The truth value of a logical expression is defined recursively as: •The truth va
ID: 3587531 • Letter: T
Question
The truth value of a logical expression is defined recursively as:
•The truth value of t is t.
•The truth value of f is f.
•The truth value of (x1 x2) is t if both x1 and x2 have truth value t, it is f otherwise.
•The truth value of (x1 x2) is f if both x1 and x2 have truth value f, it is t otherwise.
•The truth value of ¬(x) is f if x has truth value t, it is t otherwise.
Define a CFG that generates the following language over{t, f,,,¬,(,),=}:
L={w=x: w is a logical expression over{t, f},x {t, f}, and x is the truth value of w}
Thus, “t = t”, “((t f) f) = f”, and “¬(((t f) f)) = t” are in L, but “((t f) f) = t”and “(t f) f = f” are not: the former because ((t f) f) is false and not true, the latter because the expression lacks the outermost set of parentheses.
Explanation / Answer
A context-free grammar G is defined by the 4-tuple:
G = {V, T, P, S} where
For the given language:
L={w=x: w is a logical expression over{t, f},x {t, f}, and x is the truth value of w}
The corresponding Context Free Grammar (CFG) can be written as:
S -> S or T | T
T -> T and F | F
F -> not F | t | f | (B)
Here in the above CFG
V = {S, T, F}
T = {t, f}
S = {S}
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