Skill: Write DFA construc4on proofs of closure proper4es. Learning Objec4ves: This slide comes aCer we have had a mini-‐lecture introducing the Theorem, and exploring how it might be proved by performing the full construc4on on a specific working example. Students oCen struggle to transi4on from their specific working examples, which are important in the brainstorming stage of proof-‐wri4ng, to a generalized proof that holds for all possible Regular Languages. This series of slides guides them through that transi4on. OCen they start out selec4ng distractor answer choices that reference the working example, and the experience of doing that realizing that is not correct seems to help cement the dis4nc4on. During the debriefing, the visual separa4on between the leC and right halves of the slide is emphasized. Correct choice: (c) Answer choice/distractor design: (a) Has the wrong domain and range (b) Has the wrong range (c) Correct (though in subsequent 4mes I have taught this course, I have revised this
answer choice to include clarifica4on of the variables x,y,c, as follows: δ((x,y),c) = (δ1(x,c), δ2(y,c)), for c in Σ and (x,y) in Q)
(d) This correctly transcribes one of the edges in the specific working example, however, we are no longer supposed to be referring to the working example at
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Correct choice: (d) Answer choice/distractor design: (a) This is correct for the specific working example, but we are moving away from
that and now doing a generalized proof. (b) This would be correct if we were trying to show closure under Intersec4on. (c) Distrac4ng because it uses the Union operator, however, it is not correct. (d) Correct
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Skill: Trace in a DFA. Learning objec4ve: This slide reviews basic tracing in a DFA, and prepares students for the Pumping Lemma, which applies the Pigeonhole Principle to the length of strings a given DFA is capable of accep4ng without visi4ng a state more than once. Correct answer: (b) Answer choices/distractor design: The distractors are just several numbers in the range. However, 4 is typically the most “distrac4ng” because there is a fencepost coun4ng issue between the number of states (4) and the longest path (3).
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The answer is (b). Skill: Tracing in a PDA using the stack. Learning objec4ve: This slide is introducing the PDA. The main difference from NFA (which they have already done) is the addi4on of a stack, so this slide focuses squarely on that new addi4on. Answer choices/distractor design: (a) This is the input string, not a state the stack is ever in. This emphasizes drawing
the dis4nc4on between the input string, and the stack. The input is not on the stack (unless you put it there).
(b) The correct answer. (c) and (d) Tes4ng understanding of the transi4on rule nota4on—although lambda
and b appear in transi4on rules, they are never put on the stack. Design issues: My main hesita4on with this pair is possible confusion over the asterisk note. However, it actually doesn’t affect the answer—i.e. there is no way to get #lambda etc regardless of path followed or input.
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Skill: Understanding configura4on nota4on for TMs and how to apply a single transi4on rule to a configura4on. Learning Objec4ves: Typically students have several minor points of confusion with the configura4on nota4on, on things such as: (1) when the head is wrigen as between two characters, does it “read” the one to the leC or right, (2) when the head moves to the leC or right, does it write the character that used to be to its leC, or that is now to its leC? Correct choice: (c) Distractor choices address minor points of confusion described above.
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Skill: Know when to use Pumping Lemma Learning Objec4ves: As part of midterm review, we did this pair and several others, helping students understand which kind of approach is appropriate in different situa4ons. Students oCen get to a point where they can correctly execute a given approach, but may have a “lose sight of the forest for the trees” type issue in terms of not being able to iden4fy when different approaches apply. Correct choice: (d) Distractor choices are other kinds of proofs we have covered in the course.
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