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CSCI 8980, Computation and Deduction
Fall 2004, Homework 1
Comments on Grading

General Remarks

This assignment was graded out of 55 points. The points distribution was 5, 14, 6, 5, 9, 12 and 4, respectively. The mean score on the assignment was 50.18 and the standard deviation was 4.27. I put this information up only in case you are curious about your performance relative to the rest of the class. In reality, these homeworks constitute only a quarter of the total points---the term paper and the presentation are the main things in this course---and nobody's performance on the homework was such that it will adversely impact on the final grade.

Problem 1

This main thing in this problem was to prove the indicated "lemma." The only thing to comment on is the fact that the lemma as stated, i.e.
Show that for any interpretation that yields a certain truth value for F[X_1,...X_n], there is an interpretation that yields the same truth value for F[A_1,...,A_n],
should not itself become the thing that you show by induction. Rather, you have to strengthen what you show a bit. In particular, for the interpretation relative to which you have determined a truth value for F[X_1,...,X_n], you have to first define another interpretation (that assigns values to A_1,...,A_n) and then show by induction that this interpretation leads to the same truth value for F[A_1,...,A_n]. The stronger form for the statement proved by induction is required. Some of you did not realize this, got stuck as a result in the proof and then, when bailing out, used incorrect arguments.

Problem 2

The points for this problem were divided as follows: 2 points were reserved for each of the six (classical proofs) and 2 points for explaining which sequents were and were not intuitionistically provable. When a sequent is intuitionistically provable, you can simply demonstrate the proof and, to do this, it was best to have written down the intuitionistic sequent or natural deduction proof. In the case of a negative answer, an explicit argument that an intuitionistic proof does not exist must be given---we discussed this in a HyperNews post pertaining to this homework. To construct this proof, you will have to examine all the rules that could possibly have yielded the end sequent and then consider the upper sequents of that rule, etc. This is possible to do in a finitary fashion since we do not have cut (for which there are infinite manifestations) in our system. However, since we cannot throw out contraction entirely in intuitionistic logic, we will have to notice some pattern to keep our argument finite. Which formulas are not provable in intuitionistic logic? Only the last one. This formula actually has a special name: it is called Pierce's formula. If you did not actually present an argument to show that this was not provable, you may try this now and ask for help if you still cannot get it. (Derek will talk about ways of deciding if formulas have intuitionistic proofs so this exercise will be a lead into understanding what he will say.)

Problem 3

Each part in this problem carried 3 points. We have talked about this in class, so there is not much to be said about the proof that was expected. One answer avoided the inductive proof and argued using the completeness theorem instead. In the finite sequent calculus case this works since we can prove completeness directly. However, in the infinite sequent calculus case the observation in this problem was used in the proof of completeness. Thus this argument is circular. The observation was needed mainly for the completeness proof that we constructed in class so I guess I was expecting you not to use this kind of argument here.

Problem 4

This problem carried 5 points.

Problem 5

The breakup of points for this problem was 2, 6 and 1. I don't think anyone had a real difficulty with any of the parts.

Problem 6

The breakup of points for this was as follows: 2 points for characterizing the axioms, 5 points for the rules, 2 points for arguing for soundness and 3 for completeness. The only "hard" part may have been that of figuring out a measure that always decreased. The thing to note would be that negation should count for less than the other connectives and that is should not count at all next to an atomic formula. (Even the last is unnecessary if you find a proper way to characterize termination.)

Problem 7

Nothing much to be said here.

Last updated on Nov 1, 2004 by gopalan@cs.umn.edu.