{
  "translations": {
    "en": {
      "membership": {
        "type": "select",
        "description": "The <b>Membership Problem</b> is to determine if a given string belongs to a given language (as defined by a given grammar).",
        "question": "If JAVA is the set of all syntactically correct Java programs, what is a Java program?",
        "answer": "A string in the set JAVA",
        "choices": ["A string in the set JAVA", "A set of words", "A set of lines of code"]
      },
      "java": {
        "type": "true/false",
        "description": "Consider the case where G is a grammar for Java, and $w$ is a Java program. Is $w$ a syntactically correct program? Answering this is (part of) what a compiler does. You write a program, you compile it, and the compiler finds all your syntax mistakes.",
        "question": "The process of compiling the Java source code and determining that the source code is a syntactically correct Java program is an example of the membership problem",
        "answer": "True",
        "choices": []
      },
      "evena": {
        "type": "select",
        "description": "We have seen that if we can find a derivation from $G$ for $w$, then we know that $w$ is $\\in L(G)$.",
        "question": "Suppose that $G = (\\{S\\}, \\{a, b\\}, S, P),$<br/>$P = S \\rightarrow SS\\ |\\ aSa\\ |\\ b\\ |\\ \\lambda$. What is $L(G)$?",
        "answer": "The set of strings with an even number of $a$'s",
        "choices": ["The set of strings with an even number of $a$'s", "The set of strings with one $b$", "$\\{a^nba^n \\mid n \\ge 0\\}$"]
      },
      "abbab": {
        "type": "select",
        "description": "We have seen that if we can find a derivation from $G$ for $w$, then we know that $w$ is $\\in L(G)$.",
        "question": "Suppose that $G = (\\{S\\}, \\{a, b\\}, S, P),$<br/>$P = S \\rightarrow SS\\ |\\ aSa\\ |\\ b\\ |\\ \\lambda$. Is $abbab \\in L$?",
        "answer": "Yes",
        "choices": ["Yes", "No"]
      },
      "onestep": {
        "type": "select",
        "description": "The Exhaustive Search Algorithm is<br/> For all $i = 1, 2, 3, \\ldots$<br/>Examine all sentential forms yielded by $i$ substitutions.<br/>Suppose that $G = (\\{S\\}, \\{a, b\\}, S, P), \\quad P = S \\rightarrow SS\\ |\\ aSa\\ |\\ b\\ |\\ \\lambda$.",
        "question": "Use exhaustive search to see if $abbab \\in L$. First select all sentential forms that result from making one substitution from $S$.",
        "answer": ["$S \\Rightarrow SS$", "$S \\Rightarrow aSa$", "$S \\Rightarrow b$", "$S \\Rightarrow \\lambda$"],
        "choices": ["$S \\Rightarrow SS$", "$S \\Rightarrow aSa$", "$S \\Rightarrow b$", "$S \\Rightarrow \\lambda$", "$S \\Rightarrow SS \\Rightarrow SSS$", "$S \\Rightarrow SS \\Rightarrow aSaS$", "$S \\Rightarrow SS \\Rightarrow bS$", "$S \\Rightarrow SS \\Rightarrow S$", "$S \\Rightarrow SS \\Rightarrow aSSa$"]
      },
      "twostep": {
        "type": "select",
        "description": "The Exhaustive Search Algorithm is<br/> For all $i = 1, 2, 3, \\ldots$<br/>    Examine all sentential forms yielded by $i$ substitutions.<br/>Suppose that $G = (\\{S\\}, \\{a, b\\}, S, P), \\quad P = S \\rightarrow SS\\ |\\ aSa\\ |\\ b\\ |\\ \\lambda$.",
        "question": "Use exhaustive search to see if $abbab \\in L$. Now select all sentential forms that result from making two substitutions from $S$. Which of these can be reached in two substitutions?",
        "answer": ["$S \\Rightarrow SS \\Rightarrow SSS$", "$S \\Rightarrow SS \\Rightarrow aSaS$", "$S \\Rightarrow SS \\Rightarrow bS$", "$S \\Rightarrow SS \\Rightarrow S$", "$S \\Rightarrow SS \\Rightarrow SaSa$"],
        "choices": ["$S \\Rightarrow SS$", "$S \\Rightarrow aS$", "$S \\Rightarrow b$", "$S \\Rightarrow \\lambda$", "$S \\Rightarrow SS \\Rightarrow SSS$", "$S \\Rightarrow SS \\Rightarrow aSaS$", "$S \\Rightarrow SS \\Rightarrow bS$", "$S \\Rightarrow SS \\Rightarrow S$", "$S \\Rightarrow SS \\Rightarrow SaSa$", "$S \\Rightarrow SS \\Rightarrow aSaaSa$"],
        "correctFeedback": ["Of course, this does not show all of the possible Sentential forms that can be reached in two substitutions. There is a good reason why this algorithm is called brute force! And it is terribly inefficient."]
      },
      "numstep": {
        "type": "select",
        "description": "The Exhaustive Search Algorithm is<br/> For all $i = 1, 2, 3, \\ldots$<br/>    Examine all sentential forms yielded by $i$ substitutions.<br/>Suppose that $G = (\\{S\\}, \\{a, b\\}, S, P), \\quad P = S \\rightarrow SS\\ |\\ aSa\\ |\\ b\\ |\\ \\lambda$.<br/>The derivation for $abbab$ is:<br/> $S \\Rightarrow SS \\Rightarrow aSaS \\Rightarrow aSSaS \\Rightarrow abSaS \\Rightarrow abbaS \\Rightarrow abbab$",
        "question": "How many derivation steps were needed to reach the derivation of $abbab$?",
        "answer": "6",
        "choices": ["4", "5", "6", "7"]
      },
      "lambda": {
        "type": "true/false",
        "description": "If the string is in the language, the search algorithm will find it. But what happens if $w$ is not in $L(G)$?<br/><br/>When do we stop the loop in the algorithm and know for sure that $w$ is not going to be derived?",
        "question": "Suppose that<br/>$G = (\\{S\\}, \\{a, b\\}, S, P), \\quad P = S \\rightarrow SS\\ |\\ aSa\\ |\\ b\\ |\\ \\lambda$. Suppose we have the following sentential form: $S \\Rightarrow SS \\Rightarrow ... \\Rightarrow SSSSSSSSSS \\Rightarrow ...$. Substitutions can be made on $SSSSSSSSSS$ to derive $\\lambda$",
        "answer": "True",
        "choices": [],
        "correctFeedback": ["This means that the brute force algorithm can enter an infinite loop when trying to derive a string."]
      },
      "length": {
        "type": "true/false",
        "description": "$\\textbf{Theorem:}$ If CFG $G$ does not contain rules of the form<br/>$A \\rightarrow \\lambda$<br/>$A \\rightarrow B$<br/>where $A,B\\in V$, then we can determine if $w\\in L(G)$ or if $w \\not \\in L(G)$.",
        "question": "Consider the length of the sentential form. If the grammar follows the theorem, then the length of the sentential form will only increase or stay the same after every substitution.",
        "answer": "True",
        "choices": []
      },
      "terminals": {
        "type": "true/false",
        "description": "$\\textbf{Theorem:}$ If CFG $G$ does not contain rules of the form<br/>$A \\rightarrow \\lambda$<br/>$A \\rightarrow B$<br/>where $A,B\\in V$, then we can determine if $w\\in L(G)$ or if $w \\not \\in L(G)$.",
        "question": "Consider the number of terminal symbols in a sentential form. If the grammar follows the theorem, then the number of terminals in the sentential form will only increase or stay the same after every substitution.",
        "answer": "True",
        "choices": []
      },
      "maxstep": {
        "type": "select",
        "description": "Following this theorem, either the length of the sentential form or the number of terminal symbols in the sentential form will increase with each derivation",
        "question": "If $|w| = n$, how many times through the loop will the exhaustive algorithm take to find if the string $w \\in L$ or not?",
        "answer": "$\\le 2n$",
        "choices": ["$\\le 2n$", "$\\ge 2n$", "$2n$", "$n^2$"],
        "correctFeedback": ["Every substitution must either increase the number of terminals or the number of variables. And neither the number of terminals or the number of variables can exceed $n$ in a successful derivation."]
      },
      "double": {
        "type": "select",
        "description": "If $|w| = n$, how many times through the loop can the exhaustive search take to find if $w \\in L$?",
        "question": "If we go $>2|w|$ times through loop without deriving the string, then...",
        "answer": "$w \\not \\in L(G)$",
        "choices": ["$w \\in L(G)$", "$w \\not \\in L(G)$"]
      },
      "goodgram": {
        "type": "true/false",
        "description": "$\\textbf{Example}$ Let $L_2 = L_1 - \\{\\lambda\\}$. $L_2 = L(G)$ where $G$ is:<br/>$S \\rightarrow SS\\ |\\ aa\\ |\\ aSa\\ |\\ b$",
        "question": "The Grammar does not contain rules of the form<br/>$A \\rightarrow \\lambda$<br/>$A \\rightarrow B$.",
        "answer": "True",
        "choices": []
      },
      "baaba": {
        "type": "select",
        "description": "$\\textbf{Example}$ Let $L_2 = L_1 - \\{\\lambda\\}$. $L_2 = L(G)$ where $G$ is:<br/>$S \\rightarrow SS\\ |\\ aa\\ |\\ aSa\\ |\\ b$",
        "question": "Let $w = baaba$, is $w \\in L$?",
        "answer": "No",
        "choices": ["No", "Yes"]
      },
      "fail": {
        "type": "select",
        "description": "$\\textbf{Example}$ Let $L_2 = L_1 - \\{\\lambda\\}$. $L_2 = L(G)$ where $G$ is:<br/>$S \\rightarrow SS\\ |\\ aa\\ |\\ aSa\\ |\\ b$",
        "question": "We know that the string is not in the language if we have not derived it within how many substitutions?",
        "answer": "10",
        "choices": ["5", "10", "12", "8"]
      }
    }
  }
}