Quantitative Variants of Language Equations and their Applications to Description Logics

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. Together with the special case of matching, they were first investigated in detail for the DL FL0, where these problems can be reduced to solving certain language equations. In this thesis, we extend this service in two directions. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify, in order to account for potential small modelling errors. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL0 can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for particular distance measures. Furthermore, we make a first step towards integrating background knowledge, formulated in so-called TBoxes, by investigating the special case of matching in the presence of TBoxes of different forms. We acquire a tight complexity bound for the general case, while we prove that the problem becomes easier in a restricted setting. To achieve these bounds, we take advantage of an equivalence characterization of FL0 concepts that is based on formal languages. In addition, we incorporate TBoxes in computing concept distances. Even though our results on the approximate setting cannot deal with TBoxes yet, we prepare the framework that future research can build on. Before we journey to the technical details of the above investigations, we showcase our program in the simpler setting of the equational theory ACUI, where we are able to also combine the two extensions. In the course of studying the above problems, we make heavy use of automata theory, where we also derive novel results that could be of independent interest

Approximately Solving Set Equations

Unification with constants modulo the theory ACUI of an associative (A), commutative (C) and idempotent (I) binary function symbol with a unit (U) corresponds to solving a very simple type of set equations. It is well-known that solvability of systems of such equations can be decided in polynomial time by reducing it to satisfiability of propositional Horn formulae. Here we introduce a modified version of this problem by no longer requiring all equations to be completely solved, but allowing for a certain number of violations of the equations. We introduce three different ways of counting the number of violations, and investigate the complexity of the respective decision problem, i.e., the problem of deciding whether there is an assignment that solves the system with at most l violations for a given threshold value l.Submitted to 30th International Workshop on Unificatio

Approximate Unification in the Description Logic FL₀

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. It was first investigated in detail for the DL FL₀, where unification can be reduced to solving certain language equations. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL₀ can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for two particular distance measures

Approximation in Description Logics: How Weighted Tree Automata Can Help to Define the Required Concept Comparison Measures in FL₀

Recently introduced approaches for relaxed query answering, approximately defining concepts, and approximately solving unification problems in Description Logics have in common that they are based on the use of concept comparison measures together with a threshold construction. In this paper, we will briefly review these approaches, and then show how weighted automata working on infinite trees can be used to construct computable concept comparison measures for FL₀ that are equivalence invariant w.r.t. general TBoxes. This is a first step towards employing such measures in the mentioned approximation approaches.Accepted to LATA 201

Quantitative Variants of Language Equations and their Applications to Description Logics

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. Together with the special case of matching, they were first investigated in detail for the DL FL0, where these problems can be reduced to solving certain language equations. In this thesis, we extend this service in two directions. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify, in order to account for potential small modelling errors. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL0 can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for particular distance measures. Furthermore, we make a first step towards integrating background knowledge, formulated in so-called TBoxes, by investigating the special case of matching in the presence of TBoxes of different forms. We acquire a tight complexity bound for the general case, while we prove that the problem becomes easier in a restricted setting. To achieve these bounds, we take advantage of an equivalence characterization of FL0 concepts that is based on formal languages. In addition, we incorporate TBoxes in computing concept distances. Even though our results on the approximate setting cannot deal with TBoxes yet, we prepare the framework that future research can build on. Before we journey to the technical details of the above investigations, we showcase our program in the simpler setting of the equational theory ACUI, where we are able to also combine the two extensions. In the course of studying the above problems, we make heavy use of automata theory, where we also derive novel results that could be of independent interest

Quantitative Variants of Language Equations and their Applications to Description Logics

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. Together with the special case of matching, they were first investigated in detail for the DL FL0, where these problems can be reduced to solving certain language equations. In this thesis, we extend this service in two directions. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify, in order to account for potential small modelling errors. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL0 can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for particular distance measures. Furthermore, we make a first step towards integrating background knowledge, formulated in so-called TBoxes, by investigating the special case of matching in the presence of TBoxes of different forms. We acquire a tight complexity bound for the general case, while we prove that the problem becomes easier in a restricted setting. To achieve these bounds, we take advantage of an equivalence characterization of FL0 concepts that is based on formal languages. In addition, we incorporate TBoxes in computing concept distances. Even though our results on the approximate setting cannot deal with TBoxes yet, we prepare the framework that future research can build on. Before we journey to the technical details of the above investigations, we showcase our program in the simpler setting of the equational theory ACUI, where we are able to also combine the two extensions. In the course of studying the above problems, we make heavy use of automata theory, where we also derive novel results that could be of independent interest

Approximately Solving Set Equations

Unification with constants modulo the theory ACUI of an associative (A), commutative (C) and idempotent (I) binary function symbol with a unit (U) corresponds to solving a very simple type of set equations. It is well-known that solvability of systems of such equations can be decided in polynomial time by reducing it to satisfiability of propositional Horn formulae. Here we introduce a modified version of this problem by no longer requiring all equations to be completely solved, but allowing for a certain number of violations of the equations. We introduce three different ways of counting the number of violations, and investigate the complexity of the respective decision problem, i.e., the problem of deciding whether there is an assignment that solves the system with at most l violations for a given threshold value l.Submitted to 30th International Workshop on Unificatio

Approximate Unification in the Description Logic FL₀

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. It was first investigated in detail for the DL FL₀, where unification can be reduced to solving certain language equations. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL₀ can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for two particular distance measures

Approximate Unification in the Description Logic FL₀

Unification in description logics (DLs) has been introduced as a novel inference service that can be used to detect redundancies in ontologies, by finding different concepts that may potentially stand for the same intuitive notion. It was first investigated in detail for the DL FL₀, where unification can be reduced to solving certain language equations. In order to increase the recall of this method for finding redundancies, we introduce and investigate the notion of approximate unification, which basically finds pairs of concepts that “almost” unify. The meaning of “almost” is formalized using distance measures between concepts. We show that approximate unification in FL₀ can be reduced to approximately solving language equations, and devise algorithms for solving the latter problem for two particular distance measures

Approximation in Description Logics: How Weighted Tree Automata Can Help to Define the Required Concept Comparison Measures in FL₀

Recently introduced approaches for relaxed query answering, approximately defining concepts, and approximately solving unification problems in Description Logics have in common that they are based on the use of concept comparison measures together with a threshold construction. In this paper, we will briefly review these approaches, and then show how weighted automata working on infinite trees can be used to construct computable concept comparison measures for FL₀ that are equivalence invariant w.r.t. general TBoxes. This is a first step towards employing such measures in the mentioned approximation approaches.Accepted to LATA 201