{
"translations": {
"en": {
"lemma": {
"type":"select",
"description":"Let's begin by looking at configurations and combining machines.
Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question":"What does this configuration $(q_1, w_1\\underline{a_1}u_1)$ mean?",
"answer":["The machine is in state $q_1$", "The read/write head is on the symbol $a_1$"],
"choices": ["The machine is in state $q_1$", "The read/write head is on the symbol $a_1$"]
},
"yield": {
"type":"select",
"description":"Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question": "What do we mean by $\\vdash_M^*$?",
"answer": ["The machine will do zero or more transitions"],
"choices": ["The machine will do zero or more transitions", "The machine will do one transition"]
},
"step1": {
"type":"select",
"description":"Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question":"What do we mean by
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$?",
"answer":["The machine will change the state from $q_1$ to $q_2$", "The tape content will be modified from $w_1a_1u_1$ to $w_2a_2u_2$", "The head will be on the symbol $a_2$"],
"choices": ["The machine will change the state from $q_1$ to $q_2$", "The tape content will be modified from $w_1a_1u_1$ to $w_2a_2u_2$", "The head will be on the symbol $a_2$", "The machine will accept string $w_2$"]
},
"meaning1": {
"type":"select",
"description":"Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question":"In other words, the machine did some processing on the tape contents and modified it accordingly.",
"answer":"True",
"choices": []
},
"step2": {
"type":"select",
"description":"Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question":"What do we mean by
$(q_2, w_2\\underline{a_2}u_2) \\vdash_M^* (q_3, w_3\\underline{a_3}u_3)$?",
"answer": ["The machine will change the state from $q_2$ to $q_3$", "The tape content will be modified from $w_2a_2u_2$ to $w_3a_3u_3$", "The head will be on symbol $a_3$"],
"choices": ["The machine will change the state from $q_2$ to $q_3$", "The tape content will be modified from $w_2a_2u_2$ to $w_3a_3u_3$", "The head will be on symbol $a_3$", "The machine will accept string $w_3$"]
},
"meaning2": {
"type":"select",
"description":"Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question":"In other words, the machine did more processing on the tape contents and modified it accordingly.",
"answer":"True",
"choices": []
},
"meaning3": {
"type":"select",
"description":"Lemma: If
$(q_1, w_1\\underline{a_1}u_1) \\vdash_M^* (q_2, w_2\\underline{a_2}u_2)$ for strings $u_i$ and $w_i$ and symbols $a_i$, and
$(q_2, w_2\\underline{a_2}u_2) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$ then
$(q_1, w_1\\underline{a_1}u_1) \\vdash^*_M (q_3, w_3\\underline{a_3}u_3)$.",
"question":"So, in general we can say that the machine started at $q_1$ with tape content $w_1a_1u_1$, did some processing to it, and ended at $q_3$ with tape content changed to $w_3a_3u_3$.",
"answer":"True",
"choices": []
},
"sigma": {
"type":"select",
"description":"Here are some of the most basic machines and notation that we will use to build more complex machines.
• $|\\Sigma|$ symbol-writing machines (in other words, we need one for each symbol in the input alphabet)): Any given symbol $\\sigma$ has a symbol-writing machine named $\\sigma$.",
"question":"If $\\Sigma = \\{a, b\\}$, what does the TM $a$ do?",
"answer":"write the symbol $a$ to the tape",
"choices": ["write the symbol $a$ to the tape", "$a$ is just a symbol in the alphabet set $\\Sigma$"]
},
"move": {
"type":"select",
"description":"Here are some of the most basic machines and notation that we will use to build more complex machines.
• $|\\Sigma|$ symbol-writing machines (in other words, we need one for each symbol in the input alphabet)): Any given symbol $\\sigma$ has a symbol-writing machine named $\\sigma$.
• Head-moving machines, named $R$, $L$ and $S$, that move the head appropriately.",
"question":"What does the TM $R$ do?",
"answer":"This TM moves the head to the Right",
"choices": ["This TM moves the head to the Right", "This TM moves the head to the Left", "This TM makes the head Stay on the current cell"]
},
"not": {
"type":"select",
"description":"Here are some of the most basic machines and notation that we will use to build more complex machines.
• $|\\Sigma|$ symbol-writing machines (in other words, we need one for each symbol in the input alphabet)): Any given symbol $\\sigma$ has a symbol-writing machine named $\\sigma$.
• Head-moving machines, named $R$, $L$ and $S$, that move the head appropriately.
• Transitions on anything other than (for example) $\\#$ are labeled $\\overline{\\#}$",
"question":"What does a tranistion labled with $\\overline{a}$ mean?",
"answer":"Change the state if the TM machine reads anything except $a$",
"choices": ["Change the state if the TM machine reads anything except $a$", "Change the state if the TM machine reads $a$"]
},
"multi": {
"type":"select",
"description":"Here are some of the most basic machines and notation that we will use to build more complex machines.
• $|\\Sigma|$ symbol-writing machines (in other words, we need one for each symbol in the input alphabet)): Any given symbol $\\sigma$ has a symbol-writing machine named $\\sigma$.
• Head-moving machines, named $R$, $L$ and $S$, that move the head appropriately.
• Transitions on anything other than (for example) $\\#$ are labeled $\\overline{\\#}$.
• Multiple copies of a machine get a superscript.",
"question":"$R^3$ means move the read/write head to the right 3 times.",
"answer":"True",
"choices": []
},
"first": {
"type":"select",
"description":"Let's look at an example.",
"question":"Which machine will be done first?",
"answer":"$M_1$",
"choices": ["$M_1$", "$M_2$", "$M_3$"]
},
"a": {
"type":"select",
"description":"Let's look at an example:",
"question":"Once $M_1$ ends execution, the head will be on some tape symbol. What will happen once $M_1$ completes if the current tape cell contains $a$?",
"answer": ["Start $M_2$"],
"choices": ["Start $M_1$", "Start $M_2$", "Start $M_3$"]
},
"b": {
"type":"select",
"description":"Let's look at an example:",
"question":"Once $M_1$ ends execution, the head will be on some tape symbol. What will happen once $M_1$ completes if the current tape cell contains $b$?",
"answer": ["Start $M_3$"],
"choices": ["Start $M_1$", "Start $M_2$", "Start $M_3$"]
},
"find": {
"type":"select",
"description":"Consider this example:",
"question":"What does this machine do?",
"answer":"Move the head right until an empty cell is found",
"choices": ["Move the head right until an empty cell is found", "Move the head right until a non-empty cell is found"]
},
"rpound": {
"type":"select",
"description":"Consider this example
We will use the notation $R_\\#$ for this process.",
"question":"What does $R_\\#$ machine do?",
"answer":"Move the head right until an empty cell is found",
"choices": ["Move the head right until an empty cell is found", "Move the head right until a non-empty cell is found"]
},
"emptyM": {
"type":"select",
"description":"Consider this example:",
"question":"What does this machine do?",
"answer":["Move the head left until an empty cell is found", "Once the empty cell is reached, the machine $M$ will run"],
"choices": ["Move the head left until an empty cell is found", "Once the empty cell is reached, the machine $M$ will run", "Accept the string", "Halt"]
},
"lpound": {
"type":"select",
"description":"Consider this example
We will use the notation $L_\\#$ for the Machine that will go left until it finds an empty cell. Thus, the two machines below do the same thing, with one using our new notation $L_\\#$.",
"question":"What do these machines do?",
"answer":["Move the head left until an empty cell is found", "Once the empty cell is reached, the machine $M$ will run"],
"choices": ["Move the head left until an empty cell is found", "Once the empty cell is reached, the machine $M$ will run"]
}
}
}
}