Tautology

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Problems Description

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: <ul> <li>p, q, r, s, and t are WFFs</li> <li>if w is a WFF, Nw is a WFF</li> <li>if w and x are WFFs, Kwx, Awx, Cwx, and Ewx are WFFs.</li> </ul> The meaning of a WFF is defined as follows: <ul> <li>p, q, r, s, and t are logical variables that may take on the value 0 (false) or 1 (true).</li> <li>K, A, N, C, E mean and, or, not, implies, and equals as defined in the truth table below.</li> </ul> <center> <table border="1" bgcolor="#ccffff" width="80%" cols="1"> <tbody> <tr> <td><center>Definitions of K, A, N, C, and E</center></td> </tr> </tbody> </table> <table border="1" bgcolor="#ccffff" width="80%" cols="6"> <tbody> <tr> <td>     w  x</td> <td>  Kwx</td> <td>  Awx</td> <td>   Nw</td> <td>  Cwx</td> <td>  Ewx</td> </tr> <tr> <td>  1  1</td> <td>  1</td> <td>  1</td> <td>   0</td> <td>  1</td> <td>  1</td> </tr> <tr> <td>  1  0</td> <td>  0</td> <td>  1</td> <td>   0</td> <td>  0</td> <td>  0</td> </tr> <tr> <td>  0  1</td> <td>  0</td> <td>  1</td> <td>   1</td> <td>  1</td> <td>  0</td> </tr> <tr> <td>  0  0</td> <td>  0</td> <td>  0</td> <td>   1</td> <td>  1</td> <td>  1</td> </tr> </tbody> </table> </center> 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.

Input格式

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Output格式

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Sample

Sample Input 1

ApNp ApNq 0

Sample Output 1

tautology not

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