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.

Thesis (doctoral)--Universität zu Lübeck, 2006. / Includes bibliographical references (p. 161-169) and index.

The purpose of this paper is to study the basic theoretic structure of intelligent data base by means of logic. There are three parts of this paper. The first part introduces the concept and relational algebra in relational data base. The second part focuses on the relationship between logic and the structure of intelligent data base and compares the intelligent database application. The last part, a SQL (Structure Query Language) queries simulation program using logic programming language Prolog, demonstrates how logic applies to query languages. / Department of Computer Science

Data is mostly stored in relational databases today. However, most data mining algorithms are not capable of working on data stored in relational databases directly. Instead they require a preprocessing step for transforming relational data into algorithm specified form. Moreover, several data mining algorithms provide solutions for single relations only. Therefore, valuable hidden knowledge involving multiple relations remains undiscovered. In this thesis, an implementation is developed for discovering multi-relational association rules in relational databases. The implementation is based on a framework providing a representation of patterns in relational databases, refinement methods of patterns, and primitives for obtaining necessary record counts from database to calculate measures for patterns. The framework exploits meta-data of relational databases for pruning search space of patterns. The implementation extends the framework by employing Apriori algorithm for further pruning the search space and discovering relational recursive patterns. Apriori algorithm is used for finding large itemsets of tables, which are used to refine patterns. Apriori algorithm is modified by changing support calculation method for itemsets. A method for determining recursive relations is described and a solution is provided for handling recursive patterns using aliases. Additionally, continuous attributes of tables are discretized utilizing equal-depth partitioning. The implementation is tested with gene localization prediction task of KDD Cup 2001 and results are compared to those of the winner approach.

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.

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.

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.

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.

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.

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.