Problem D: Tautology
WFF 'N PROOF is a logic game played with dice. Each die has six faces representing
some subset of the possible symbols K, A, N, C, E, p, q, r, s, t.
A Well-formed formula (WFF) is any string of these symbols obeying the following
rules:
- p, q, r, s, and t are WFFs
- if w is a WFF, Nw is a WFF
- if w and x are WFFs, Kwx, Awx, Cwx,
and Ewx are WFFs.
The meaning of a WFF is defined as follows:
- p, q, r, s, and t are logical variables that may take on the value 0 (false) or
1 (true).
- K, A, N, C, E mean and, or, not, implies, and equals as defined
in the truth table below.
|
Definitions of K, A, N,
C, and E
|
| w
x |
Kwx |
Awx |
Nw |
Cwx |
Ewx |
| 1 1 |
1 |
1 |
0 |
1 |
1 |
| 1 0 |
0 |
1 |
0 |
0 |
0 |
| 0 1 |
0 |
1 |
1 |
1 |
0 |
| 0 0 |
0 |
0 |
1 |
1 |
1 |
A tautology is a WFF that has value 1 (true) regardless of the values of
its variables. For example, ApNp is a tautology because it is true regardless
of the value of p. On the other hand, ApNq is not, because it has the value 0 for
p=0, q=1.
You must determine whether or not a WFF is a tautology.
Input consists of several test cases. Each test case is a single line containing a
WFF with no more than 100 symbols. A line containing 0 follows the
last case. For each test case, output a line containing
tautology or not as appropriate.
Sample Input
ApNp
ApNq
0
Possible Output for Sample Input
tautology
not
Gordon V. Cormack