## DWITE '11 R3 #5 - Tautology

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Points: 10
Time limit: 1.0s
Memory limit: 64M

Problem type
Allowed languages
Ada, Assembly, Awk, Brain****, C, C#, C++, COBOL, CommonLisp, D, Dart, F#, Forth, Fortran, Go, Groovy, Haskell, Intercal, Java, JS, Kotlin, Lisp, Lua, Nim, ObjC, OCaml, Octave, Pascal, Perl, PHP, Pike, Prolog, Python, Racket, Ruby, Rust, Scala, Scheme, Sed, Swift, TCL, Text, Turing, VB, Zig
##### DWITE, December 2011, Problem 5 We define a propositional formula as follows:

• are atomic propositions, representing either true or false.
• If and are propositional formulae, then so are:
•  is boolean "and"
•  is boolean "or"
•  is boolean "not"

For example, is a propositional formula. A tautology is a propositional formula that equates to true for all possible value assignments to the atomic propositions. Our previous example is not a tautology because for the assignments , and , the formula evaluates to a false. However is a tautology because no matter what the atomic proposition is this equates to true; (true or not-true) == true, (false or not-false) == true.

The input will contain 5 test cases, each three lines (not more than 255 characters) with a propositional formula per line.

The output will contain 5 lines of output, each three characters long. Y for tautology, N for not tautology.

#### Sample Input

((a v b) ^ (~c v a))
(a v ~a)
~(a ^ ~a)
a
~b
((a ^ b) v ~(c ^ ~c))

Note that ~ is used for , ^ for , and v for .

#### Sample Output

NYY
NNY