#
CSCI 8980, Computation and Deduction

Fall 2004, Homework 2

Comments on Grading

## General Remarks

This assignment was graded out of 20 points. The points distribution
was 6, 3, 2, 2, 2 and 5 respectively. The mean on the homework was 19
and the standard deviation was 0.88.

## Problem 1

Most of you got this problem, but the argument was sometimes
laborious. Part of this may have been that you took me too literally
when I showed you how to construct a semantic argument in class. The
point is that you can be a bit more "loose" in the application of the
semantic ideas and do not have to talk explicitly in terms of
"x-variants" of assignments each time.
To take an example, here is the way I may argue for the first of the
formulas. Given any interpretation, consider the set corresponding to
*P* under it. If this set includes the entire domain, then *(all
x) P(x)* must be satisfied by any interpretation and it quickly follows
that the entire formula is satisfied. On the other hand, suppose that
the interpretation of *P* does not include at least one element
from the domain. Let us call this *a*. Then the interpretation
together with the assignment that maps *x* to *a* satisfies
the formula *(P(x) => (all x) P(x))*. But then it follows that
the given formula is satisfied by the interpretation.

## Problems 2-6

There isn't much to be said about these problems, especially since
most of you had the general feel for them.
As a particular application of the last problem, consider showing the
following: A formula *F* in negation normal form that does not
contain existential quantifiers or the equality symbol is
unsatisfiable if and only if it is unsatisfiable in the term model.
This observation is central to the refutation approach to theorem
proving.

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