Marking Comments for Problem Set 1   (Graded by Matei David)
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Question 1
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Most people got lots of marks for this one. I do not have a Turing Machine
verification mechanism built inside my brain, so there was no way I could
have completely verified your otherwise very ingenious transition
functions. In general, I was looking for two details. One was a mechanism
to carry the addition bit, the other was a mechanism to provide for
potentially zero padded binary strings. Expect to lose 2 marks for omitting
either of those. There were 10/20 points allocated for informal
descriptions and one which would not convince me would as well lose some
marks. This applies particularly to those squeezed on the right side of the
picture with the transition function. I was displeased with exorbitant
numbers of states, like 20 or more.


Question 2
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You were asked for a "complete formal description of the simulation". I
didn't expect anything less than that. As a result, some people received at
most half of the marks (5/10) for only giving correct and even clear
informal descriptions. Due to some misunderstanding between the TAs, there
were two versions of this problem which people have tried to solve, one
with a 1-way infinite tape, the other with a 2-way infinite tape. I have
taken that situation into account when marking.


Question 3
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You were asked for an *injective* mapping from N^3 into N *only*. There
were a number of simple answers to this. In general I was suspicious of
solutions which were trying to give a bijective mapping simply because that
meant you haven't really understood the simplicity of the question. Some
were able to give a correct description of the latter type of function, but
most who attempted it have lost marks for incomplete details. Pictorial
representations of lines going around a grid with no other explanations
would in general be lucky to get about 3 marks. A solution which would
propose a function which clearly does not work (I usually put down an
example) would get 0.


Question 4
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(a) There were 2 points for saying it's not recursive; 2 points for saying
it's re; 2 points for arguing it's re; and 4 points for proving non
recursiveness in some way. You would lose marks for omitting to talk about
the case a "bad" input is feeded to f.

(b) 5 points for saying it's recursive, 5 points for proving it. Only 2
points for saying that it's re. You will lose some of the latter 5 if there
are errors/omissions in the proof, you will even lose all of them if you
give a bogus argument which does not convince me that you really know how
to prove it. I also gave 2 marks for proof of re-ness but only if your
proof worked. In many such cases it didn't, remember the machine has to
halt and accept any input in the re language, not only accept if it halts
on such an input.

(c) 5 points for simply stating it's not re (that's a lot...) The other 5
towards the proof. You might lose marks for not using the reduction
mechanism and instead mixing up Turing recognition with Turing acceptance.
You would also lose 1 mark for not treating the "bad" input case.

(d) A total of 3 bonus points only, for observing part of the problem with
the listing - see the discussion in the solutions.

(e) Mostly like (b), you might get no marks at all for the proof if you do
not talk about running the machines in the list - which shows you didn't
get the question. Proving only re-ness is very strange, as it will
automatically give you recursiveness, whenever you do it right.


Question 5
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Generally done well, I took off 2 marks for not explaining how exactly to
do the simulation and instead using terms such as "in parallel",
"simlutaneously" etc. I was looking for dove-tailing or some variant of it.
You would lose more marks (at least 4) for not even hinting at the need to
run both machines somehow together, ie, not observing that we cannot afford
to wait until one of them halts.