Representing and Reasoning about Temporal Granularities

C. Combi
2004 Journal of Logic and Computation  
In this paper, we propose a new logical approach to represent and to reason about different time granularities. We identify a time granularity as an infinite sequence of time points properly labelled with proposition symbols marking the starting and ending points of the corresponding granules, and we symbolically model sets of granularities by means of linear time logic formulas. Some real-world granularities are provided, from a clinical domain and from the Gregorian Calendar, to motivate and
more » ... xemplify our approach. Different formulas are introduced, which represent relations between different granularities. The proposed framework permits to algorithmically solve the consistency, the equivalence, and the classification problems in a uniform way, by reducing them to the validity problem for the considered linear time logic. Context and Motivation Any time granularity can be viewed as the partitioning of a temporal domain in groups of elements, where each group is perceived as an indivisible unit (a granule). The description of a fact can use these granules to provide it with a temporal qualification, at the appropriate abstraction level. However, adding the concept of time granularity to a formalism does not merely mean that one can use different temporal units to represent temporal quantities in a unique flat model, but it involves semantic issues related to the problem of assigning a proper meaning to the association of statements with the different temporal domains of a layered model and of switching from one domain to a coarser/finer one. The ability of providing and relating temporal representations at different 'grain levels' of the same reality is an important research theme in computer science. In particular, it is a major requirement for formal specifications, temporal databases, data mining, problem solving, and natural language understanding. • As for formal specifications, there exists a large class of reactive systems whose components have dynamic behavior regulated by very different time constants (granular reactive systems). A good specification language must enable one to specify and verify the behavior of the components of a granular reactive system and their interactions in a simple and intuitively clear way [13, 19, 20, 29, 40, 46, 47, 48, 49] . • With regard to temporal databases, the common way to represent temporal information is to timestamp either attributes (attribute timestamping) or tuples/objects (tuple-timestamping). Timestamping is performed taking time values over some fixed granularity. However, it may happen that differently-grained timestamps have to be associated with data, for example, when information is collected from different sources which are not under the same control. Furthermore, users and applications may also require the flexibility of viewing and querying the temporal information stored in the database in terms of different granularities. To guarantee consistency either the data must be converted into a uniform representation that is independent of time granularity or temporal operations must be generalized to cope with data associated with different temporal domains. In both cases, a precise semantics for time granularity is needed [3, 12, 18, 26, 37, 38, 45, 51, 54, 58, 59] . • With regard to data mining, a huge amount of data is collected every day in the form of event-time sequences. These sequences represent valuable sources of information, not only for what is explicitly registered, but also for deriving implicit information and predicting the future behavior of the process that we are monitoring. 2 The latter activity requires an analysis of the frequency of certain events, the discovery of their regularity, and the identification of sets of events that are linked by particular temporal relationships. Such frequencies, regularity, and relationships are very often expressed in terms of multiple granularities, and thus analysis and discovery tools must be able to deal with these granularities [1, 4, 6, 25, 42] . • With regard to problem solving, several problems in scheduling, planning, and diagnosis can be formulated as temporal constraint satisfaction problems, often involving multiple time granularities. In a temporal constraint satisfaction problem, variables are used to represent event occurrences and constraints are used to represent their granular temporal relationships [8, 5, 21, 28, 36, 39, 50, 53, 55] . • Finally, shifts in the temporal perspective occur very often in natural language communication, and thus the ability of supporting and relating a variety of temporal models, at different grain sizes, is a relevant feature for the task of natural language understanding [11, 30, 34] . A further distinction we have to introduce is between the representation and reasoning on time granularities and the representation and reasoning on facts/statements associated with times specified at different granularities. The requirements for reasoning on facts at different levels of granularity are often related and specific to the different research areas mentioned above: e.g., supporting for different time granularities for database query languages [26] , supporting the specification of real-time systems with different granularities [20], providing algorithms for pattern discovery on time series with several time units [4] . Nevertheless, the need for formalisms allowing the specification and the reasoning on granularities is common to all the mentioned research areas and originated several different proposals [9, 30, 35, 44, 51, 52, 60] . More specifically, most approaches proposed in the literature for representing and reasoning about time granularity can be classified into algebraic approaches and logical ones. In the algebraic (or operational) framework, a bottom granularity is assumed, and a finite set of calendar operators are exploited to create new granularities by suitably manipulating other granularities [9, 30, 51, 52] . In the logical (or descriptive) framework for time granularity, the different granularities and their interconnections are represented by means of mathematical structures called layered structures, consisting of a possibly infinite set of related differently-grained temporal domains. Suitable operators make it possible to move within a given temporal domain and across temporal domains. Logical formulas allow one to specify properties involving different time granularities in a single formula by mixing such operators [33, 44, 46, 48, 49] . Algebraic and logical frameworks stem from different research areas calling for different focuses. For instance, in the database context, where the algebraic framework is usually adopted, granule conversion plays a major role 3 because it allows the user to view the temporal information contained in the database in terms of different granularities, while in the context of verification, where logical frameworks have been proposed, decision procedures are unavoidable to automatically validate the system (for example, to establish whether two different representations define the same granularity). Abstracting away from the research areas of the two frameworks, it is possible to identify their main limitations and advantages. The main advantage of the algebraic framework is its naturalness: by applying user-friendly operations to existing standard granularities like 'days', 'weeks', and 'months', a quite large class of new granularities, like 'business weeks', 'business months', and 'years since 2000', can be easily generated. The major weakness of the algebraic framework is that reasoning methods basically reduce to granule conversions and semantic translations of statements. Little attention has received the investigation of algorithms to check whether some meaningful relation holds between granularities (e.g., to verify whether the granularity G 1 is finer than granularity G 2 or G 1 is equivalent to G 2 ). Moreover, only a finite number of time granularities can be represented. On the contrary, reasoning methods have been extensively investigated in the logical framework,
doi:10.1093/logcom/14.1.51 fatcat:jrecrdzmxfcedgokp2xpl3prqe