Summary of Recent Approaches

Temporal representation and reasoning is an essential feature in any activities that involve changes. This explains, why temporal representation and reason- ing services are so important and appear in so many areas, including planning, natural language understanding, and knowledge representation.

Recent articles describe approaches in the area of Temporal Constraint Programming, an important area of temporal reasoning [102, 37]. Gennari describes a temporal reasoning system as a temporal knowledge base. It also contains a procedure to check its consistency, and inference mechanisms, which are able to derive new information and get a solution or all solutions to queries. Temporal reasoning tasks are mainly formulated as constraint satisfaction problems; therefore, the constraint satisfaction techniques can be used to check consistency, to search for solutions or all solutions for the given problem.

Events are the primitive entities in the knowledge base. They are character- ized in temporal constraint programming by means of their time of occurrence, which can be given by time points or intervals (see above).

Temporal information can constrain events to happen at a particular time (e.g., “Coffee time is at 3:30 pm”) or to hold during a time interval (e.g., “A class lasts 90 minutes”); moreover it can state relations between events of a

qualitative type (e.g., is before or of a metric one (e.g.,

has started at least three hours before

Constraints can be either extensionally characterized by real or rational numbers, or intensionally represented as (finite) sets or relations of some al- gebra (e.g., Allen’s interval algebra [2]). According to the formalization of constraints and the time unit chosen, the approaches can be classified into three main streams4:

Temporal reasoning with metric information: In the quantitative approach to temporal reasoning with constraints, variables range over real or rational numbers. Originally finite sets of real intervals, constraints are lately represented by unions of interval-sets. A temporal constraint is

explicitly given as a set of intervals where The

Other authors such as [102] and [123] describe these three main streams as met- ric point (for metric information), qualitative point and qualitative interval (for qualitative approaches based on Allen’s interval algebra), and combinations (for mixed approaches).

constraints can be unary or binary and are represented by

An unary constraint restricts the domain of a vari- able to the given set of intervals. Thus, it is represented by the dis-

junction The binary constraint

restricts the values for the distance of the variables and represents

the disjunction [23]. The

authors assume that all the intervals are pairwise disjoint.

Constraint propagation algorithms are based on metric properties of the continuous variable domain. Since the satisfiability problem of general tem- poral constraints is NP-hard, research if focussed on particular classes of temporal constraint problems such as single temporal constraint problems, backtracking algorithms, and constraint propagation algorithms in order to achieve local consistency or at least a good approximation of local con- sistency (e.g., [101]).

In principle, these methods can be used for reasoning services on the Se- mantic Web. However, the adaptation for their use implies a large modeling effort.

Qualitative approaches based on Allen’s interval algebra: The most fun- damental and well-known theory about reasoning with time intervals has been formulated by [2]. This approach has been revised over the years and is based on interval structures, which are used as primitives.5

Allen motivates his approach with the problem that much of our tempo- ral knowledge is relative, and hence cannot be described by a date (or even a fuzzy date). As Allen further argues in his paper, his framework is particularly designed for these reasons:

it allows “significant imprecision”: much temporal knowledge is relative and sometimes it has no relation to absolute dates;

“uncertainty of information” can be represented by means of disjunc- tions of relations between two intervals;

because of the qualitative representation of constraints one has a cer- tain freedom when modeling knowledge and can choose the grain of rea- soning, for instance expressing time in terms of days, weeks or business- days;

the reasoning engine allows for default reasoning of the type “If I parked my car in lot A this morning, then it should still be there now”. In Allen’s framework, variables range over real or rational valued intervals. Constraints are specified as unions of atomic (basic) relations, which are pairwise disjoint. Variables represent time intervals and the basic temporal relations are

There is a difference to the intervals described above since those intervals are composed by time points. Here, time intervals are primitives.

The class of all possible unions of the atomic relations forms a boolean al- gebra, Allen’s interval algebra. There are 13 atomic relations and thus relations in total. Checking consistency for this algebra turned out to be NP-hard. Allen introduces a path-consistency algorithm to deal with the problems that propagates relations between intervals by means of compo- sition. The algebra consists of relations which means that there are possible subsets in that algebra, which make them intractable. Therefore, research in that area is concentrating on tractable and recently

maximal tractable subalgebras. Some of the most important subalgebras of Allen’s interval algebra are obtained by “translating” metric point rela- tions into Allen relations. This means that there have to be languages to describe sets of qualitative or quantitative relations between points, and that these have to be translated in tractable subalgebras.

An exhaustive search by computers is a key technique to prove the maxi- mality of the algebras that up to now have been discovered; this machine case analysis was firstly introduced by [86]. A different approach to this problem in a geometric and not a logic apparatus, is given in Ligozat’s work [75, 74]. Some of the studied subalgebras are the Point Algebra [124, 5] and the NB algebra [86]. To compute a solution, backtracking search is used. It has been shown that the search gets more effective with the addi- tional use of path-consistency checking such as a forward-checking method within the backtracking algorithm [102].

These mentioned arguments hold true also for the Semantic Web. Thus, interval-based approaches are valuable when discussing methods and tech- niques for temporal reasoning on the Web.

Mixed approach based on metric and qualitative constraints: In this frame- work, the other approaches are combined in order to gain expressiveness, while trying not to loose the tractability of the problem; however, the com- plexity results are not always optimal. The ontological entities in the first approach are time points only, and the primitive entities in the second approach are time intervals. This third approach involves both points and intervals as primitive objects of the language; therefore new relations are introduced in order to “relate” time points and time intervals.

Some authors have studied particular metric temporal constraint problems in order to find new sub-algebras of interval algebra. This can be seen as a qualitative approach because its main goal is an interval algebra. An approach is “mixed” when it aims at using both the expressive power of the qualitative and of the quantitative approaches to create “new” temporal frameworks, of which the satisfiability can be decided in polynomial time. The research in this direction is one of the most promising [107], however, the relative literature is still scarce.