Section: New Results

Construction of Hanf sentences

A classical result by Hanf from the 60s states that first-order formulas over structures of bounded degree are equivalent to boolean combinations of statements of the form: “pattern P occurs at least n times”. Hanf's theorem has many model-theoretic applications, in particular in automata theory and database query answering.

However, until recently, no elementary construction was known. In [49] , we present the first algorithm that computes a Hanf normal in elementary time. More precisely, our algorithm is triply exponential, which we also show to be optimal.