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TEST 3 COMMENTS
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1 a) 4 for a reasonable algorithm. 2 for justifying that the algorithm
is polytime--usually, the complexity of the code is enough.

1 b) 2 for showing that 3-colour is in P. 2 for saying that 3-colour in P
and NPC implies N=P. Note, it is necessary that 3-colour is in NPC and
not just NP. Furthermore, NP neq P implies that P is a proper subset of
NP.  So any language can still be in P and NP.

2 a) 1 for guess, 1 for check.

b) 2 for B even case. For B odd case, 1 each for V',E',B'. 1 for
correctness condition.

3 a) 1 for saying the statement is true. 1 for stating what f is. 2 for
proper construction of the TM.

3 b) 1 for saying the statement is false. 3 for a valid counter-example.
Unfortunately, only one student got full marks for this question.

3 c) 1 for true. 2 for running the TMs in parallel. 1 for correct
acceptance and rejection.

There are no marks awarded for proving something that is false or giving a
counter-example to something that is true.