Three Variables Suffice for Real-Time Specification

A natural framework for real-time specification is monadic first-order logic over the structure (R, <,+1)—the ordered real line with unary +1 function. Our main result is that (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 certain unary arithmetic functions on the reals in place of +1. Finally we exhibit a bijection f : R → R such that (R,<, f) does not have the k-variable property for any k.