Arity and Alternation: A Proper Hierarchy in Higher Order Logics

Abstract

We study the effect of simultaneously bounding the maximal arity of the higher-order variables and the alternation of quantifiers in higher-order logics, as to their expressive power on finite structures (or relational databases). Let AA ⁱ(r,m) be the class of (i + 1)-th order logic formulae where all quantifiers are grouped together at the beginning of the formulae, forming m alternating blocks of consecutive existential and universal quantifiers, and such that the maximal arity of the higher-order variables is bounded by r. Note that, the order of the quantifiers in the prefix may be mixed. We show that, for every i ≥ 1, the resulting AA ⁱ(r,m) hierarchy of formulae of (i + 1 )-th order logic is proper. From the perspective of database query languages this means that, for every i ≥ 2, if we simultaneously increase the arity of the quantified relation variables by one and the number of alternating blocks of quantifiers by four in the fragment of higher-order relational calculus of order i, AA ^{i − − 1}, then we can express more queries. This extends a result by J. A. Makowsky and Y. B. Pnueli who proved that the same hierarchy in second-order logic is proper. In both cases the strategy used to prove the results consists in considering formulae which, represented as finite structures, satisfy themselves. As the well known diagonalization argument applies here, this gives rise, for each order i and for each level of the AA ⁱ(r,m) hierarchy of arity and alternation, to a class of formulae which is not definable in that level, but which is definable in a higher level of the same hierarchy. We then use a similar argument to prove that the classes of Σ$^{i}_{m}$ ∪ Π$^{i}_{m}$ formulae in which the higher-order variables of all orders up to i + 1 have maximal arity at most r, also induce a proper hierarchy in each higher-order logic of order i ≥ 3. It is not known whether the correspondent hierarchy in second-order is proper.