Shrinking games and local formulas

Abstract Gaifman's normal form theorem showed that every first-order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7 n −1 . This bound was improved by Lifsches and Shelah to 3×4 n −1 . We use Ehrenfeucht–Fraisse type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and Shelah, with a more easily understood proof and with the bound on the radius improved to 4 n −1 . We also obtain bounds for a normal form theorem of Schwentick and Barthelmann.