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Logic Journal of IGPL Advance Access originally published online on September 26, 2007
Logic Journal of IGPL 2007 15(5-6):503-526; doi:10.1093/jigpal/jzm037
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

On Minimal Models

Francicleber Martins Ferreira 1

Departamento de Computação, Universidade Federal do Ceará, CP 12.166, Fortaleza-CE, Brasil, CEP 60455-760. E-mail: fran{at}lia.ufc.br

Ana Teresa Martins 2

Departamento de Computaçãao, Universidade Federal do Ceará, CP 12.166, Fortaleza-CE, Brasil, CEP 60455-760. E-mail: ana{at}lia.ufc.br


  Abstract

We investigate some logics which use the concept of minimal models in their definition. Minimal objects are widely used in Logic and Computer Science. They are applied in the context of Inductive Definitions, Logic Programming and Artificial Intelligence. An example of logic which uses this concept is the MIN(FO) logic due to van Benthem [20]. He shows that MIN(FO) is equivalent to the Least Fixed Point logic (LFP) in expressive power. In [6], we extended MIN(FO) to the MIN Logic and proved it is equivalent to second-order logic in expressive power. Here, we exhibit a fragment of MIN, the MIN{Delta} logic, which is more expressive than LFP, less expressive than MIN and closed under boolean connectives and first-order quantification. In order to do this, in the Section 2, we prove that the Downward Löwenheim-Skolem Theorem holds for arbitrary countable sets of LFP-formulas by showing that every infinite structure has a countable LFP-substructure. The method may be used to generalize this theorem to any set of LFP-formulas. We also analyse the expressive power of the Nested Abnormality Theories (NATs) of Lifschitz, another formalism based on minimal models used in Artificial Intelligence, and we demonstrate that for each second-order theory {Gamma} there is a NAT which is a conservative extension of {Gamma}. We give a translation from second-order sentences into such NATs which is linear in the size of the sentence in prenex normal form. Finally, we establish a hierarchy of expressiveness of these logics that deal with the concept of minimal models.

Key Words: minimal models • fixed points • expressiveness.

Received for publication 13 September 2006.

This research is partially supported by CNPq and FUNCAP.

This research is partially supported by PADCT/CNPq, PRONEX/FUNCAP-CNPq, CNPq and FUNCAP.


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