Three Variables Suffice for Real-Time Logic.

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 the structure (R,<, f) for any fixed linear function f : R -> R. On the other hand, we exhibit a countable dense linear order (E,<) and a bijection f : E -> E such that (E,<, f) does not have the k-variable property for any k.