Second order logic and the weak exponential hierarchies

Abstract

Second order logic over finite structures is well-known to capture the levels of the polynomial hierarchy PH. Recently, it has been shown that Θ ¹ _k — the first order closure of second order Σ ¹ _m — captures the class Θ ^P _k = $L^{\Sigma _k^P }$, a natural intermediate class of the polynomial hierarchy [12].

In this paper we show that with respect to expression complexity, second order logic characterizes the levels of the weak exponential hierarchy EH. Moreover, we extend these results to intermediate classes EΘ ^P _k in EH which correspond to the Θ ^P _k classes in PH.

To this end, in extending previous results, we show completeness under projection translations of certain quantified propositional formula languages for Θ ^P _k . Those, as well as quantified Boolean formulas are applied to improved complexity upgrade techniques based on the ”succinct input” paradigm.

Thus, we obtain a uniform treatment for obtaining expression complexity results for a large number of natural languages. We exhibit examples from database theory and nonmonotonic reasoning. In particular, we investigate the expression complexity of first order logic with Henkin quantifiers and default logic.

Short abridged version, omitting several proofs. The full paper [13] is available from the authors.