PBnot

PBnot is a formal operator used in probabilistic and Boolean logic contexts to perform probabilistic negation on a binary state. It is defined for a Boolean variable X that has probability p of being true. Under the simplest interpretation, applying PBnot yields a new distribution in which the probability of X being true after negation is q = 1 − p. Thus, an input with p = 0.7 would produce a negated distribution with q = 0.3 for the true state.

Variants of PBnot introduce bias or noise. A common generalization uses a parameter θ to model imperfect negation, with q = θ · (1 − p) + (1 − θ) · p, or, more generally, q = f(p, θ) mapping [0,1] to [0,1]. PBnot is typically used as a lightweight transform in models that must represent uncertainty about binary states, such as probabilistic Boolean networks, gene regulatory models, and fault-tolerant digital circuits.

In practice, PBnot is an algebraic operator that can be composed with other probabilistic operators. It is easy to implement in software by applying a simple map to a probability value. Limitations include the need to specify how p is defined when variables are correlated, and the possibility that different modeling choices for negation yield different results.

See also: negation, Boolean algebra, probabilistic Boolean networks, fuzzy logic, probability theory.