{ "translations": { "en": { "nfadfa": { "type": "multiple", "description": "The two fundamental differences between Non-Deterministic Finite Automata (NFA) and Deterministic Finite Automata (DFA) are that (1) the NFA may have multiple edges coming out of a given node with the same symbol, and (2) the NFA may have a transition on $\\lambda$ (the empty string).", "question": "Is every DFA an NFA?", "answer": [ "Yes" ], "choices": [ "Yes", "No" ], "correctFeedback": ["Just because an NFA is permitted to have a state in which there are choices on what to do when it sees a given letter, that does not mean that it must do that. So, every DFA is also an NFA. This will be important later."] }, "def": { "type": "multiple", "description": "Define a NFA as $(Q, \\Sigma, \\delta, Q_0, F)$ where
$Q$ is a finite set of states
$\\Sigma$ is the input alphabet (a finite set)
$\\delta: Q \\times(\\Sigma \\cup \\{\\lambda\\}) \\rightarrow 2^Q$ ($2^Q$ here means the power set of $Q$)
$Q_0$ is the intial state ($q_0 \\in Q$)
$F \\subseteq Q$ is a set of final states. ", "question": "What part of the definition for the NFA is different from definition for the DFA?", "answer": [ "$\\delta$" ], "choices": [ "$Q$", "$\\Sigma$", "$F$", "$Q_0$", "$\\delta$", "nothing" ], "correctFeedback": ["$\\delta$ changes, so that instead of transitioning to a single state when we see a symbol, the machine is allowed to transition to any set of states. It also changes to allow $\\lambda$ as a possible transition."] }, "multi": { "type": "multiple", "description": "While the result of $\\delta$ for the DFA is some state $q \\in Q$, the result of $\\delta$ for the NFA is any subset of Q.", "question": "The NFA can have a state with 2 transitions for the same input symbol.", "answer": [ "True" ], "choices": [ "True", "False" ] }, "moretrans": { "type": "multiple", "description": "While the result of $\\delta$ for the DFA is some state $q \\in Q$, the result of $\\delta$ for the NFA is any subset of Q.", "question": "The NFA can have 3 or more transitions from a state using the same input symbol.", "answer": [ "True" ], "choices": [ "True", "False" ] }, "choices": { "type": "multiple", "description": "Non-deterministic means that it allows choices. Given input $b$ from state $q$, $\\delta$ might include transitions to more than one state.", "question": "The NFA's state can always be determined.", "answer": [ "False" ], "choices": [ "True", "False" ], "correctFeedback": ["Though it depends a little bit on what you mean by 'determined'. You can't know at the time which state the machine will transition to when there are choices. But you can know the set of possible choices that the machine is allowed to transition to."] }, "nfaexamp": { "type": "select", "description": "In the NFA below, there are two choices when the machine sees 'a' while in state q0. So, $\\delta(q0, a) = \\{q1, q2\\}$.", "question": "What are the possible states to transition to when the machine is in q0 and sees the symbol 'a'?", "answer": [ "q1", "q2" ], "choices": [ "q0", "q1", "q2", "q3", "the trap state (reject the string)" ] }, "badb": { "type": "select", "description": "In the NFA below, there are two choices when the machine sees 'a' while in state q0. So, $\\delta(q0, a) = \\{q1, q2\\}$.", "question": "What are the possible states to transition to when the machine is in q0 and sees the symbol 'b'?", "answer": [ "the trap state (reject the string)" ], "choices": [ "q0", "q1", "q2", "q3", "the trap state (reject the string)" ] }, "lambda": { "type": "multiple", "description": "A $\\lambda$ transition is a 'free ride' to another state without any input.", "question": "A $\\lambda$ transition does not consume a symbol when it moves to the new state.", "answer": [ "True" ], "choices": [ "True", "False" ] }, "multilambda": { "type": "multiple", "description": "A $\\lambda$ transition is a 'free ride' to another state without any input.", "question": "A $\\lambda$ transition cannot be immediately followed by another $\\lambda$ transition to yet another state.", "answer": [ "False" ], "choices": [ "True", "False" ], "correctFeedback": ["Potentially, the machine might allow a whole series of 'free ride' $\\lambda$ transitions before the next symbol is processed."] }, "starting": { "type": "select", "description": "A $\\lambda$ transition is a 'free ride' to another state without any input.", "question": "Which state(s) might this NFA be in when it processes the first input symbol?", "answer": [ "q0", "q1", "q2" ], "choices": [ "q0", "q1", "q2", "q3", "q4", "q5", "q6", "the trap state (reject the string)" ], "incorrectFeedback": ["The answer we want is all states that the $\\lambda$ transitions might take you to before processing the first input symbol. Beware: Even if a state does not show a transition out of a given symbol, there might be a default transition to the trap state."] }, "firstletter": { "type": "select", "description": "A $\\lambda$ transition is a 'free ride' to another state without any input.", "question": "Which state(s) can the NFA be in after processing the first letter of a string that starts with 'a'?", "answer": [ "q3", "q4", "the trap state (reject the string)" ], "choices": [ "q0", "q1", "q2", "q3", "q4", "q5", "q6", "the trap state (reject the string)" ] }, "walk": { "type": "select", "description": "Definition: $q_j \\in \\delta^*(q_i, w)$ if and only if there exists some walk from $q_i$ to $q_j$ using $w$ where $w$ is a string. Note that in this definition, while $w$ is a string, it is not necessarily the entire input to the machine. This definition just refers to states that can be reached from $q_i$ when processing $w$. And if $w$ is three symbols, then that means transitioning over (at least) three edges (one for each symbol, and maybe some $\\lambda$ transitions).", "question": "What is $\\delta^*(q_0, ab)$?", "answer": [ "q5", "q6", "q1", "the trap state (reject the string)" ], "choices": [ "q0", "q1", "q2", "q3", "q4", "q5", "q6", "the trap state (reject the string)" ], "incorrectFeedback": ["The answer includes every state that the machine can be in after processing 'ab'. Note that this includes the possibility of taking $\\lambda$ transitions, either before, during, or after processing 'ab'."] }, "formalaccept": { "type": "multiple", "description": "We now have the notation and concepts that we need to provide a formal definition for the language of a NFA. For any NFA $M$,
$L(M)= \\{w \\in {\\Sigma}^{*} \\mid \\delta^{*}(q_0,w) \\cap F \\neq \\emptyset \\}$.
This means that the machine accepts all strings $w$ from the set of all possible strings (generated from the alphabet $\\Sigma$) such that some state reachable on $w$ from the start state ($q_0$) is a final state.", "question": "When $w =ab$, how many accepting paths exist?", "answer": [ "2" ], "choices": ["0", "1", "2","3", "4", "5", "6", "7", "8"], "correctFeedback": ["The machine can follow the path $q0, q1, q3, q5$ or it can follow the path $q0, q2, q4, q6$. It could also follow the path $q0, q1, q3, q5, q1$, but that is not an accepting path."] }, "acceptreject": { "type": "multiple", "description": "The NFA tries every acceptable walk from the start state for the given input. From that set of possible paths, if any ends in a state that is final, then the NFA will ACCEPT the input. If that set is empty (not possible to make a path using this input), or those path(s) do not end in a final state, the NFA will REJECT.", "question": "A NFA ACCEPTS when every possible walk for the input string ends in a final state.", "answer": [ "False" ], "choices": [ "True", "False" ], "correctFeedback": ["Correct! If 10 possible walks exist, and only one ends in a final state, then the NFA will ACCEPT. It ignores all failure walks."] }, "nfaex": { "type": "multiple", "description": "Let's start to figure out how to convert a NFA to a DFA. Consider this NFA.", "question": "What language does this NFA accept?", "answer": "Strings of alternating a's and b's that start with a.", "choices": [ "Strings of alternating a's and b's that start with a.", "Strings of alternating a's and b's that start with b.", "Strings of a's and b's.", "Strings of alternating a's and b's.", "Strings of alternating a's and b's that start and end with a.", "Strings of alternating a's and b's that start with a and end with b." ] } } } }