Problem B
"Expression"
It is known that Sheffer stroke function (NOT-AND) can be used to
construct any Boolean function. The truth table for this function is
given below:
Truth table for Sheffer stroke function
| x | y | x|y
|
| 0 | 0 | 1
|
| 0 | 1 | 1
|
| 1 | 0 | 1
|
| 1 | 1 | 0
|
Consider the problem of adding two binary numbers A and
B, each containing N bits. The individual bits
of A and B are numbered from 0 (the least
significant) to N-1 (the most significant). The sum of
A and B can always be represented by N+1
bits. Let's call most significant bit of the sum (bit number
N) the overflow bit.
Your task is to construct a logical expression using the Sheffer
stroke function that computes the value of the overflow bit for
arbitrary values of A and B. Your expression shall
be constructed according to the following rules:
- Ai is an expression that denotes value of
ith bit of number A.
- Bi is an expression that denotes value of
ith bit of number B.
- (x|y) is an expression that denotes the result
of Sheffer stroke function for x and y, where
x and y are expressions.
When writing the index, i, for bits in A and
B, the index shall be written as a decimal number without
leading zeros. For example, bit number 12 of A must be
written as A12. The expression should be completely parenthesized
(according to the 3rd rule). No blanks are allowed inside
the expression.
Input
The first line of the input contains an integer indicating the number of test cases in the input. After that there is a blank line and the test cases separated by a blank line. Each test case consists of a single integer N
(1 ≤ N ≤ 100) on a line by itself.
Output
Write to the output file an expression for calculating overflow bit of
the addition of two N-bit numbers A and B
according to the rules given in the problem statement. Print a blank line between test cases.
Note: The stroke symbol ( | ) is an ASCII character
with code 124 (decimal).
The output file size shall not exceed 50*N bytes.
Sample input
1
2
Sample output for the sample input
((A1|B1)|(((A0|B0)|(A0|B0))|((A1|A1)|(B1|B1))))