The Complexity and Expressive Power of Second-Order Extended Logic
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We study the expressive powers of SO-HORN^*, SO-HORN^r and SO-HORN^*r on all finite structures. We show that SO-HORN^r, SO-HORN^*r, FO(LFP) coincide with each other and SO-HORN^* is proper sublogic of SO-HORN^r. To prove this result, we introduce the notions of DATALOG^* program, DATALOG^r program and their stratified versions, S-DATALOG^* program and S-DATALOG^r program. It is shown that, on all structures, DATALOG^r and S-DATALOG^r are equivalent and DATALOG^* is a proper sublogic of DATALOG^r. SO-HORN^* and SO-HORN^r can be treated as the negations of DATALOG^* and DATALOG^r, respectively. We also show that SO-EHORN^r logic which is an extended version of SO-HORN captures co-NP on all finite structures.
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