Pseudo-Boolean function vs Boolean function

Question

What is the difference between a pseudo-Boolean function and a Boolean function?

Wikipedia states that a Boolean function is a special case of a pseudo-Boolean function, but I do not understand in what way. It seems like both are of the form $$ f \colon \{0,1\}^n \to \mathbb{R} $$ However, this does not really help me in distinguishing the two...

Answer 1

For a Boolean function, the range is a subset of $\{0,1\}$.

For a pseudo-Boolean function, the range is a subset of $\mathbb{R}$, but not necessarily a subset of $\{0,1\}$.

For example . . .

• The function $f:\{0,1\}^n\to\mathbb{R}\;\,$given by $$f(x_1,...,x_n)=\max(x_1,...,x_n)$$ is a Boolean function.$\\[10pt]$
• The function $g:\{0,1\}^n\to\mathbb{R}\;\,$given by $$g(x_1,...,x_n)=x_1 + \cdots + x_n$$ is a pseudo-Boolean function, but not a Boolean function (unless $n=1$).