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Second order logic and the weak exponential hierarchies

  • Georg Gottlob
  • Nicola Leone
  • Helmut Veith
Invited Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 969)

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 k 1 — the first order closure of second order Σ m 1 — captures the class Θ k P = \(L^{\Sigma _k^P }\), 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 k P in EH which correspond to the Θ k P classes in PH.

To this end, in extending previous results, we show completeness under projection translations of certain quantified propositional formula languages for Θ k P . 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.

Keywords

Order Logic Expression Complexity Propositional Formula Default Theory Finite Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Georg Gottlob
    • 1
  • Nicola Leone
    • 2
  • Helmut Veith
    • 1
  1. 1.Christian Doppler Laboratories for Expert Systems, Information Systems DepartmentTU ViennaWienAustria
  2. 2.ISI-CNRRendeItaly

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