all strings that end in a final state.",
        "question": "Are the stack contents relevant if the machine ends in a final state?",
        "answer": "No",
        "choices": ["Yes", "No"]
      },
      "language": {
        "type": "select",
        "description": "Consider the following PDA.",
        "question": "What is the language represented by the PDA?",
        "answer": "$\\{a^nb^n\\mid n> 0\\}$",
        "choices": ["$\\{a^nb^n\\mid n> 0\\}$", "$\\{a^nb^m\\mid n> 0, m>0\\}$", "$\\{a^nb^n\\mid n\\ge 0\\}$"]
      },
      "aaabbb": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.",
        "question": "Is string $aaabbb$ in the language?",
        "answer": "Yes",
        "choices": ["Yes", "No"]
      },
      "memory": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.",
        "question": "What memory will we use to help determine if the a's and b's match?",
        "answer": "Stack",
        "choices": ["Integer", "Stack", "Queue"]
      },
      "a1machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine's state?",
        "answer": "Move to $q_1$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "a1stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
        "answer": "Push $a$",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$"]
      },
      "a2machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Stay at $q_1$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "a2stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
        "answer": "Push $a$",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$"]
      },
      "a3machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Stay at $q_1$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "a3stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the Stack?",
        "answer": "Push $a$",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$"]
      },
      "b1machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Move to $q_2$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "b1stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the Stack?",
        "answer": "Pop the top",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$"]
      },
      "b2machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Stay at $q_2$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "b2stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
        "answer": "Pop the top",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$"]
      },
      "b3machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Stay at $q_2$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "b3stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
        "answer": "Pop the top",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$"]
      },
      "emptystring": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack the fact that we are at the end of the string, what will happen to the machine?",
        "answer": "Move to $q_3$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "emptystack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the Stack?",
        "answer": "Push $Z$ back",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$", "Push $Z$ back"]
      },
      "final": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Will the string be accepted?",
        "answer": "Yes, the machine ended in a final state",
        "choices": ["Yes, the machine ended in a final state", "No", "Yes, the stack reached $Z$ again"]
      },
      "ignorestack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $aaabbb$.",
        "question": "Does the stack content affect the acceptance of the string?",
        "answer": "No, in this model of acceptance we use the stack only as memory",
        "choices": ["No, in this model of acceptance we use the stack only as memory", "Yes"],
        "correctFeedback": ["Note that we had to have the final transition from $q_2$ to $q_3$ on an empty string and empty stack. If instead we simply made $q_2$ a final state and skipped this transition, then yes this string would be accepted. But so would any string that had at least as many a's as b's, which would be incorrect."]
      },
      "reject": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Now consider the string $abab$.",
        "question": "What about the string $abab$, will it be accepted?",
        "answer": "No, it is not in the language",
        "choices": ["No, it is not in the language", "Yes, the string has the same number of a's and b's"]
      },
      "ababa1machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $abab$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Move to $q_1$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "ababa1stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $abab$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
        "answer": "Push $a$",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$", "Push $Z$ back"]
      },
      "ababb1machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $abab$.",
        "question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
        "answer": "Move to $q_2$",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$"]
      },
      "ababb1stack": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $abab$.",
        "question": "Based on the top of the stack and the highlighted input letter, what will happen to the stack?",
        "answer": "Pop the top",
        "choices": ["Push $a$", "Push $b$", "Pop the top", "Push $\\lambda$", "Push $Z$ back"]
      },
      "ababa2machine": {
        "type": "select",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $abab$.",
        "question": "Based on the top of the stack and the highlighted input letter, what will happen to the machine?",
        "answer": "There is no possible transition",
        "choices": ["Move to $q_0$", "Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Stay at $q_1$", "Stay at $q_2$", "There is no possible transition"]
      },
      "hang": {
        "type": "true/false",
        "description": "$L = \\{a^nb^n\\ |\\ n \\ge 0\\},\\ \\Sigma = \\{a, b\\},\\ \\Gamma = \\{Z, a\\}$.<br/>Let us trace processing the string $abab$.",
        "question": "Since the machine did not find any possible transition, it will <b>hang</b> on a non-final state and reject the string.",
        "answer": "True",
        "choices": []
      }
    }
  }
}