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 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 there is a NAT which is a conservative extension of. 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.
We investigate the following problem: given a sample of classified strings, find a first-order sentence of minimal quantifier rank that is consistent with the sample. We represent strings as successor string structures, that is, finite structures with unary predicates to denote symbols in an alphabet, and a successor relation. We use results of the Ehrenfeucht-Fraïssé game over successor string structures in order to design an algorithm to find such sentence. We use conditions characterizing the winning strategies for the Spoiler on successor strings structures in order to define formulas which distinguish two strings. Our algorithm returns a boolean combination of such formulas.
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