Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

s’ ╞ ¬q. (6)
But by the first two assumptions,
s’ ╞ pq (7)
and s’ ╞ p. (8)
So by Modes Ponens from (7) & (8)
s’ ╞ q. (9)
Now, (9) is a contradiction to (6).
Hence, the initial assumption s ╞ ¬ Ā(q) is false. Consequently s ╞ Ā(q) is
true. Thus
Ā (pq) (Ā(p) Ā(q) ).
is a valid expression.
11.6.2. The ‘Next-Time’ Operator
Normally, time in a computer system is discrete; i.e., if the current instant is
defined the third, next will be the fourth. This is due to the hardwired clocking
of the system resources including processor, memory and peripheral circuit
modules. Thus it makes sense to express the ‘next’ instant of time. The next
instant of time or next-time is denoted by the ‘O’ symbol. Thus to represent
that proposition p follows from the next instant of s1 we write
s1 ╞ O p.
From the meaning of the ‘O’ operator, it is thus clear that
╞ Ā(p) O p and
╞ O p S p.
Further, linear-time temporal logic is characterized by
O p ╞ ¬O¬p.
We now present a few elementary axioms using O operator.
i) ├ O (A B) (O A O B)
ii)├ Ā (A OA) (A Ā A).
Further, for any proposition p
iii) ├ Ā p O p
iv) ├ O p S p.