{ "translations": { "en": { "language": { "type": "select", "description": "We will show how to convert the following right-regular grammr to a left-regular grammar.", "question": "What is L(G)?", "answer": [ "$a^*cbb^*$", "$\\{a^ncb^m \\mid n\\ge 0, m>0\\}$" ], "choices": [ "$a^*cbb^*$", "$\\{a^ncb^m \\mid n\\ge 0, m>0\\}$" ] }, "reverse": { "type": "select", "description": "The next step in our algorithm is to REVERSE the NFA.", "question": "How can we reverse an NFA?", "answer": [ "This means that we need to filp all transitions", "Swap the start and final states" ], "choices": [ "This means that we need to filp all transitions", "Swap the start and final states" ] }, "loops": { "type": "select", "description": "Lest us start by reversing the transitions.", "question": "What is the result of reversing the transitions $\\delta(S, a) = S$, and $\\delta(B, b) = B$?", "answer": [ "The edge that represent $\\delta(S, a) = S$ will not change as it is a loop.", "The edge that represent $\\delta(B, b) = B$ will not change as it is a loop." ], "choices": [ "The edge that represent $\\delta(S, a) = S$ will not change as it is a loop.", "The edge that represent $\\delta(B, b) = B$ will not change as it is a loop." ] }, "SB": { "type": "multiple", "description": "Lest us start by reversing the transitions.", "question": "What is the result of reversing the transition $\\delta(S, c) = B$?", "answer": "$\\delta(B, c) = S$", "choices": [ "$\\delta(S, a) = S$", "$\\delta(B, b) = B$", "$\\delta(B, c) = S$", "$\\delta(S, c) = B$", "$\\delta(B, b) = F$", "$\\delta(F, b) = B$" ] }, "BF": { "type": "multiple", "description": "Lest us start by reversing the transitions.", "question": "What is the result of reversing the transition $\\delta(B, b) = F$?", "answer": "$\\delta(F, b) = B$", "choices": [ "$\\delta(S, a) = S$", "$\\delta(B, b) = B$", "$\\delta(B, c) = S$", "$\\delta(S, c) = B$", "$\\delta(B, b) = F$", "$\\delta(F, b) = B$" ] }, "swapSF": { "type": "multiple", "description": "All edges are reversed.", "question": "What is the next step?", "answer": "Swap the start and Final states", "choices": [ "Done, the NFA is reversed.", "Swap the start and Final states" ] }, "onefinal": { "type": "multiple", "description": "Convert the original final state to be the initital state, and the original initial state to be final.", "question": "In case that the NFA has multiple final states. How can we convert them to a start state?", "answer": "Convert the NFA with multiple final states to an NFA with one final state. Then make the new final state a start state.", "choices": [ "Convert every final state to a start state", "Convert the NFA with multiple final states to an NFA with one final state. Then make the new final state a start state." ] }, "nfa2rr": { "type": "multiple", "description": "We already have seen an algorithm to convert from a NFA to a right-regular grammar. We will use it to convert this NFA to the right-regular grammar for $L^R$.", "question": "What is the production equivelent to the transition $\\delta(S, a) = S$? (Keep in mind that we are making a right-regular grammar.)", "answer": "$S\\rightarrow aS$", "choices": [ "$S\\rightarrow aS$", "$S\\rightarrow Sa$" ] }, "rrBB": { "type": "multiple", "description": "We already have seen an algorithm to convert from a NFA to a right-regular grammar. We will use it to convert this NFA to the right-regular grammar for $L^R$.", "question": "What is the production equivelent to the transition $\\delta(B, b) = B$?", "answer": "$B\\rightarrow bB$", "choices": [ "$B\\rightarrow bB$", "$B\\rightarrow Bb$" ] }, "rrBS": { "type": "multiple", "description": "We already have seen an algorithm to convert from a NFA to a right-regular grammar. We will use it to convert this NFA to the right-regular grammar for $L^R$.", "question": "What is the production equivelent to the transition $\\delta(B, c) = S$?", "answer": "$B\\rightarrow cS$", "choices": [ "$B\\rightarrow Sc$", "$B\\rightarrow cS$" ] }, "rrFB": { "type": "multiple", "description": "We already have seen an algorithm to convert from a NFA to a right-regular grammar. We will use it to convert this NFA to the right-regular grammar for $L^R$.", "question": "What is the production equivelent to the transition $\\delta(F, b) = B$?", "answer": "$F\\rightarrow bB$", "choices": [ "$F\\rightarrow bB$", "$F\\rightarrow Bb$" ] }, "result": { "type": "multiple", "description": "We already have seen an algorithm to convert from a NFA to a right-regular grammar. We will use it to convert this NFA to the right-regular grammar for $L^R$.", "question": "The resulting grammar is for what language?", "answer": "The resulting grammar is the right-regular grammar for the reverse of the original Language", "choices": [ "The resulting grammar is the right-regular grammar for the original Language", "The resulting grammar is the right-regular grammar for the reverse of the original Language" ] }, "revSS": { "type": "multiple", "description": "Reversing each production in a right-regular grammar will give a left-regular grammar for the reverse language. So, doing that here will yield the result that we are looking for: a left-regular grammar for $(L^R)^R = L$.", "question": "What is the result of reversing $S\\rightarrow aS$?", "answer": "$S\\rightarrow Sa$", "choices": [ "$S\\rightarrow aS$", "$S\\rightarrow Sa$" ] }, "revBB": { "type": "multiple", "description": "By reversing each production in the resulting Right Linear grammar, we will get the the Left Linear grammar for the reverse of $L^R$. So, we have the Left Linear grammar for $(L^R)^R = L$.", "question": "What is the result of reversing $B\\rightarrow bB$?", "answer": "$B\\rightarrow Bb$", "choices": [ "$B\\rightarrow bB$", "$B\\rightarrow Bb$" ] }, "revBS": { "type": "multiple", "description": "By reversing each production in the resulting Right Linear grammar, we will get the the Left Linear grammar for the reverse of $L^R$. So, we have the Left Linear grammar for $(L^R)^R = L$.", "question": "What is the result of reversing $B\\rightarrow cS$?", "answer": "$B\\rightarrow Sc$", "choices": [ "$B\\rightarrow Sc$", "$B\\rightarrow cS$" ] }, "revFB": { "type": "multiple", "description": "By reversing each production in the resulting Right Linear grammar, we will get the the Left Linear grammar for the reverse of $L^R$. So, we have the Left Linear grammar for $(L^R)^R = L$.", "question": "What is the result of reversing $F\\rightarrow bB$?", "answer": "$F\\rightarrow Bb$", "choices": [ "$F\\rightarrow bB$", "$F\\rightarrow Bb$" ] }, "resulttype": { "type": "multiple", "description": "Done. We reversed all productions", "question": "What is the type of the resulting grammar?", "answer": "Left-Regular Grammar", "choices": [ "Right-Regular Grammar", "Left-Regular Grammar" ] } } } }