{
"translations": {
"en": {
"reachC":{
"type": "select",
"description": "Consider this grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
What can you say about this grammar in terms of the usefulness of its variables?",
"question": "First think about the variable $C$. Can we reach variable $C$ from $S$ directly or indirectly?",
"answer": "No",
"choices": ["No", "Yes"]
},
"useless":{
"type": "select",
"description": "Consider this grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
What can you say about this grammar in terms of the usefulness of its variables?",
"question": "Is $C$ a useful or useless variable?",
"answer": "useless",
"choices": ["useless","useful"]
},
"remove":{
"type": "select",
"description": "Consider this grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
What can you say about this grammar in terms of the usefulness of its variables?",
"question": "Should we keep the variable $C$?",
"answer": "No, because it is useless variable.",
"choices": ["Yes, it might be useful later.", "No, because it is useless variable."]
},
"impossible":{
"type": "select",
"description": "Consider this grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
What can you say about this grammar in terms of the usefulness of its variables?",
"question": "Can you generate any string from this grammar?",
"answer": "No",
"choices": ["Yes", "No"],
"correctFeedback": ["There is no way to replace all of the variables with terminals, so we can never actually derive a string."],
"incorrectFeedback": ["How can we remove all of the variables during a derivation?"]
},
"alluseless":{
"type": "true/false",
"description": "Consider this grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
What can you say about this grammar in terms of the usefulness of its variables?",
"question": "$A$ , $S$, and $B$ are useless variables since they can’t derive a string of terminals.",
"answer": "True",
"choices": []
},
"nochange":{
"type": "true/false",
"description": "Theorem: (useless productions): Let $G$ be a CFG. Then there exists $G'$ that does not contain any useless variables or productions such that $L(G) = L(G')$.",
"question": "Removing useless productions will affect the grammar's language.",
"answer": "False",
"choices": []
},
"unreachable":{
"type": "select",
"description": "We have seen two types of useless productions.
1. Productions that are unreachable. These are the productions for variables that we cannot reach from the start variable.",
"question": "Consider the grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
Which variable is unreachable?",
"answer": "C",
"choices": ["A","S","C","B"]
},
"nonterm":{
"type": "select",
"description": "We have seen two types of useless productions.
1. Productions that are unreachable. These are the productions for variables that we cannot reach from the start variable.
2. Productions that are nonterminating, meaning that these productions will keep looping with the same/other productions without termination (so, no strings can be generated from a sentential form that contains these variables).",
"question": "Consider the grammar:
$S \\rightarrow aB \\mid bA$
$A \\rightarrow aA$
$B \\rightarrow Sa$
$C \\rightarrow cBc \\mid a$
Which variables are nonterminating?",
"answer": ["A","S","B"],
"choices": ["A","S","C","B"]
},
"initv1":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$.
1. Compute $V_1 =$ {Variables that can derive strings of terminals}
$\\quad 1.\\ V_1 = \\emptyset$
$\\quad 2.$ Repeat until no more variables are added:
$\\quad\\quad$ For every $A \\in V$ with $A\\rightarrow x_1x_2\\ldots x_n$, $x_i \\in (T^* \\cup V_1)$, add $A$ to $V_1$.
In other words, on each iteration add all variables with at least one production that contains only terminals and variables already in $V_1$.",
"question": "For the grammar on the left, what is the initial value for $V_1$?",
"answer": "$\\emptyset$",
"choices": ["A","S","C","B","$\\emptyset$","E","D"]
},
"firstiter":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$.
1. Compute $V_1 =$ {Variables that can derive strings of terminals}
$\\quad 1.\\ V_1 = \\emptyset$
$\\quad 2.$ Repeat until no more variables are added:
$\\quad\\quad$ For every $A \\in V$ with $A\\rightarrow x_1x_2\\ldots x_n$, $x_i \\in (T^* \\cup V_1)$, add $A$ to $V_1$.
In other words, on each iteration add all variables with at least one production that contains only terminals and variables already in $V_1$.",
"question": "What variable(s) should we add to $V_1$ in the first iteration?",
"answer": ["B","C","E"],
"choices": ["A","S","C","B","$\\emptyset$","E","D"],
"incorrectFeedback": ["We want to add any variables with at least one production whose RHS contains only terminals and variables that are already in $V_1$."]
},
"seconditer":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$.
1. Compute $V_1 =$ {Variables that can derive strings of terminals}
$\\quad 1.\\ V_1 = \\emptyset$
$\\quad 2.$ Repeat until no more variables are added:
$\\quad\\quad$ For every $A \\in V$ with $A\\rightarrow x_1x_2\\ldots x_n$, $x_i \\in (T^* \\cup V_1)$, add $A$ to $V_1$.
In other words, on each iteration add all variables with at least one production that contains only terminals and variables already in $V_1$.",
"question": "What should we add to $V_1$ in the next iteration?",
"answer": ["S","D"],
"choices": ["A","S","C","B","$\\emptyset$","E","D"],
"incorrectFeedback": ["We want to add any variables with at least one production whose RHS contains only terminals and variables that are already in $V_1$. Look at what is in $V_1$ now."]
},
"lastiter":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$.
1. Compute $V_1 =$ {Variables that can derive strings of terminals}
$\\quad 1.\\ V_1 = \\emptyset$
$\\quad 2.$ Repeat until no more variables are added:
$\\quad\\quad$ For every $A \\in V$ with $A\\rightarrow x_1x_2\\ldots x_n$, $x_i \\in (T^* \\cup V_1)$, add $A$ to $V_1$.
In other words, on each iteration add all variables with at least one production that contains only terminals and variables already in $V_1$.",
"question": "What should we add to $V_1$ in the next iteration?",
"answer": ["$\\emptyset$"],
"choices": ["A","S","C","B","$\\emptyset$","E","D"],
"correctFeedback": ["Since there is new variable to add, we add the empty set."],
"incorrectFeedback": ["Is there any new variable to add? If not, add the empty set."]
},
"doneiter":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$.
1. Compute $V_1 =$ {Variables that can derive strings of terminals}
$\\quad 1.\\ V_1 = \\emptyset$
$\\quad 2.$ Repeat until no more variables are added:
$\\quad\\quad$ For every $A \\in V$ with $A\\rightarrow x_1x_2\\ldots x_n$, $x_i \\in (T^* \\cup V_1)$, add $A$ to $V_1$.
In other words, on each iteration add all variables whose productions contain only terminals and variables already in $V_1$.",
"question": "Should we continue?",
"answer": "No because we did not add any variables in the last iteration",
"choices": ["Yes", "No because we did not add any variables in the last iteration"]
},
"whichprods":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$.
1. Compute $V_1 =$ {Variables that can derive strings of terminals}
$\\quad 1.\\ V_1 = \\emptyset$
$\\quad 2.$ Repeat until no more variables are added:
$\\quad\\quad$ For every $A \\in V$ with $A\\rightarrow x_1x_2\\ldots x_n$, $x_i \\in (T^* \\cup V_1)$, add $A$ to $V_1$.
$\\quad 3.\\ P_1=$ all productions in $P$ whose RHS contains only $(V_1\\cup T)^*$.",
"question": "Which productions should be added to $P_1$?",
"answer": ["$S\\rightarrow aB$", "$B\\rightarrow Sa$", "$B\\rightarrow b$", "$C\\rightarrow cBc$", "$C\\rightarrow a$", "$D\\rightarrow bCb$", "$E\\rightarrow b$"],
"choices": ["$S\\rightarrow aB$", "$S\\rightarrow bA$", "$A\\rightarrow aA$", "$B\\rightarrow Sa$", "$B\\rightarrow b$", "$C\\rightarrow cBc$", "$C\\rightarrow a$", "$D\\rightarrow bCb$", "$E\\rightarrow Aa$", "$E\\rightarrow b$"],
"incorrectFeedback": ["$P_1=$ all productions in $P$ whose RHS contains only $(V_1\\cup T)^*$. Hint: There are 7 correct productions."]
},
"doneyet":{
"type": "select",
"description": "The resulting grammar $G_1=(V_1,T,S,P_1)$ has no variables that can’t derive strings.",
"question": "Did we finish removing useless productions?",
"answer": "No",
"choices": ["No", "Yes"],
"correctFeedback": ["There are variables that we cannot reach in a derivation."],
"incorrectFeedback": ["Can we derive every one of the variables still in the set of production rules?"]
},
"howmanyvar":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$
2. Remove unreachable productions by using a Variable Dependency Graph (VDG)
First draw a node for every variable.",
"question": "How many nodes does the VDG have?",
"answer": "5",
"choices": ["1", "2","3","4","5","6","7"],
"incorrectFeedback": ["This is the same as asking: How many variables are in the grammar?"]
},
"sb":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$
2. Remove unreachable productions by using a Variable Dependency Graph (VDG)
First draw a node for every variable.
For each production $A\\rightarrow xBy$, draw edge $A \\rightarrow B$.",
"question": "What is the edge for the highlighted production?",
"answer": "S to B",
"choices": ["S to B", "B to S", "C to B", "D to C"]
},
"bs":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$
2. Remove unreachable productions by using a Variable Dependency Graph (VDG)
First draw a node for every variable.
For each production $A\\rightarrow xBy$, draw edge $A \\rightarrow B$.",
"question": "What is the edge for the highlighted production?",
"answer": "B to S",
"choices": ["S to B", "B to S", "C to B", "D to C"]
},
"cb":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$
2. Remove unreachable productions by using a Variable Dependency Graph (VDG)
First draw a node for every variable.
For each production $A\\rightarrow xBy$, draw edge $A \\rightarrow B$.",
"question": "What is the edge for the highlighted production?",
"answer": "C to B",
"choices": ["S to B", "B to S", "C to B", "D to C"]
},
"dc":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$
2. Remove unreachable productions by using a Variable Dependency Graph (VDG)
First draw a node for every variable.
For each production $A\\rightarrow xBy$, draw edge $A \\rightarrow B$.",
"question": "What is the edge for the highlighted production?",
"answer": "D to C",
"choices": ["S to B", "B to S", "C to B", "D to C"]
},
"whichremove":{
"type": "select",
"description": "An algorithm to remove useless productions: Let $G=(V,T,S,P)$
2. Remove unreachable productions by using a Variable Dependency Graph (VDG)
First draw a node for every variable.
For each production $A\\rightarrow xBy$, draw edge $A \\rightarrow B$.
Remove productions for any $V$ that has no path from $S$ to $V$ in the dependency graph.",
"question": "Which variables must be removed?",
"answer": ["E", "C", "D"],
"choices": ["E", "S", "B", "C", "D"],
"incorrectFeedback": ["Remove productions for any $V$ that has no path from $S$ to $V$ in the dependency graph."]
}
}
}
}