Keywords

1 Introduction

In the Semantic Web field, representing and reasoning on imprecise temporal information is a common requirement. Indeed, temporal information given by users is often imprecise. For instance, if they give the information “Alexandre was married to Nicole by 1981 to late 90” two measures of imprecision are involved. On the one hand, the information “by 1981” is imprecise in the sense that it could mean approximately from 1980 to 1982; on the other hand, the information “late 90” is imprecise in the sense that it could mean, with an increasingly possibility, from 1995 to 2000. When an event is characterized by a gradual beginning and/or ending, it is usual to represent the corresponding time span as an imprecise time interval.

In OWL, many works have been proposed to represent and reason on precise temporal information; however, to the best of our knowledge, there is no work devoted to represent and reason on imprecise time intervals.

In this paper, we propose a fuzzy-based approach for representing and reasoning on imprecise time intervals in ontology. It is based on fuzzy sets theory and fuzzy tools. It is based on Fuzzy-OWL 2 [1] which is an extension of OWL 2 that deals with fuzzy information. To represent imprecise time intervals in Fuzzy-OWL 2, we extend the 4D-fluents model [2] in two ways: (1) It is enhanced with new fuzzy components to be able to model imprecise time intervals. (2) It is enhanced with qualitative temporal expressions representing fuzzy relations between imprecise temporal intervals. To reason on imprecise time intervals, we extend Allen’s work to compare imprecise time intervals in a fuzzy gradual personalized way. Our Allen’s extension introduces gradual fuzzy interval relations e.g., “long before”. It is personalized in the sense that it is not limited to a given number of interval relations. It is possible to determinate the level of precision that should be in a given context. For instance, the classic Allen relation “before” may be generalized in N interval relations, where “before(1)” means “just before” and gradually the time gap between the two imprecise intervals increases until “before(N)” which means “long before”. The resulting fuzzy interval relations are inferred from the introduced imprecise time intervals using the FuzzyDL reasoner [3], via a set of Mamdani IF-THEN rules, in Fuzzy-OWL 2.

The current paper is organized as follows: Sect. 2 is devoted to present some preliminary concepts and related work in the field of temporal information representation in OWL and reasoning on time intervals. In Sect. 3, we introduce our fuzzy-based approach for representing and reasoning on imprecise time intervals. Section 4 draws conclusions and future research directions.

2 Preliminaries and Related Work

In this section, we introduce some preliminary concepts and related work in the field of temporal information representation in OWL and reasoning on time intervals.

2.1 Representing Temporal Information in OWL

Five main approaches are proposed to represent time information in OWL: Temporal Description Logics [4], Versioning [5], N-ary relations [6] and 4D-fluents [2]. All these approaches represent only crisp temporal information in OWL. Temporal Description Logics extend the standard description logics with additional temporal constructs e.g., “sometime in the future”. N-ary relations approach represents an N-ary relation using an additional object. The N-ary relation is represented as two properties each related with the new object. The two objects are related to each other with an N-ary relation. Reification is “a general purpose technique for representing N-ary relations using a language such as OWL that permits only binary relations” [7]. Versioning approach is described as “the ability to handle changes in ontologies by creating and managing different variants of it” [5]. When an ontology is modified, a new version is created to represent the temporal evolution of the ontology. 4D-fluents approach represents temporal information and the evolution of the last ones in OWL. Concepts varying in time are represented as 4-dimensional objects with the 4th dimension being the temporal dimension.

Based on the present related work, we choose the 4D-fluents approach. Indeed, compared to related work, it minimizes the problem of data redundancy as the changes occur only on the temporal parts and keeping therefore the static part unchanged. It also maintains full OWL expressiveness and reasoning support [7]. We extend this approach in two ways. It is extended with new fuzzy components to represent (1) imprecise time intervals and (2) fuzzy interval relations.

2.2 Allen’s Interval Algebra

In [8], Allen has proposed 13 mutually exclusive primitive relations that may hold between two precise time intervals. Their semantics is illustrated in Table 1. Let I = [I−, I+] and J = [J−, J+] two time intervals; where I− (respectively J−) is the beginning time-step of the event and I+ (respectively J+) is the ending.

Table 1. Allen’s temporal interval relations
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A number of works fuzzify Allen’s temporal interval relations. We classify these works into (1) works focusing on fuzzifying Allen’s interval algebra to compare precise time intervals and (2) works focusing on fuzzifying Allen’s interval algebra to compare imprecise time intervals.

Three approaches have been proposed to fuzzify Allen’s interval algebra in order to compare two precise time intervals: [9], [10] and [11]. In [9], the authors propose fuzzy Allen relations viewed as fuzzy sets of ordinary Allen relationship taking into account a neighborhood structure, a notion originally introduced in [12]. In [10], the authors represent a time interval as a pair of possibility distributions that define the possible values of the endpoints of the crisp interval. Using possibility theory, the possibility and necessity of each of the interval relations can then be calculated. This approach also allows modeling imprecise relations such as “long before”. In [11], the authors propose a fuzzy extension of Allen’s work, called IAfuz where degrees of preference are associated to each relation between two precise time intervals.

Four approaches have been proposed to fuzzify Allen’s interval algebra to compare two imprecise time intervals: [13, 14, 15 and 16]. In [13], the authors propose a temporal model based on fuzzy sets to extend Allen relations with imprecise time intervals. The authors introduce a set of auxiliary operators on intervals and define fuzzy counterparts of these operators. The compositions of these relations are not studied by the authors. In [14], the authors propose an approach to handle some gradual temporal relations as “more or less finishes”. However, this work cannot take into account gradual temporal relations such as “long before”. Furthermore, many of the symmetry, reflexivity, and transitivity properties of the original temporal interval relations are lost in this approach; thus it is not suitable for temporal reasoning. In [15], the authors propose a generalization of Allen’s relations with precise and imprecise time intervals. This approach allows handling classical temporal relations, as well as other imprecise relations. Interval relations are defined according to two fuzzy operators comparing two time instants: “long before” and “occurs before or at approximately the same time”. In [16], the authors generalize the definitions of the 13 Allen’s classic interval relations to make them applicable to fuzzy intervals in two ways (conjunctive and disjunctive). Gradual temporal interval relations are not taken into account.

3 Our Fuzzy-Based Approach for Representing and Reasoning on Imprecise Time Intervals in Ontology

In this section, we propose our fuzzy-based approach to represent and reason on imprecise time intervals. This approach is based on a fuzzy environment. We extend the 4D-fluents model to represent imprecise time intervals and their relationships in Fuzzy-OWL 2. To reason on imprecise time intervals, we extend the Allen’s interval algebra in a fuzzy gradual personalized way. We infer the resulting fuzzy interval relations in Fuzzy-OWL 2 using a set of Mamdani IF-THEN rules.

3.1 Representing Imprecise Time Intervals and Fuzzy Qualitative Interval Relations in Fuzzy-OWL 2

We represent an imprecise time interval using fuzzy sets. We represent the imprecise beginning interval bound as a fuzzy set which has the L-function membership function and the ending interval bound as a fuzzy set which has the R-function membership function. Let I = [I−, I+] be an imprecise time interval. We represent the binging bound I− as a fuzzy set which has the L-function membership function (A = I−(1) and B = I−(N)). We represent the ending bound I+ as a fuzzy set which has the R-function membership function (A = I+(1) and B = I+(N)). For instance, if we have the information “Alexandre was starting his PhD study in 1973 and was graduated in late 80”, the beginning bound is crisp. The ending bound is imprecise and it is represented by L-function membership function (A = 1976 and B = 1980). For the rest of the paper, we use the membership functions defined in [17] and shown in Fig. 1.

Fig. 1.
figure 1

R-Function, L-Function and Trapezoidal membership functions [17].

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The classic 4D-fluents model introduces two crisp classes “TimeSlice” and “TimeInterval” and four crisp properties “tsTimeSliceOf”, “tsTimeInterval”, “hasBegining” and “hasEnd”. The class “TimeSlice” is the domain class for entities representing temporal parts (i.e., “time slices”). The property “tsTimeSliceOf” connects an instance of class “TimeSlice” with an entity. The property “tsTimeInterval” connects an instance of class “TimeSlice” with an instance of class “TimeInterval”. The instance of class “TimeInterval” is related with two temporal instants that specify its starting and ending points using, respectively, the “hasBegining” and “hasEnd” properties. Figure 2 illustrates the use of the 4D-fluents model to represent the following example: “Alexandre was started his PhD study in 1975 and he was graduated in 1978”.

Fig. 2.
figure 2

An instantiation of the classic the 4D-fluents model.

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We extend the original 4D-fluents model to represent imprecise time intervals in the following way. We add two fuzzy datatype properties “FuzzyHasBegining” and “FuzzyHasEnd” to the class “TimeInterval”. “FuzzyHasBegining” has the L-function membership function (A = I−(1) and B = I−(N)). “FuzzyHasEnd” has the R-function membership function (A = I+(1) and B = I+(N)).

The 4D-fluents approach is also enhanced with qualitative temporal relations that may hold between imprecise time intervals. We introduce the “FuzzyRelationIntervals”, as a fuzzy object property between two instances of the class “TimeInterval”. “FuzzyRelationIntervals” represent fuzzy qualitative temporal relations. “FuzzyRelationIntervals” has the L-function membership function (A = 0 and B = 1). Figure 3 represents our extended 4D-fluents model in Fuzzy-OWL 2.

Fig. 3.
figure 3

The extended 4D-fluents model in Fuzzy-OWL 2.

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We can see in Fig. 4 an instantiation of the extended 4D-fluents model in Fuzzy-OWL2. On this example, we consider the following information: “Alexandre was married to Nicole just after he was graduated with a PhD. Alexandre was graduated with a PhD in 1980. Their marriage lasts 15 years. Alexandre was remarried to Béatrice since about 10 years and they were divorced in 2016”. Let I = [I−, I+] and J = [J−, J+] be two imprecise time intervals representing, respectively, the duration of the marriage of Alexandre with Nicole and the one with Béatrice. I− is represented with the fuzzy datatype property “FuzzyHasBegining” which has the L-function membership function (A = 1980 and B = 1983). I+ is represented with the fuzzy datatype property “FuzzyHasEnd” which has the R-function membership function (A = 1995 and B = 1998). J− is represented with the fuzzy datatype property “FuzzyHasBegining” which has the L-function membership function (A = 2005 and B = 2007). J+ is represented with the crisp datatype property “HasEnd” which has the value “2016”.

Fig. 4.
figure 4

An instantiation of the extended 4D-fluents model in Fuzzy-OWL 2.

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3.2 Representing Imprecise Time Intervals and Fuzzy Qualitative Interval Relations in Fuzzy-OWL 2

We propose a set of fuzzy gradual personalized comparators that may hold between two time instants T1 and T2. Based on these operators, we present our fuzzy gradual personalized extension of Allen’s work to compare two imprecise time intervals. Then, we infer, in Fuzzy OWL 2, the resulting temporal interval relations via a set of Mamdani IF-THEN rules using the fuzzy reasoner FuzzyDL.

We generalize the crisp time instants comparators “Follow”, “Precede” and “Same”, introduced in [18]. Let α and β two parameters allowing the definition of the membership function of the following comparators (∈]0, +∞[); N is the number of slices; T1 and T2 are two time instants; we define the following comparators (illustrated in Fig. 5):

Fig. 5.
figure 5

Fuzzy gradual personalized time instants comparators. (A) Fuzzy sets of {Follow(1) (α, β) (T1, T2) . . . Follow(N) (α, β) (T1, T2)}. (B) Fuzzy sets of {Precede(1) (α, β) (T1, T2) . . . Precede(N) (α, β) (T1, T2)}. (C) Fuzzy set of Same(α, β) (T1, T2).

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  • {Follow(1) (α, β) (T1, T2) . . . Follow(N) (α, β) (T1, T2)} are a generalization of the crisp time instants relation “Follows”. Follow(1) (α, β) (T1, T2) means that T1 is “just after or approximately at the same time” T2 w.r.t. (α, β) and gradually the time gap between T1 and T2 increases until Follow(N) (α, β) (T1, T2) which means that T1 is “long after” T2 w.r.t. (α, β). N is set by the expert domain. {Follow(1) (α, β) (T1, T2) . . . Follow(N) (α, β) (T1, T2)} are defined as fuzzy sets. Follow(1) (α, β) (T1, T2) has R-Function membership function which has as parameters A = α and B = (α + β). All comparators {Follow(2) (α, β) (T1, T2) … Follow(N−1) (α, β) (T1, T2)} have trapezoidal membership function which has as parameters A = ((K − 1) α) and B = ((K − 1) α + (K − 1) β), C = (K α + (K − 1) β) and D = (K α + K β); where 2 ≤ K ≤ N − 1. Follow(N) (α, β) (T1, T2) has L-Function membership function which has as parameters A = ((N − 1) α + (N − 1) β) and B = ((N − 1) α + (N − 1) β);

  • {Precede(1) (α, β) (T1, T2) … Precede(N) (α, β) (T1, T2)} are a generalization of the crisp time instants relation “Precede”. Precede(1) (α, β) (T1, T2) means that T1 is “just before or approximately at the same time” T2 w.r.t. (α, β) and gradually the time gap between T1 and T2 increases until Precede(N) (α, β) (T1, T2) which means that T1 is “long before” T2 w.r.t. (α, β). N is set by the expert domain. {Precede(1) (α, β) (T1, T2) … Precede(N) (α, β) (T1, T2)} are defined as fuzzy sets. Precede(i) (α, β) (T1, T2) is defined as:

    $$ \begin{aligned} Precede_{(i)} \,{}^{(\alpha , \, \beta )}\left( {T_{1} , \, T_{2} } \right) = \,1 - Follow_{(i)} {}^{(\alpha , \, \beta )}\left( {T_{1} , \, T_{2} } \right) \hfill \\ \hfill \\^{{}} \hfill \\ \end{aligned} $$
    (1)

  • We define the comparator Same(α, β) (T1, T2) which is a generalization of the crisp time instants relation “Same”. Same(α, β) (T1, T2) means that T1 is “approximately at the same time” T2 w.r.t. (α, β). It is defined as:

    $$ Same{}^{{^{(\alpha , \, \beta )} }}\left( {T_{1} , \, T_{2} } \right)\, = \,Min(Follow_{(1)} {}^{(\alpha , \, \beta )}(T_{1} , \, T_{2} ),Precede_{(1)} {}^{(\alpha , \, \beta )}(T_{1} , \, T_{2} )) $$
    (2)

Then, we extend Allen’s work to compare imprecise time intervals with a fuzzy gradual personalized view. We provide a way to model gradual, linguistic-like description of temporal interval relations. Compared to related work, our work is not limited to a given number of imprecise relations. It is possible to determinate the level of precision that should be in a given context. For instance, the classic Allen relation “before” may be generalized in N imprecise relations, where “Before(1) (α, β) (I, J)” means that I is “just before” J w.r.t. (α, β) and gradually the time gap between I and J increases until “Before(N) (α, β) (I, J)” which means that I is long before J w.r.t. (α, β). The definition of our fuzzy interval relations is based on the fuzzy gradual personalized time instants compactors. Let I = [I−, I+] and J = [J−, J+] two imprecise time intervals; where I− has the L-function membership function (A = I−(1) and B = I−(N)); I+ is a fuzzy set which has the R-function membership function (A = I+(1) and B = I+(N)); J− is a fuzzy set which has the L-function membership function (A = J−(1) and B = J−(N)); J+ is a fuzzy set which has the R-function membership function (A = J+(1) and B = J+(N)). For instance, the fuzzy interval relation “Before(1) (α, β) (I, J)” is defined as:

$$ \forall I^{ + \left( i \right)} \in I^{ + } ,\forall J^{ - \left( j \right)} \in J^{ - } : \, Precede_{(1)} {}^{(\alpha , \, \beta )}(I^{ + (i)} ,J^{ - (j)} ) $$
(3)

This means that the most recent time instant of I+ (I+(N)) ought to proceed the oldest time instant of J− (J−(1)):

$$ Precede_{(1)} {}^{(\alpha , \, \beta )}(I^{ + (N)} ,J^{ - (1)} ) $$
(4)

In the similar way, we define the others temporal interval relations, as shown in Table 2.

Table 2. Fuzzy gradual personalized temporal interval relations upon imprecise time intervals.
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Finally, we have implemented our fuzzy gradual personalized extension of Allen’s work in Fuzzy-OWL 2. We use the ontology editor PROTÉGÉ version 4.3 and the fuzzy reasoner FuzzyDL. We propose a set of Mamdani IF-THEN rules to infer the temporal interval relations from the introduced imprecise time intervals which are represented using the extended 4D-fluents model in Fuzzy-OWL2. For each temporal interval relation, we associate a Mamdani IF-THEN rule. For instance, the Mamdani IF-THEN rule to infer the “Overlaps(1) (α, β) (I, J)” relation is the following:

(define-concrete-feature Precede(1/1) real) (define-concrete-feature Precede(1/2) real) (define-concrete-feature Precede(1/3) real) (define-concrete-feature Overlaps(1) real)

(define-fuzzy-concept Fulfilled Right-shoulder(0,-α-β,-α,0)) (define-fuzzy-concept True Right-shoulder(0,0,1,1))

(define-concept Rule0 (g-and (some Precede(1/1) Fulfilled) (some Precede(1/2) Fulfilled) Fulfilled) (some Precede(1/3) Fulfilled) (some Overlaps(1) True))) //Fuzzy rule

(instance facts (= Precede(1/1) (I−(N) - J−(1)))) (instance facts (= Precede(1/2) (J−(N) - I+(1)))) (instance facts (= Precede(1/3) (I+(N) - J+(1)))) //Instantiations

We define three input fuzzy variables, named “Precede(1/1)”, “Precede(1/2)” and “Precede(1/3)”, which have the same membership function than that of “Precede(1) (α, β)”. We define one output variable “Overlaps(1)” which has the same membership than that of the fuzzy object property “FuzzyRelationIntervals”. “Precede(1/1)”, “Precede(1/2)” and “Precede(1/3)” are instantiated with, respectively, (I−(N) − J−(1)), (J−(N) − I+(1)) and (I+(N) − J+(1)).

4 Conclusion

In this paper, we proposed a fuzzy-based approach to represent and reason on imprecise time intervals in ontology. It is entirely based only on fuzzy environment. We extended the 4D-fluents model to represent imprecise time intervals and fuzzy interval relations in Fuzzy-OWL 2. To reason on imprecise time intervals, we extend the Allen’s interval algebra in a fuzzy gradual personalized way. We infer the resulting fuzzy interval relations in Fuzzy-OWL2 using a set of Mamdani IF-THEN rules.

The works presented in this paper have been tested in two projects, having in common to manage life logging data: (1) In the VIVAFootnote 1 project, we aim to design the Captain Memo memory prosthesis [19] [20] for Alzheimer Disease patients. Among other functionalities, this prosthesis manages a knowledge base on the patient’s family tree, using an OWL ontology. Imprecise inputs are especially numerous when given by an Alzheimer Disease patient. Furthermore, dates are often given in reference to other dates or events. Thus, we have been using our approach reported in this paper. One interesting point in this solution is to deal with a personalized slicing of the person’s life in order to sort the different events. (2) The ANR DAPHNE project aims to allow Middle Ages specialized historians to deal with prosopographical data bases storing Middle age academic’s career histories. Data come from various archives among Europe and data about a same person are very difficult to align. Representing and ordering imprecise time interval is required to redraw the careers across the different European universities who hosted the person.

Future work will be devoted to propose a crisp-based approach to represent and reason on imprecise time intervals in ontology. This approach uses only crisp standards and tools and is modeled in OWL 2.