We investigate the expressive power of different fragments of higher-order logics over finite relational structures (or equivalently, relational databases) with special emphasis in higher-order logics of order greater than or equal three. Our main results concern the study of the effect on the expressive power of higher-order logics, of simultaneously bounding the arity of the higher-order variables and the alternation of quantifiers. Let AAi(r,m) be the class of (i + 1)-th order logic formulae where all quantifiers are grouped together at the beginning of the formulae, forming m alternating blocks of consecutive existential and universal quantifiers, and such that the maximal-arity (a generalization of the concept of arity, not just the maximal of the arities of the quantified variables) of the higher-order variables is bounded by r. Note that, the order of the quantifiers in the prefix may be mixed. We show that, for every i [greater than or equal to] 1, the resulting AAi hierarchy of formulae of (i + 1)-th order logic is proper. This extends a result by Makowsky and Pnueli who proved that the same hierarchy in second-order logic is proper. In both cases the strategy used to prove the results consists in considering the set AUTOSAT(F) of formulae in a given logic F which, represented as finite structures, satisfy themselves. We then use a similar strategy to prove that the classes of [Sigma superscript i subscript m union Pi superscript i subscript m] formulae in which the higher-order variables of all orders up to i+1 have maximal-arity at most r, also induce a proper hierarchy in each higher-order logic of order i [greater than or equal to] 3. It is not known whether the correspondent hierarchy in second-order logic is proper. Using the concept of finite model truth definitions introduced by M. Mostowski, we give a sufficient condition for that to be the case. We also study the complexity of the set AUTOSAT(F) and show that when F is one of the prenex fragments [Sigma superscript 1 subscript m] of second-order logic, it follows that AUTOSAT(F) becomes a complete problem for the corresponding prenex fragment [Sigma superscript 2 subscript m] of third-order logic. Finally, aiming to provide the background for a future line of research in higher-order logics, we take a closer look to the restricted second-order logic SO[superscript w] introduced by Dawar. We further investigate its connection with the concept of relational complexity studied by Abiteboul, Vardi and Vianu. Dawar showed that the existential fragment of SO[superscript w] is equivalent to the nondeterministic inflationary fixed-point logic NFP. Since NFP captures relational NP, it follows that the existential fragment of SO[superscript w] captures relational NP. We give a direct proof, in the style of the proof of Fagin’s theorem, of this fact. We then define formally the concept of relational machine with relational oracle and prove the exact correspondence between the prenex fragments of SO[superscript w] and the levels of the relational polynomial-time hierarchy. This allows us to stablish a direct connection between the relational polynomial hierarchy and SO without using the Abiteboul and Vianu normal form for relational machines.
| Identifer | oai:union.ndltd.org:ADTP/280452 |
| Date | January 2008 |
| Creators | Ferrarotti, Flavio Antonio |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
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