Arity and alternation in second-order logic

J.A. Makowsky, Y.B. Pnueli
1996 Annals of Pure and Applied Logic  
We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA@, 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 + cl,
more » ... n + d,) for some cl, d,. The method to show this is to introduce the set AUTOSAT 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 PSpacrcomplete. For first-order logic FOL, with alternation of quantifiers bounded by n, AUTOSAT (FOL,) is definable in AA(3, n + 4). AUTOSAT(AA(k, n)) is definable in AA(k + cl, n + d,) for some cl, d,.
doi:10.1016/0168-0072(95)00013-5 fatcat:b3iwj3l4sze6plwzp7jw2nmvsu