Counting Modulo Quantifiers on Finite Structures

Abstract

We give a combinatorial method for proving elementary equivalence in first-order logic FO with counting modulo n quantifiers D n . Inexpressibility results for FO ( D n ) with built-in linear order are also considered. For instance, the class of linear orders of length divisible by n +1 cannot be expressed in FO ( D n ). Using this result we prove that comparing cardinalities or connectivity of ordered graphs are not definable in FO ( D n ). We also show that the height of complete n -ary trees cannot be expressed in FO ( D n ) with linear order. Interpreting the predicate y = nx as a complete n -ary tree, we show that the predicate y = px cannot be defined in FO ( D n ) with linear order, whenever p has a prime factor that does not divide n . This solves the problem raised by Niwiński and Stolboushkin (LICS '93). We also discuss a connection between our results and the well-known open problem in circuit complexity theory, whether ACC = NC 1 .