Temporal logics are a very popular family of logical languages, used to specify properties of abstracted systems. In the last few years, many extensions of temporal logics have been proposed, in order to address the need to express more than just abstract properties. In our work we study temporal logics extended by local constraints, which allow to express quantitative properties on data values from an arbitrary relational structure called the concrete domain. An example of concrete domain can be (Z, <, =), where the integers are considered as a relational structure over the binary order relation and the equality relation. Formulas of temporal logics with constraints are evaluated on data-words or data-trees, in which each node or position is labeled by a vector of data from the concrete domain. We call the constraints local because they can only compare values at a fixed distance inside such models. Several positive results regarding the satisfiability of LTL (linear temporal logic) with constraints over the integers have been established in the past years, while the corresponding results for branching time logics were only partial. In this work we prove that satisfiability of CTL* (computation tree logic) with constraints over the integers is decidable and also lift this result to ECTL*, a proper extension of CTL*. We also consider other classes of concrete domains, particularly ones that are \"tree-like\". We consider semi-linear orders, ordinal trees and trees of a fixed height, and prove decidability in this framework as well. At the same time we prove that our method cannot be applied in the case of the infinite binary tree or the infinitely branching infinite tree. We also look into extending the expressiveness of our logic adding non-local constraints, and find that this leads to undecidability of the satisfiability problem, even on very simple domains like (Z, <, =). We then find a way to restrict the power of the non-local constraints to regain decidability.

Temporal logics are a very popular family of logical languages, used to specify properties of abstracted systems. In the last few years, many extensions of temporal logics have been proposed, in order to address the need to express more than just abstract properties. In our work we study temporal logics extended by local constraints, which allow to express quantitative properties on data values from an arbitrary relational structure called the concrete domain. An example of concrete domain can be (Z, <, =), where the integers are considered as a relational structure over the binary order relation and the equality relation. Formulas of temporal logics with constraints are evaluated on data-words or data-trees, in which each node or position is labeled by a vector of data from the concrete domain. We call the constraints local because they can only compare values at a fixed distance inside such models. Several positive results regarding the satisfiability of LTL (linear temporal logic) with constraints over the integers have been established in the past years, while the corresponding results for branching time logics were only partial. In this work we prove that satisfiability of CTL* (computation tree logic) with constraints over the integers is decidable and also lift this result to ECTL*, a proper extension of CTL*. We also consider other classes of concrete domains, particularly ones that are \"tree-like\". We consider semi-linear orders, ordinal trees and trees of a fixed height, and prove decidability in this framework as well. At the same time we prove that our method cannot be applied in the case of the infinite binary tree or the infinitely branching infinite tree. We also look into extending the expressiveness of our logic adding non-local constraints, and find that this leads to undecidability of the satisfiability problem, even on very simple domains like (Z, <, =). We then find a way to restrict the power of the non-local constraints to regain decidability.

Description logics are knowledge representation and reasoning formalisms which represent conceptual knowledge on an abstract logical level. Concrete domains are a theoretically well-founded approach to the integration of description logic reasoning with reasoning about concrete objects such as numbers, time intervals or spatial regions. In this paper, the complexity of combined reasoning with description logcis and on concrete domains is investigated. We extend ALC(D), which is the basic description logic for reasoning with concrete domains, by the operators 'feature agreement' and 'feature disagreement'. For the extended logic,called ALCF(D), an algorithm for deciding the ABox consistency problem is devised. The strategy employed by this algorithm is vital for the efficient implementation of reasoners for description logics incorporating concrete domains. Based on the algorithm, it is proved that the standard reasoning problems for both logics ALC(D) and ALCF(D) are PSpace-complete - provided that the satisfiability test of the concrete domain used is in PSpace. / This is an extended version of the article in: Proceedings of IJCAI-99, Stockholm, Sweden, July 31-August 6, Morgan Kaufmann Publ. In ., San Mateo, CA, 1999

Concrete domains have been introduced in the area of Description Logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL ALC in the presence of GCIs, quite strong restrictions, called ω-admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of ω-admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate ω-admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield ω-admissible concrete domains. This allows us to show ω-admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the EL family, achieving decidability of reasoning is not enough. One wants the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although ω-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs. DL systems that can handle concrete domains allow their users to employ a fixed set of predicates of one or more fixed concrete domains when modelling concepts. They do not provide their users with means for defining new predicates, let alone new concrete domains. The good news is that finitely bounded homogeneous structures offer precisely that. We show that integrating concrete domains based on finitely bounded homogeneous structures into ALC yields decidable DLs even if we allow predicates specified by first-order formulas. This class of structures also provides effective means for defining new ω-admissible concrete domains with at most binary predicates. The bad news is that defining ω-admissible concrete domains with predicates of higher arities is computationally hard. We obtain two new lower bounds for this meta-problem, but leave its decidability open. In contrast, we prove that there is no algorithm that would facilitate defining p-admissible concrete domains already for binary signatures.:1. Introduction . . . 1 2. Preliminaries . . . 5 3. Description Logics with Concrete Domains . . . 9 3.1. Basic definitions and undecidability results . . . 9 3.2. Decidable and tractable DLs with concrete domains . . . 16 4. A Model-Theoretic Analysis of ω-Admissibility . . . 23 4.1. Homomorphism ω-compactness via ω-categoricity . . . 23 4.2. Patchworks via homogeneity . . . 24 4.3. JDJEPD via decomposition into orbits . . . 27 4.4. Upper bounds via finite boundedness . . . 28 4.5. ω-admissible finitely bounded homogeneous structures . . . 32 4.6. ω-admissible homogeneous cores with a decidable CSP . . . 34 4.7. Coverage of the developed sufficient conditions . . . 36 4.8. Closure properties: homogeneity & finite boundedness . . . 39 5. A Model-Theoretic Analysis of p-Admissibility . . . 47 5.1. Convexity via square embeddings . . . 47 5.2. Convex ω-categorical structures . . . 50 5.3. Convex numerical structures . . . 52 5.4. Ages defined by forbidden substructures . . . 54 5.5. Ages defined by forbidden homomorphic images . . . 56 5.6. (Non-)closure properties of convexity . . . 59 6. Towards user-definable concrete domains . . . 61 6.1. A proof-theoretic perspective . . . 65 6.2. Universal Horn sentences and the JEP . . . 66 6.3. Universal sentences and the AP: the Horn case . . . 77 6.4. Universal sentences and the AP: the general case . . . 90 7. Conclusion . . . 99 7.1. Contributions and future outlook . . . 99 A. Concrete Domains without Equality . . . 103 Bibliography . . . 107 List of figures . . . 115 Alphabetical Index . . . 117

We show that extending description logics by simple aggregation functions as available in database systems may lead to undecidability of inference problems such as satisfiability and subsumption.

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL ALC in the presence of GCIs, quite strong restrictions, in sum called ω-admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of ω-admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate ω-admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield ω-admissible concrete domains. This allows us to show ω-admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the E L family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate padmissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Althoughω-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs.

Concrete domains have been introduced in the area of Description Logic to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. One contribution of this paper is to strengthen the existing undecidability results further by showing that concrete domains even weaker than the ones considered in the previous proofs may cause undecidability. To regain decidability in the presence of GCIs, quite strong restrictions, in sum called w-admissiblity, need to be imposed on the concrete domain. On the one hand, we generalize the notion of w-admissiblity from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate w-admissiblity to well-known notions from model theory. In particular, we show that finitely bounded, homogeneous structures yield w-admissible concrete domains. This allows us to show w-admissibility of concrete domains using existing results from model theory.

Concrete domains have been introduced in Description Logics (DLs) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. To retain decidability when integrating a concrete domain into a decidable DL, the domain must satisfy quite strong restrictions. In previous work, we have analyzed the most prominent such condition, called w-admissibility, from an algebraic point of view. This provided us with useful algebraic tools for proving w-admissibility, which allowed us to find new examples for concrete domains whose integration leaves the prototypical expressive DL ALC decidable. When integrating concrete domains into lightweight DLs of the EL family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. In the present paper, we investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical padmissible concrete domain based on the rational numbers. Although w-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs.

As estruturas de dados concretas (cds) são quaternas (C, V, E, l-) que contêm um conjunto C de células, um conjunto V de valores, um conjunto E de eventos e uma relação de habilitação l-. O conjunto de estados de uma cds é um domínio concreto que pode ser considerada a parte "abstrata" das cds. Da mesma maneira tem-se que os domínios de eventos (que são generalizações dos domínios concretos) são a parte abstrata das estruturas de eventos. Mostra-se a relação dos domínios concretos e domínios de eventos com os espaços coerentes, assim como também das teias de espaços coerentes com as cds e estruturas de eventos. Intuitivamente, uma cds é uma teia de um espaço coerente se toda célula c de C não é habilitada por nenhum evento (ou equivalentemente, é habilitada pelo conjunto vazio), isto é, V C E C, 0 F c. Outra forma de expressar isto é dizer que uma cds e uma teia de um espaço coerente se o conjunto de estados da cds é um espaço coerente. Definem-se os algoritmos lineares como sendo estados de uma cds no estilo dos algoritmos seqüenciais do Curien ([CUR 86]). Em particular as cds consideradas são teias de espaços coerentes. Mostra-se como obter a cds !A—>B, a partir de uma função estável f. A —> B. O algoritmo linear desta cds possui todas as estratégias de computação (seqüenciais e paralelas) que computam a função subjacente f, o que implica que os algoritmos lineares podem ser considerados meta-algoritmos. Mostra-se que para toda estratégia de computação seqüencial de um algoritmo linear, existe um algoritmo seqüencial de Curien que computa a mesma função, e vice-versa. A definição de estratégia de computação é dada de maneira tal que permite se dar semântica a segmentos de programas. Define-se uma operação de composição de estratégias, de forma tal que se pode obter uma estratégia de computação de um programa, a partir da composição das estratégias dos segmentos. / The concrete data structures, or cds, (C, V, E, l-) consists of a set C of cells, a set V of values. a set E of events and an enabling relation l-. The set of states of a cds is a concrete domain, that can be considered the "abstract" counterpart of the cds. In the same way we have that the events domains (that are more general that the concretes domains) are the abstract counterpart of the events structures. We show the relation between the concretes domains and events domains with the coherence spaces, as just as the relation between the cds and events structures with webs of coherence spaces. Intuitivelly, a cds is a web of a coherence space if any cell c is not enabled for any event, i.e. Vce C, 0 F c. We can say that a cds is a web of a coherence space if the set of states of the cds is a coherence space. We define the linear algorithms as states of a cds following the Curien's sequential algorithms ([CUR 86]). In particular the cds considered are webs of coherence spaces. We show how to obtain a cds !A—>B from a stable function f. A —> B. The linear algorithm of this cds contain all the computational strategies (sequentials and parallels) that compute the subjacent function f; this implies that the linear algorithms can be considered a kind of meta-algorithms. We show that for all sequential computational strategy of a linear al gorithm exists a Curien's sequential algorithm that compute the same function and conversely. We define the computational strategies in such a way that we can give semantic of segments of programs. We define a composition operation for strategies. This operation has the advantage that we can obtain the computational strategy of a program as the composition of the segments of it.

As estruturas de dados concretas (cds) são quaternas (C, V, E, l-) que contêm um conjunto C de células, um conjunto V de valores, um conjunto E de eventos e uma relação de habilitação l-. O conjunto de estados de uma cds é um domínio concreto que pode ser considerada a parte "abstrata" das cds. Da mesma maneira tem-se que os domínios de eventos (que são generalizações dos domínios concretos) são a parte abstrata das estruturas de eventos. Mostra-se a relação dos domínios concretos e domínios de eventos com os espaços coerentes, assim como também das teias de espaços coerentes com as cds e estruturas de eventos. Intuitivamente, uma cds é uma teia de um espaço coerente se toda célula c de C não é habilitada por nenhum evento (ou equivalentemente, é habilitada pelo conjunto vazio), isto é, V C E C, 0 F c. Outra forma de expressar isto é dizer que uma cds e uma teia de um espaço coerente se o conjunto de estados da cds é um espaço coerente. Definem-se os algoritmos lineares como sendo estados de uma cds no estilo dos algoritmos seqüenciais do Curien ([CUR 86]). Em particular as cds consideradas são teias de espaços coerentes. Mostra-se como obter a cds !A—>B, a partir de uma função estável f. A —> B. O algoritmo linear desta cds possui todas as estratégias de computação (seqüenciais e paralelas) que computam a função subjacente f, o que implica que os algoritmos lineares podem ser considerados meta-algoritmos. Mostra-se que para toda estratégia de computação seqüencial de um algoritmo linear, existe um algoritmo seqüencial de Curien que computa a mesma função, e vice-versa. A definição de estratégia de computação é dada de maneira tal que permite se dar semântica a segmentos de programas. Define-se uma operação de composição de estratégias, de forma tal que se pode obter uma estratégia de computação de um programa, a partir da composição das estratégias dos segmentos. / The concrete data structures, or cds, (C, V, E, l-) consists of a set C of cells, a set V of values. a set E of events and an enabling relation l-. The set of states of a cds is a concrete domain, that can be considered the "abstract" counterpart of the cds. In the same way we have that the events domains (that are more general that the concretes domains) are the abstract counterpart of the events structures. We show the relation between the concretes domains and events domains with the coherence spaces, as just as the relation between the cds and events structures with webs of coherence spaces. Intuitivelly, a cds is a web of a coherence space if any cell c is not enabled for any event, i.e. Vce C, 0 F c. We can say that a cds is a web of a coherence space if the set of states of the cds is a coherence space. We define the linear algorithms as states of a cds following the Curien's sequential algorithms ([CUR 86]). In particular the cds considered are webs of coherence spaces. We show how to obtain a cds !A—>B from a stable function f. A —> B. The linear algorithm of this cds contain all the computational strategies (sequentials and parallels) that compute the subjacent function f; this implies that the linear algorithms can be considered a kind of meta-algorithms. We show that for all sequential computational strategy of a linear al gorithm exists a Curien's sequential algorithm that compute the same function and conversely. We define the computational strategies in such a way that we can give semantic of segments of programs. We define a composition operation for strategies. This operation has the advantage that we can obtain the computational strategy of a program as the composition of the segments of it.