Three Variables Suffice for Real-Time Specification

A natural framework for real-time specification is monadic first-order logic over the structure $(\mathbb{R},<,+1)$the ordered real line with unary $+1$ function. Our main result is that $(\mathbb{R},<,+1)$ has the 3-variable property: every monadic first-order formula with at most 3 free variables is equivalent over this structure to one that uses 3 variables in total. As a corollary we obtain also the 3-variable property for the structure $(\mathbb{R},<,f)$ for any fixed linear function $f:\mathbb{R}\rightarrow\mathbb{R}$. On the other hand, we exhibit a countable dense linear order $(E,<)$ and a bijection $f:E\rightarrow E$ such that $(E,<,f)$ does not have the $k$-variable property for any $k$.