{
"translations": {
"en": {
"different": {
"type": "select",
"description": "Definition: Let NPDA $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, \\emptyset)$.
The last two symbols in this definition mean that the stack is initialized to $Z$ (as usual for a PDA), and the set of final states is the empty set.
Then, $M$ accepts $L(M)$ by Empty Stack acceptance if
$L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$.",
"question": "Look carefully at the parameter set of $M$. What is different?",
"answer": "The set of final states",
"choices": ["The start state", "The set of final states", "The input alphabet", "The stack alphabet", "The set of stack symbols", "The set of states", "The stack start symbol"],
"correctFeedback": ["There are no final states, so this is defined to be the empty set: $\\emptyset$."]
},
"p": {
"type": "select",
"description": "Definition: Let NPDA $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, \\emptyset)$.
The last two symbols in this definition mean that the stack is initialized to $Z$ (as usual for a PDA), and the set of final states is the empty set.
Then, $M$ accepts $L(M)$ by Empty Stack acceptance if
$L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$.",
"question": "To understand $(q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$, what does it mean that $q_0$ changed to $p$?",
"answer": "The machine moved from the start state to state $p$.",
"choices": [ "The machine moved from the start state to state $p$.", "The string is consumed by the machine.", "The stack contents are consumed by the machine."]
},
"pfinal": {
"type": "select",
"description": "Definition: Let NPDA $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, \\emptyset)$.
The last two symbols in this definition mean that the stack is initialized to $Z$ (as usual for a PDA), and the set of final states is the empty set.
Then, $M$ accepts $L(M)$ by Empty Stack acceptance if
$L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$.",
"question": "Is $p$ a final state?",
"answer": "No, we do not have a final state in this type of acceptance",
"choices": ["No, we do not have a final state in this type of acceptance", "Yes"]
},
"w": {
"type": "select",
"description": "Definition: Let NPDA $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, \\emptyset)$.
The last two symbols in this definition mean that the stack is initialized to $Z$ (as usual for a PDA), and the set of final states is the empty set.
Then, $M$ accepts $L(M)$ by Empty Stack acceptance if
$L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$.",
"question": "To understand $(q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$, what does it mean that $w$ changed to $\\lambda$?",
"answer": "The string is consumed by the machine.",
"choices": [ "The machine moved from the start state to state $p$.", "The string is consumed by the machine.", "The stack contents are consumed by the machine."]
},
"Z": {
"type": "select",
"description": "Definition: Let NPDA $M = (Q, \\Sigma, \\Gamma, \\delta, q_0, Z, \\emptyset)$.
The last two symbols in this definition mean that the stack is initialized to $Z$ (as usual for a PDA), and the set of final states is the empty set.
Then, $M$ accepts $L(M)$ by Empty Stack acceptance if
$L(M) = \\{w \\in \\Sigma^*\\ |\\ (q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$.",
"question": "To understand $(q_0, w, Z) \\stackrel{*}{\\vdash} (p, \\lambda, \\lambda)\\}$, what does it mean that $Z$ changed to $\\lambda$?",
"answer": "The stack contents are consumed by the machine.",
"choices": [ "The machine moved from the start state to state $p$.", "The string is consumed by the machine.", "The stack contents are consumed by the machine."]
},
"language": {
"type": "select",
"description": "Consider the following PDA",
"question": "What is the language represented by the PDA?",
"answer": "$L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$",
"choices": ["$L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$", "$L = \\{a^nb^mc^{2n}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$"]
},
"aabccc": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.",
"question": "Is string $aabccc$ in the language?",
"answer": "Yes",
"choices": ["Yes", "No"]
},
"a1machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"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 $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"a1stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
"answer": "Push $0$",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"]
},
"a2machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"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 $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"a2stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
"answer": "Push $0$",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"]
},
"b1machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"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 $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"b1stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
"answer": "Push $0$",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"]
},
"c1machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
"answer": "Move to $q_3$",
"choices": ["Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"c1stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"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 $0$", "Pop the top", "Push $\\lambda$"]
},
"c2machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
"answer": "Stay at $q_3$",
"choices": ["Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"c2stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"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 $0$", "Pop the top", "Push $\\lambda$"]
},
"c3machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
"answer": "Stay at $q_3$",
"choices": ["Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"c3stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"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 $0$", "Pop the top", "Push $\\lambda$"]
},
"aabcccend": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the fact that we are at the end of the string, what will happen to the machine?",
"answer": "Move to $q_4$",
"choices": ["Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"aabcccstack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Based on the top of the stack and the fact that we are at the end of the string, what will happen to the stack?",
"answer": "Pop the top",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"],
"correctFeedback": ["In this case, popping the top pops $Z$ off the stack."]
},
"accept": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabccc$.",
"question": "Will the string be accepted?",
"answer": "Yes, the machine ends with an empty stack",
"choices": ["Yes, the machine ends with an empty stack", "No, the machine ends in a non-final state"]
},
"aabcaccept": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"question": "Should string $aabc$ be accepted?",
"answer": "No, it is not in the language",
"choices": ["No, it is not in the language", "Yes"]
},
"aabca1machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"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 $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"aabca1stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
"answer": "Push $0$",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"]
},
"aabca2machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"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 $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"aabca2stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
"answer": "Push $0$",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"]
},
"aabcb1machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"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 $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"aabcb1stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the stack?",
"answer": "Push $0$",
"choices": ["Push $0$", "Pop the top", "Push $\\lambda$"]
},
"aabcc1machine": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"question": "Based on the top of the stack and the highlighted input symbol, what will happen to the machine?",
"answer": "Move to $q_3$",
"choices": ["Move to $q_1$", "Move to $q_2$", "Move to $q_3$", "Move to $q_4$", "Stay at $q_1$", "Stay at $q_2$", "Stay at $q_3$"]
},
"aabcc1stack": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"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 $0$", "Pop the top", "Push $\\lambda$"]
},
"reject": {
"type": "select",
"description": "Example}: $L = \\{a^nb^mc^{n+m}\\ |\\ n, m > 0\\}$, $\\Sigma = \\{a, b, c\\}$, $\\Gamma =\\{0, Z\\}$.
Let us trace processing the string $aabc$.",
"question": "Will the string be accepted?",
"answer": "No, the machine is finished all steps and ended with a non-empty stack",
"choices": ["Yes", "No, the machine is finished all steps and ended with a non-empty stack"]
}
}
}
}