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...ral predicates is to employ concepts of temporal logic. We will first mention some related work and afterwards explain why we will take a partly different approach in this paper. Based on his work in =-=[1]-=- where he has identified thirteen possible relationships (like before, equal, meets, overlaps, during) between intervals, Allen in [2] defines a predicate Holds(p; i) which asserts that a property p i...

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...e a partly different approach in this paper. Based on his work in [1] where he has identified thirteen possible relationships (like before, equal, meets, overlaps, during) between intervals, Allen in =-=[2]-=- defines a predicate Holds(p; i) which asserts that a property p is true during a time interval i. This predicate implies that p holds at every subinterval of i as well. Logical operators like Boolean...

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...n features in the (especially two-dimensional) Euclidean space have been intensively studied in the literature. Using point sets and point set topology as a formal framework, the 9-intersection model =-=[13, 7]-=- provides a canonical collection of topological predicates for each combination of spatial types. Based on the nine topologically invariant intersections of boundaries, interiors, and exteriors of the...

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... and the process of building such a section is called sectioning. Sectioning is a popular syntactic tool in functional programming that allows to denote certain functions in a highly concise way (see =-=[4]-=-). For example, (1+) fixes the first argument of the addition function with 1 and thus denotes the successor function. Likewise, (\Gamma1) denotes the predecessor function (note that this is different...

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...endent) distance of an airplane and a storm. This could be achieved by an operation: Distance : (point) \Theta (region) ! (real ) This example demonstrates the important principle of temporal lifting =-=[29] avoiding -=-an inflation of operation definitions: we can, in principle, take almost any flat (that is, non-temporal) operation and "lift" it so that it works on temporal objects returning also a tempor...

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...orts a more integrated view of space and time and incorporates the treatment of continuous spatial changes. It will be the basis of this paper and introduces the concept of spatio-temporal data types =-=[15, 16]-=-. These data types are designed as abstract data types whose values can be integrated as complex entities into databases [36, 35] and whose definition and integration into databases is independent of ...

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...n features in the (especially two-dimensional) Euclidean space have been intensively studied in the literature. Using point sets and point set topology as a formal framework, the 9-intersection model =-=[13, 7]-=- provides a canonical collection of topological predicates for each combination of spatial types. Based on the nine topologically invariant intersections of boundaries, interiors, and exteriors of the...

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...es on spatio-temporal objects (Section 5). Second, we are interested in designing a canonical collection of spatio-temporal predicates (Section 6). A first hint to a possible design gives the work in =-=[12] which consider-=-s possible topological transitions (that is, changes) between topological relationships that are represented in a so-called "conceptual-neighborhood-graph" or "closest-topological-relat...

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...ime intervals and not on time points. Galton [25] has extended Allen's approach to the treatment of temporally changing topological relationships. Topological relationships are based on the RCC model =-=[10, 31]-=- which comes to similar results as Egenhofer's 9-intersection model. In contrast to Allen, Galton also takes time points into account, as we do. In specifying changes of spatial situations he uses the...

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...ch. 2.1 Spatio-Temporal Objects So far, only a few data models for spatio-temporal data have been proposed. One approach has been to appropriately extend a spatial data model by temporal concepts. In =-=[41]-=- spatio-temporal objects are defined as so-called spatio-bitemporal complexes. Their spatial features are described by simplicial complexes, and their temporal features are given by bitemporal element...

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..., line, and region have turned out to be a fundamental abstraction for modeling the two-dimensional structure of geometric entities, their properties, relationships, and operations (see, for example, =-=[28, 30, 32, 33, 37]-=-). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this context are called bou...

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...to formally express this requirement and to avoid the anomalies just described, the formalization of a data type for regions is based on point set topology [23] and on a process called regularization =-=[40]-=-. Point set topology also rests on the point set paradigm but additionally distinguishes different parts of an (arbitrary) point set. Given such a point set, say A, these parts identify its boundary @...

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...ime intervals and not on time points. Galton [25] has extended Allen's approach to the treatment of temporally changing topological relationships. Topological relationships are based on the RCC model =-=[10, 31]-=- which comes to similar results as Egenhofer's 9-intersection model. In contrast to Allen, Galton also takes time points into account, as we do. In specifying changes of spatial situations he uses the...

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..., line, and region have turned out to be a fundamental abstraction for modeling the two-dimensional structure of geometric entities, their properties, relationships, and operations (see, for example, =-=[28, 30, 32, 33, 37]-=-). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this context are called bou...

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..., line, and region have turned out to be a fundamental abstraction for modeling the two-dimensional structure of geometric entities, their properties, relationships, and operations (see, for example, =-=[28, 30, 32, 33, 37]-=-). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this context are called bou...

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...bitemporal complexes. Their spatial features are described by simplicial complexes, and their temporal features are given by bitemporal elements attached to all components of simplicial complexes. In =-=[39]-=-, in a discrete snapshot model, a spatio-temporal object o is given as a time-evolving spatial object, that is, its evolution is represented by a set of triples (o id ; s i ; t i ) where o id is the o...

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...tions [13] have been identified which lead to the eight predicates called equal, disjoint, coveredBy, covers, overlap, meet, inside, and contains. For a line and a region 19 topological relationships =-=[14]-=- can be distinguished. For two lines 33 relationships [11] can be found. For a point and a region we obtain the three predicates disjoint, meet, and inside. For two points we get the two predicates di...

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...of this paper and introduces the concept of spatio-temporal data types [15, 16]. These data types are designed as abstract data types whose values can be integrated as complex entities into databases =-=[36, 35]-=- and whose definition and integration into databases is independent of a particular DBMS data model. The definition of a temporal object [22] in general is motivated by the observation that anything t...

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...ed in spatio-temporal query languages. The concept for spatio-temporal predicates presented in this paper is a hybrid approach taking into account elements from temporal logic (similar to the work in =-=[3]-=-) and elements from point set theory and point set topology. The main reason for not taking a purely logic approach is the intended integration into spatio-temporal databases and query languages. Thes...

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...ll-defined and regular. In order to be able to formally express this requirement and to avoid the anomalies just described, the formalization of a data type for regions is based on point set topology =-=[23]-=- and on a process called regularization [40]. Point set topology also rests on the point set paradigm but additionally distinguishes different parts of an (arbitrary) point set. Given such a point set...

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...e, and region have turned out to be a fundamental abstraction for modeling the twodimensional structure of geometric entities, their properties, relationships, and operations (see, for example, [28], =-=[30]-=-, [32], [33], [37]). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this cont...

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...odels integrating spatial aspects include variants of Gadia's temporal model [24], which are described in [6, 5]. A third approach has been to use linear constraints for modeling spatio-temporal data =-=[27]-=-. This model allows to efficiently represent and manipulate infinite point sets in arbitrary dimension. But in [27] time and geometry are treated as different and independent categories of data. Unfor...

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..., line, and region have turned out to be a fundamental abstraction for modeling the two-dimensional structure of geometric entities, their properties, relationships, and operations (see, for example, =-=[28, 30, 32, 33, 37]-=-). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this context are called bou...

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... approach has been to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute timestamped models (described in =-=[8, 9, 24, 34]-=-, for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently modeled as funct...

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...orts a more integrated view of space and time and incorporates the treatment of continuous spatial changes. It will be the basis of this paper and introduces the concept of spatio-temporal data types =-=[15, 16]-=-. These data types are designed as abstract data types whose values can be integrated as complex entities into databases [36, 35] and whose definition and integration into databases is independent of ...

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.... The concepts for spatio-temporal predicates and developments presented in the following sections have already been used in two different but related applications. The first application described in =-=[18]-=- relates to the identification of an important new class of spatio-temporal queries which is concerned with developments of spatial objects over time. This means queries ask especially for changes in ...

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... approach has been to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute timestamped models (described in =-=[8, 9, 24, 34]-=-, for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently modeled as funct...

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...mestamped models have the potential to fit very well with our purposes. Examples of temporal models integrating spatial aspects include variants of Gadia's temporal model [24], which are described in =-=[6, 5]-=-. A third approach has been to use linear constraints for modeling spatio-temporal data [27]. This model allows to efficiently represent and manipulate infinite point sets in arbitrary dimension. But ...

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...of this paper and introduces the concept of spatio-temporal data types [15, 16]. These data types are designed as abstract data types whose values can be integrated as complex entities into databases =-=[36, 35]-=- and whose definition and integration into databases is independent of a particular DBMS data model. The definition of a temporal object [22] in general is motivated by the observation that anything t...

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...n how developments of spatial objects can be concisely expressed by building 3 This number has been determined by a small Haskell program on the basis of a graph library developed by the first author =-=[15]-=-. REFERENCES 35 sequences of elementary spatio-temporal and spatial predicates. The compositional design of the framework encourages the modular, structured, and step-by-step creation of complex predi...

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...ta types including a comprehensive collection of spatio-temporal operations [16]. Nevertheless, behavioral time sequences could be used as one possible representation for our temporal objects. Galton =-=[25, 26]-=- stresses continuity as an important feature of movement. His goal is to devise a framework for formalizing common-sense knowledge of the world, and he particularly focuses on how continuity fits into...

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..., a region or a point. We will use this view for a comparison of topological predicates on spatio-temporal objects with those on 3D objects (see Sections 2.2, 3.2, and 5). Similar to our approach, in =-=[42, 43]-=- so-called behavioral time sequences have been introduced which are based on the work in [34]. Each element of such a sequence contains a geometric value, a date, and a behavioral function; the latter...

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...ed directly as a visual query interface to spatio-temporal databases, or it can provide predicate specifications that can be integrated into textual query languages leading to heterogeneous languages =-=[17]-=-. 2.3 Logic-Based Approaches An obvious approach to deal with temporal predicates is to employ concepts of temporal logic. We will first mention some related work and afterwards explain why we will ta...

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..., a region or a point. We will use this view for a comparison of topological predicates on spatio-temporal objects with those on 3D objects (see Sections 2.2, 3.2, and 5). Similar to our approach, in =-=[42, 43]-=- so-called behavioral time sequences have been introduced which are based on the work in [34]. Each element of such a sequence contains a geometric value, a date, and a behavioral function; the latter...

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...mestamped models have the potential to fit very well with our purposes. Examples of temporal models integrating spatial aspects include variants of Gadia's temporal model [24], which are described in =-=[6, 5]-=-. A third approach has been to use linear constraints for modeling spatio-temporal data [27]. This model allows to efficiently represent and manipulate infinite point sets in arbitrary dimension. But ...

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... approach has been to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute timestamped models (described in =-=[8, 9, 24, 34]-=-, for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently modeled as funct...

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...s can be integrated as complex entities into databases [36, 35] and whose definition and integration into databases is independent of a particular DBMS data model. The definition of a temporal object =-=[22]-=- in general is motivated by the observation that anything that changes over time can be expressed as a function over time. A temporal version of an object of type ff is then given by a function from t...

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...paper and introduces the concept of spatio-temporal data types [15], [16]. These data types are designed as abstract data types whose values can be integrated as complex entities into databases [36], =-=[35]-=- and whose definition and integration into databases is independent of a particular DBMS data model. The definition of a temporal object [22] in general is motivated by the observation that anything t...

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... realized in a relational model and how SQL can be appropriately extended to a spatio-temporal query language called STQL to enable the querying of developments. 3 The second application described in =-=[20, 21]-=- deals with the user's problem of comfortably specifying temporally changing topological situations and predicates. For this purpose, we have proposed a visual language which extends existing concepts...

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...ime intervals and not on time points. Galton [25] has extended Allen's approach to the treatment of temporally changing topological relationships. Topological relationships are based on the RCC model =-=[10]-=-, [31] which comes to similar results as Egenhofer's 9-intersection model. In contrast to Allen, Galton also takes time points into account, as we do. In specifying changes of spatial situations, he u...

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...edicates called equal, disjoint, coveredBy, covers, overlap, meet, inside, and contains. For a line and a region 19 topological relationships [14] can be distinguished. For two lines 33 relationships =-=[11]-=- can be found. For a point and a region we obtain the three predicates disjoint, meet, and inside. For two points we get the two predicates disjoint and meet (which corresponds to equality). For each ...

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... approach has been to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute timestamped models (described in =-=[8, 9, 24, 34]-=-, for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently modeled as funct...

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... realized in a relational model and how SQL can be appropriately extended to a spatio-temporal query language called STQL to enable the querying of developments. 3 The second application described in =-=[20, 21]-=- deals with the user's problem of comfortably specifying temporally changing topological situations and predicates. For this purpose, we have proposed a visual language which extends existing concepts...

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... spatial objects possibly satisfying some constraints over time. For example, an approach to modeling spatiotemporal partitions for describing the temporal evolution of maps has already been given in =-=[19]-=-. 2.2 Spatio-Temporal Predicates In the past, topological relationships like meet, overlap, or inside between point, line, and region features in the (especially two-dimensional) Euclidean space have ...

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...bitemporal complexes. Their spatial features are described by simplicial complexes, and their temporal features are given by bitemporal elements attached to all components of simplicial complexes. In =-=[39]-=-, in a discrete snapshot model, a spatio-temporal object o is given as a time-evolving spatial object, that is, its evolution is represented by a set of triples …o id; si;ti†, where o id is the object...

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...ta types including a comprehensive collection of spatio-temporal operations [16]. Nevertheless, behavioral time sequences could be used as one possible representation for our temporal objects. Galton =-=[25, 26]-=- stresses continuity as an important feature of movement. His goal is to devise a framework for formalizing common-sense knowledge of the world, and he particularly focuses on how continuity fits into...

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...n to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute time-stamped models (described in [8], [9], [24], =-=[34]-=-, for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently modeled as funct...

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...oral predicates is to employ concepts of temporal logic. We will first mention some related work and afterward explain why we will take a partly different approach in this paper. Based on his work in =-=[1]-=- where he has identified 13 possible relationships (like before, equal, meets, overlaps, during) between intervals, Allen in [2] defines a predicate Holds…p; i† which asserts that a property p is true...

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...ch. 2.1 Spatio-Temporal Objects So far, only a few data models for spatio-temporal data have been proposed. One approach has been to appropriately extend a spatial data model by temporal concepts. In =-=[41]-=-, spatio-temporal objects are defined as so-called spatio-bitemporal complexes. Their spatial features are described by simplicial complexes, and their temporal features are given by bitemporal elemen...

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...ll take a partly different approach in this paper. Based on his work in [1] where he has identified 13 possible relationships (like before, equal, meets, overlaps, during) between intervals, Allen in =-=[2]-=- defines a predicate Holds…p; i† which asserts that a property p is true during a time interval i. This predicate implies that p holds at every subinterval of i as well. Logical operators like Boolean...

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...n features in the (especially two-dimensional) Euclidean space have been intensively studied in the literature. Using point sets and point set topology as a formal framework, the 9-intersection model =-=[13]-=-, [7] provides a canonical collection of topological predicates for each combination of spatial types. Based on the nine topologically invariant intersections of boundaries, interiors, and exteriors o...

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...ures in the (especially two-dimensional) Euclidean space have been intensively studied in the literature. Using point sets and point set topology as a formal framework, the 9-intersection model [13], =-=[7]-=- provides a canonical collection of topological predicates for each combination of spatial types. Based on the nine topologically invariant intersections of boundaries, interiors, and exteriors of the...

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...approach has been to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute time-stamped models (described in =-=[8]-=-, [9], [24], [34], for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently...

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...n have turned out to be a fundamental abstraction for modeling the twodimensional structure of geometric entities, their properties, relationships, and operations (see, for example, [28], [30], [32], =-=[33]-=-, [37]). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this context are call...

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...mped models have the potential to fit very well with our purposes. Examples of temporal models integrating spatial aspects include variants of Gadia's temporal model [24], which are described in [6], =-=[5]-=-. A third approach has been to use linear constraints for modeling spatio-temporal data [27]. This model allows us to efficiently represent and manipulate infinite point sets in arbitrary dimension. B...

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...icates called equal, disjoint, coveredBy, covers, overlap, meet, inside, and contains. For a line and a region, 19 topological relationships [14] can be distinguished. For two lines, 33 relationships =-=[11]-=- can be found. For a point and a region, we obtain the three predicates disjoint, meet, and inside. For two points, we get the two predicates disjoint and meet (which corresponds to equality). For eac...

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...ed directly as a visual query interface to spatio-temporal databases, or it can provide predicate specifications that can be integrated into textual query languages leading to heterogeneous languages =-=[17]-=-. 2.3 Logic-Based Approaches An obvious approach for dealing with temporal predicates is to employ concepts of temporal logic. We will first mention some related work and afterward explain why we will...

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.... The concepts for spatio-temporal predicates and developments presented in the following sections have already been used in two different but related applications. The first application described in =-=[18]-=- relates to the identification of an important new class of spatio-temporal queries which is concerned with developments of spatial objects over time. This means queries ask especially for changes in ...

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...be realized in a relational model and how SQL can be appropriately extended to a spatio-temporal query language called STQL to enable the querying of developments. The second application described in =-=[20]-=-, [21] deals with the user's problem of comfortably specifying temporally changing topological situations and predicates. For this purpose, we have proposed a visual language which extends existing co...

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...lized in a relational model and how SQL can be appropriately extended to a spatio-temporal query language called STQL to enable the querying of developments. The second application described in [20], =-=[21]-=- deals with the user's problem of comfortably specifying temporally changing topological situations and predicates. For this purpose, we have proposed a visual language which extends existing concepts...

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...can be integrated as complex entities into databases [36], [35] and whose definition and integration into databases is independent of a particular DBMS data model. The definition of a temporal object =-=[22]-=- in general is motivated by the observation that anything that changes over time can be expressed as a function over time. A temporal version of an object of type is then given by a function from time...

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...as been to appropriately extend a temporal data model based on attribute timestamps by spatial concepts. In contrast to tuple timestamped models, attribute time-stamped models (described in [8], [9], =-=[24]-=-, [34], for example) aim at gathering information about an object in one tuple and allow complex attribute values. These complex values incorporate the temporal dimension and are frequently modeled as...

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...ta types, including a comprehensive collection of spatiotemporal operations [16]. Nevertheless, behavioral time sequences could be used as one possible representation for our temporal objects. Galton =-=[25]-=-, [26] stresses continuity as an important feature of movement. His goal is to devise a framework for formalizing common-sense knowledge of the world, and he particularly focuses on how continuity fit...

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...es, including a comprehensive collection of spatiotemporal operations [16]. Nevertheless, behavioral time sequences could be used as one possible representation for our temporal objects. Galton [25], =-=[26]-=- stresses continuity as an important feature of movement. His goal is to devise a framework for formalizing common-sense knowledge of the world, and he particularly focuses on how continuity fits into...

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...els integrating spatial aspects include variants of Gadia's temporal model [24], which are described in [6], [5]. A third approach has been to use linear constraints for modeling spatio-temporal data =-=[27]-=-. This model allows us to efficiently represent and manipulate infinite point sets in arbitrary dimension. But, in [27], time and geometry are treated as different and independent categories of data. ...

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...t, line, and region have turned out to be a fundamental abstraction for modeling the twodimensional structure of geometric entities, their properties, relationships, and operations (see, for example, =-=[28]-=-, [30], [32], [33], [37]). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in thi...

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...time-dependent) distance of an airplane and a storm. This could be achieved by an operation: Distance : …point† …region†! …real†: This example demonstrates the important principle of temporal lifting =-=[29]-=- in avoiding an inflation of operation definitions: We can, in principle, take almost any flat (that is, nontemporal) operation and ªliftº it so that it works on temporal objects returning also a temp...

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... region have turned out to be a fundamental abstraction for modeling the twodimensional structure of geometric entities, their properties, relationships, and operations (see, for example, [28], [30], =-=[32]-=-, [33], [37]). Points are elements of the Euclidean plane. Lines are two-dimensional curves. Regions describe point sets with a two-dimensional extent and are bounded by lines which in this context ar...

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... this paper and introduces the concept of spatio-temporal data types [15], [16]. These data types are designed as abstract data types whose values can be integrated as complex entities into databases =-=[36]-=-, [35] and whose definition and integration into databases is independent of a particular DBMS data model. The definition of a temporal object [22] in general is motivated by the observation that anyt...

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..., a region or a point. We will use this view for a comparison of topological predicates on spatio-temporal objects with those on 3D objects (see Sections 2.2, 3.2, and 5). Similar to our approach, in =-=[42]-=-, [43], so-called behavioral time sequences have been introduced which are based on the work in [34]. Each element of such a sequence contains a geometric value, a date, and a behavioral function; the...

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...gion or a point. We will use this view for a comparison of topological predicates on spatio-temporal objects with those on 3D objects (see Sections 2.2, 3.2, and 5). Similar to our approach, in [42], =-=[43]-=-, so-called behavioral time sequences have been introduced which are based on the work in [34]. Each element of such a sequence contains a geometric value, a date, and a behavioral function; the latte...