Tautology

Interview

Time limit1sMemory limit128 MB

Summary

Given propositional formulas in Polish notation, decide for each whether it is a tautology by parsing and evaluating it over every truth assignment.

Level: Medium4 of 10
Topics: String, Recursion, Brute force, Implementation
Solved: No attempts yet

Problem

WFF 'N PROOF is a logic game played with dice. Each die has six faces showing some subset of the 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, then Nw is a WFF.
If w and x are WFFs, then 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 the value 0 (false) or 1 (true).
K, A, N, C, and E mean and, or, not, implies, and equals, as defined in the truth table below.

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 value 0 when p=0 and q=1.

Determine whether or not a given WFF is a tautology.

Input

The 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 only 0 follows the last case.

Output

For each test case, output a single line containing tautology or not, as appropriate.

Examples3

Example 1
Input
```
ApNp
ApNq
0
```
Expected output
```
tautology
not
```
Example 2
Input
```
Cpp
p
0
```
Expected output
```
tautology
not
```

Example 3

Input

Epp
CpApq
CKpqp
0

Expected output

tautology
tautology
tautology