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CSC 363            Homework Exercise 4 -- Marking Scheme          Fall 2008
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NOTE TO STUDENTS:

You will find below the marking scheme used for your homework, including
the meaning of marking codes and number of marks associated with each one.
This file also contains my instructions to the marker (so you can get an
idea of how the homework was marked) and the marker's comments about each
question.

Please take the time to read this carefully before you ask questions about
the grading of your homework.


NOTE TO MARKER:

Be picky!  On any homework, it is the responsibility of students to show
that they understand how to solve each problem and to write up their
answers carefully.  At the same time, keep in mind that exercises are worth
only 1.5% of the final grade, so your marking should be somewhat coarse.
And remember that marking is not just about evaluating the students's
performances, but also about giving them feedback so that they can learn
from their mistakes.  This is especially important for students who made
numerous or more serious mistakes, as they are likely to need more feedback
in order to understand why their answers were incorrect.

For each question, I list solution elements with an associated code for
writing on student papers (the letter(s) between underscores _) and a
number of marks.  There are also general errors (with associated codes)
given below, with a maximum number of marks to take off for each type of
general error (as a percentage of the value of the question).  You will
likely encounter other common errors, or maybe decide to break down the
marking scheme further.  Simply make note of these changes/additions to
the marking scheme, and introduce new code letters (or short words) to
allow you to quickly give accurate feedback to the students (both in terms
of what they did wrong and how many marks it cost them).


GENERAL ERRORS (marked negatively, in addition to any other errors):

    _N_otation [up to 20%]:  incorrect/ambiguous notation

    _V_agueness [up to 20%]:  incorrect/unjustified/vague claim

Marker's Comments:  Nothing general for this exercise; see individual
    comments on your papers.


1.  [5 marks]

    _S_tructure [1 mark]:  correctly structured attempt to show A <=m C
        (starting from arbitrary string w, describe explicit construction
        of string f(w) and show that w (- A iff f(w) (- C); correct use of
        assumptions that A <=m B and B <=m C

        NOTE: both directions of the biconditional are necessary for the
        proof, i.e., your reduction must be described for all w (- A and
        also for all w !(- A (and you must argue about both).

    _R_eduction [2 marks]:  correct reduction function for A <=m C

    _C_orrectness [2 marks]:  good argument that reduction is correct


2.  [5 marks]

    _S_tructure [1 mark]:  correctly structured attempt to show L <=m A_TM
        (starting from arbitrary string w, describe explicit construction
        of string f(w) and show that w (- L iff f(w) (- A_TM); correct use
        of assumption that L is recognizable

    _R_eduction [2 marks]:  correct reduction function for L <=m A_TM

    _C_orrectness [2 marks]:  good argument that reduction is correct


3.  [5 marks]

    _S_tructure [2 marks]:  clear attempt to give a counter-example by
        exhibiting specific languages A, B such that A is non-regular,
        B is regular, and A <=m B (including correct structure for <=m)
    OR
        clear and well-structured attempt to give a proof of the statement
        (worth at most 1 mark)

    _L_anguages [1 mark]:  correct languages A, B clearly defined

    _R_eduction [2 marks]:  correct reduction A <=m B