Consider truth assignments involving only the propositional variables x0, x1, x2
ID: 3885006 • Letter: C
Question
Consider truth assignments involving only the propositional variables x0, x1, x2, x3 and y0, y1, y2, y3. Every such truth assignment gives a value of 1 (representing true) or 0 (representing false) to each variable. We can therefore think of a truth assignment as determining a four-bit integer x depending on the values given to x0, x1, x2 and x3, and a four-bit integer y depending on the values given to y0, y1, y2 and y3. More precisely, with (xi) being the truth value assigned to xi , we can define the integers x = 2^3 · (x3) + 2^2 · (x2) + 2^1 · (x1) + (x0) and y = 2^3 · (y3) + 2^2 · (y2) + 2^1 · (y1) + (y0).
Write a formula that is satisfied by exactly those truth assignments for which x > y . Your formula may use any of the Boolean connectives introduced so far. Explain how you obtained your formula, and justify its correctness. Note: This can be done by writing down a truth table for eight propositional variables — i.e., a truth table with 28 = 256 rows. It can be done more easily (and interestingly) by identifying the precise condition under which the four-bit number x3x2x1x0 is greater than the four-bit number y3y2y1y0. (Hint: If the high-order bits, x3 and y3, are not equal, which of the two numbers is greater? If the two high-order bits are equal, how do we proceed to determine which of the two numbers is greater?) Solutions based on large truth tables will receive zero credit.
Explanation / Answer
Sometimes our model isn’t quite right. As computer scientists, we build models of real-
world phenomena, based on limited data or on the opinions of sometimes-fallible experts.
We verify or begin to use the models and discover that they are not quite correct. Rather
than beginning the model-building phase again, we would prefer to quickly and simply
revise the current model, and continue our project. If the initial model is nearly correct,
this
should
be more efficient.
The revision of an initial theory, represented by a formula, consists of applying
syntactic
revision operators
, such as the deletion or the addition of a literal. For instance, the
CUP
theory,
1
presented in Fig. 1, might be revised to become more accurate by deleting the
literal
white
.The
revision distance
of the target theory from the initial theory is defined
to be the minimal number of revision operations from a specified fixed set needed to
produce a theory equivalent to the target, star
ting from the initial theory. As in our previous
work [30] we consider two sets of revision operators: deletions-only operators, which allow
the deletion of literals and of clauses and/or terms, and general operators, which also allow
the addition of literals. Others have also implicitly or explicitly considered both of those
models [32,40].
If the target th