Arity vs. Alternation in second order logic

Abstract

We investigate the expressive power of second order logic over finite structures, when two limitations are imposed. Let SAA(k, n) (AA(k, n)) be the set of second order formulas such that the arity of the relation variables is bounded by k and the number of alternations of (both first order and) second order quantification is bounded by n. We show that this imposes a proper hierarchy on second order logic, i.e. for every k, n there are problems not definable in AA(k, n) but definable in AA(k+c1, n+d1) for some c1, d1. The method to show this is to introduce the set AUTOSAT(F) of formulas in F which satisfy themselves. We study the complexity of this set for various fragments of second order logic. For first order logic FOL with unbounded alternation of quantifiers AUTOSAT(FOL) is PSpace-complete. For first order logic FOLn with alternation of quantifiers bounded by n AUTOSAT(FOLn) is definable in AA(3, n+ 4). AUTOSAT(AA(k, n)) is definable in AA(k+c1, n+d1) for some c1, d1.