Representing and reasoning fuzzy spatio- temporal knowledge with description logics: A survey

Abstract

Description logic, as a logical foundation of knowledge representation and reasoning, plays an important role in the Semantic Web. In practical applications, many fields contain a large number of fuzzy spatio-temporal knowledge. With a large amount of fuzzy spatio-temporal knowledge and many corresponding applications being incorporated into the Semantic Web, description logic becomes an effective method to solve the problem of fuzzy spatio-temporal knowledge representation and reasoning. Currently, many efforts have been done on fuzzy spatio-temporal extensions of description logics, and the literature on fuzzy spatio-temporal description logic has been booming. To address these issues and more importantly, in this paper, we provide a comprehensive survey of the research literature that applies description logics techniques in fuzzy spatio-temporal representation and reasoning. The paper serves as helping readers grasp the main results and highlighting the direction of fuzzy spatio-temporal representation and reasoning based on description logics.

Keywords

Fuzzy description logic fuzzy spatio-temporal knowledge representation reasoning Semantic Web

1. Introduction

Time and space are the most basic two properties in the real world. In many research areas, the study of time and space is seen as two separate branches, such as temporal database and spatial database. However, in practical applications, time and space are closely linked, and some spatial information changes over time. With the increasing demand for time and space applications, human being is no longer confined to separate time and space information, but spatio-temporal combination information. Currently, spatio-temporal information can be found in the areas of spatio-temporal databases, dynamic social network, Geographic Information Systems (GIS), meteorological monitoring systems, and Semantic Web [75, 21].

As an important research area of knowledge representation and reasoning [26, 17], spatio-temporal knowledge representation and reasoning have successfully solved many problems related to spatio-temporal representation, modeling, reasoning, and queries in many spatio-temporal applications [13, 36]. In recent years, spatio-temporal knowledge representation and reasoning have already attracted the attention of many researchers. They have also been extensively used in the areas of GIS, spatio-temporal database and Artificial Intelligence (AI) [24]. Spatio-temporal knowledge representation is a research field that uses formal symbols to represent the facts in spatio-temporal domain, while spatio-temporal reasoning is a a formalized operation of symbols focusing on the production of new spatio-temporal knowledge representations [13]. It should be noted that, since knowledge is based on information, the spatio-temporal knowledge representation and reasoning also involve the representation and reasoning of spatio-temporal information.

However, there exist the problems of information imprecision and uncertainty in many spatio-temporal applications. In other words, fuzziness can be found in spatial, temporal, and spatio-temporal information [63, 7]. With the increasing demand for spatio-temporal applications, fuzzy spatio-temporal representation and reasoning are in great need [30, 64]. Fuzzy spatio-temporal representation and reasoning have become a hot research topic in the areas of robot vision, visual object tracking, spatio-temporal knowledge base, environmental monitoring, GIS, and Semantic Web [68, 61].

With the latest development and advance in Semantic Web technologies, descriptive logics (DLs) have become a logical basis for knowledge representation and reasoning of the Semantic Web. DLs are a decidable subset of first-order logic which possess powerful function of knowledge representation and reasoning [3, 48]. Currently, a large amount of fuzzy spatio-temporal knowledge and many corresponding applications are incorporated into the Semantic Web. DLs provide an effective method to solve the problem of fuzzy spatio-temporal knowledge representation and reasoning.

In our survey, we focus on the fuzzy spatio-temporal description logics and present an up-to-date review of the current state of the art in representing and reasoning fuzzy spatio-temporal knowledge with description logics. The paper investigates a survey from four main aspects: fuzzy spatio-temporal representation models, fuzzy spatial description logics, fuzzy temporal description logics, and fuzzy spatio-temporal description logics. The goal of this paper is two-fold. The first is to provide a generic literature review of the approaches that have been proposed to representing and reasoning fuzzy spatio-temporal knowledge with description logics. The second is to identify possible interesting directions of research in the field of fuzzy spatio-temporal knowledge management. It should be noted that, since crisp spatio-temporal description logic is a special fuzzy spatio-temporal description logic, it is impossible to include all of them in this paper. This paper only focuses on fuzzy spatio-temporal logics, and discusses crisp spatio-temporal logics in a small amount in order to compare research approaches. Therefore, this paper reviews the main concepts and results of fuzzy spatio-temporal description logics.

The rest of this article is organized as follows. Section 2 presents the preliminary knowledge about fuzzy set theory, crisp/fuzzy spatial knowledge, and fuzzy description logics. Section 3 provides different detailed techniques in modeling fuzzy spatio-temporal objects and their topological relations. Section 4 reviews the works on fuzzy spatial description logics. Section 5 provides different detailed techniques in fuzzy temporal description logics. Section 6 presents different detailed techniques in fuzzy spatio-temporal description logics. Section 7 summarizes and discusses the paper and provides some possible interesting directions for the future research.

2. Preliminaries

2.1 Fuzzy set theory

Fuzzy set theory proposed by [85] is widely used to deal with fuzzy/vague concepts existing in practical applications [46].

Let $U$ be a universe, a fuzzy set $A$ [85] in $U$ can be defined by a membership function $\mu_{A}(x):U\rightarrow$ [0, 1], or simply $A(x):U\rightarrow$ [0, 1]. The function assigns each element $x\in U$ to a value in the unit interval [0, 1], i.e., $A(x)\in$ [0, 1]. The value of $A(x)$ represents the degree of truth of the fact that $x$ belongs to $U$ . For example, $A(x)=$ 0.6 means that $x$ belongs to $U$ with a degree of 0.6. If the value of $A(x)$ is closer to 1, it means that the degree to which the element $x$ belongs to the fuzzy set $A$ is higher, and vice versa. If $A(x)=$ 0 or 1, the fuzzy set becomes a normal set. The support of a fuzzy set $A$ is a crisp set $\textit{sup}(A)=\{x\mid A(x)>0\}$ . If there exists an element $x\in U$ s.t $A(x)=$ 1, the fuzzy set $A$ is called normal. Thus, a normal set is a special case of a fuzzy set.

A fuzzy binary relation $R$ in $U\times U$ is a function $R:U\times U\rightarrow$ [0, 1]. For $x$ and $y$ in $U$ , $R(x,y)$ is the degree of strength of the association between $x$ and $y$ . For instance, Is-a ( $x$ , $y$ ) $=$ 0.7 means that the object $x$ is an object $y$ to a degree of 0.7. For the detail of some properties of the fuzzy binary relation, we refer the reader to [46].

In fuzzy set theory, Zadeh defined three standard fuzzy operations on fuzzy sets: Intersection, Union, and Complement. Suppose that $A$ and $B$ are two fuzzy subsets of $U$ . For all $x\in U$ , those operations can be defined as:

$\displaystyle(A\cap B)(x)=\min\{A(x),B(x)\},$ $\displaystyle(A\cup B)(x)=\max\{A(x),B(x)\},\text{and}$ $\displaystyle A^{c}(x)=1-A(x)$

Currently, there are four popular families of fuzzy logic called Zadeh [85], Łukasiewicz, Gödel, and Product logics [33]. Table 1 introduces four corresponding logical operators, namely, $t$ -norm ( $\otimes$ ), $t$ -conorm ( $\oplus$ ), negation ( $\ominus$ ), and implication ( $\Rightarrow$ ). In Table 1, we use the letters $L$ , $G$ , and $P$ to represent Łukasiewicz, Gödel, and Product logics, respectively. Fuzzy implication as a common operator of fuzzy logic plays a very important role in fuzzy reasoning. Choosing different fuzzy implication operators will have different inference results. In addition to the fuzzy implications mentioned above, the residual fuzzy implication is also typically used in terms of $t$ -norm. It is defined as $a\Rightarrow_{r}b=\text{sup}\{\lambda\in[0,1]\mid a\otimes\lambda\leqslant b\}$ for $a,b\in[0,1]$ .

Table 1
Four popular families of fuzzy logics ( $a,b\in[0,1]$ )

Fuzzy logic	$t$ -norm	$t$ -conorm	Negation	Implication
Zadeh logic	$a\otimes b=\min(a,b)$	$a\oplus b=\max(a,b)$	$\ominus a=1-a$	$a\Rightarrow b=\min(1-a+b,1)$
Łukasiewicz logic	$a\otimes_{L}b=$	$a\oplus_{L}b=\min(a+b,1)$	$\ominus_{L}\,a=1-a$	$a\Rightarrow_{L}b=\min(1-a+b,1)$
	$\max(a+b-1,0)$
Gödel logic	$a\otimes_{G}b=\min(a,b)$	$a\oplus_{G}b=\max(a,b)$	$\ominus_{G}\,a=\left\{\begin{array}[]{ll}1&\text{if }a=0\\ 0&\text{otherwise}\\ \end{array}\right.$	$a\Rightarrow_{G}b=\left\{\begin{array}[]{ll}1&\text{if }a<b\\ b&\text{otherwise}\\ \end{array}\right.$
Product logic	$a\otimes_{P}b=a\cdot b$	$a\oplus_{P}b=a+b-a\cdot b$	$\ominus_{P}\,a=\left\{\begin{array}[]{ll}1&\text{if }a=0\\ 0&\text{otherwise}\\ \end{array}\right.$	$a\Rightarrow_{P}b=\left\{\begin{array}[]{ll}1&\text{if }a<b\\ b/a&\text{otherwise}\\ \end{array}\right.$

2.2 Crisp/Fuzzy spatial knowledge

In general, there are three kinds of spatial relationships between spatial objects, namely, topological, distance, and directional relationships. Spatial topological relationships are the most basic and important spatial relationships which are commonly used in fields such as Geographic Information Systems (GIS) and spatial database systems [60, 75].

Currently, many achievements have been gained on the representation of spatial topological relations between different spatial objects. Among them, the most representative and widely used is the 9-Intersection model [43] and the Region Connection Calculus Calculus (short for RCC) [65]. Theoretically, 512 possible topological relationship models can be generated on the basis of 9-intersection model. However, most of the topological relationships do not exist in reality. By eliminating the topological relations which have no meaning, we can distinguish: 8 kinds of topological relations between spatial regions [43], namely, disjoint, contains, equal, overlap, covers, meet, coveredby, and inside; 2 topological relations between spatial points, disjoint and meet; 3 topological relations between spatial point and region, disjoint, inside, and meet; 33 topological relations between spatial lines [42]; 19 topological relations between spatial line and regions [44]. In RCC, there are two determinable subsets (fragments) (RCC-8 and RCC-5) which are widely used in spatial qualitative representation and reasoning. The RCC-8 model describes eight spatial relationships between spatial regions, namely, $\{DC,\textit{TPP},\textit{NTPP},EQ,PO,\textit{TPPi},EC,\textit{NTPPi}\}$ . These eight RCC-8 topological relations correspond to the eight relations of the 9-intersection model which are graphically shown in Fig. 1. The RCC-8 formula has a good computational property. [15] have discussed the logical reasoning ability of RCC-8 and proves that RCC-8 is determinable. [67] have shown that RCC-8 formula satisfiability is NP-complete.

Figure 1.

Graphical representation of RCC-8 spatial topological relations.

In real-world applications, the spatial region is fuzzy/ambiguous, and the resulting RCC topological relationship is also fuzzy/ambiguous. Fuzzy RCC-8 presented by [72, 74] is an extension of crisp RCC-8. Specifically, using the t-norm and residual fuzzy implication in the Łukasiewicz logic, the RCC topological relations can be extended to fuzzy RCC relationships. The definition of fuzzy RCC topological relations is shown in Table 2. More details about fuzzy RCC relationships can be found in [72, 73, 74].

Table 2

Definitions of RCC and fuzzy RCC for regions $a$ and $b$

Name	Relation	RCC definition	Fuzzy RCC definition
Disconnected	$DC(a,b)$	$\neg C(a,b)$	$\ominus C(a,b)$
Part	$P(a,b)$	$\forall.c(C(c,a)\Rightarrow C(c,b))$	$\textit{inf}_{c\in U}(C(c,a)\Rightarrow_{r}C(c,b))$
Proper part	$PP(a,b)$	$P(a,b)\land\neg P(b,a)$	$\ominus P(b,a)\otimes P(a,b)$
Equals	$EQ(a,b)$	$P(a,b)\land P(b,a)$	$P(a,b)\otimes P(b,a)$
Overlaps	$O(a,b)$	$\exists c.(P(c,a)\land P(c,b))$	$\textit{sup}_{c\in U}(P(c,a)\otimes P(c,b))$
Discrete	$DR(a,b)$	$\neg O(a,b)$	$\ominus O(a,b)$
Partially overlaps	$PO(a,b)$	$O(a,b)\land\neg P(a,b)\land\neg P(b,a)$	$\ominus P(a,b)\otimes O(a,b)\otimes\ominus P(b,a)$
Externally connected	$EC(a,b)$	$C(a,b)\land\neg O(a,b)$	$\ominus O(a,b)\otimes C(a,b)$
Non-tangential part	$\textit{NTP}(a,b)$	$\forall.c(C(c,a)\Rightarrow O(c,b))$	$\textit{inf}_{c\in U}(C(c,a)\Rightarrow_{r}O(c,b))$
Tangential $P P$	$\textit{TPP}(a,b)$	$PP(a,b)\land\neg\textit{NTP}(a,b)$	$\ominus\textit{NTP}(a,b)\otimes PP(a,b)$
Non-tangential $P P$	$\textit{NTPP}(a,b)$	$PP(a,b)\land\textit{NTP}(a,b)$	$\textit{NTP}(a,b)\otimes PP(a,b)$

2.3 Fuzzy description logics

A fuzzy description logic (called fuzzy $\mathcal{ALC}$ ) is considered as a basic fuzzy DL. We recall the syntax and semantics of fuzzy $\mathcal{ALC}$ [77].

Let $\textbf{C},\textbf{R}$ and I be a disjoint set of concept names, roles names, roles names, and individual names. Also, let $R\in\textbf{R}$ be a role. Concepts (denoted by $C$ or $D\in\textbf{C}$ ) of fuzzy $\mathcal{ALC}$ can be defined by:

$C,D=\top\mid\bot\mid A\mid\neg C\mid C\sqcap D\mid C\sqcup D\mid\forall R.C% \mid\exists R.C$ , where $A$ is an atomic concept of $C$ .

The semantics of fuzzy $\mathcal{ALC}$ is a pair $\mathcal{I}=(\triangle^{\mathcal{I}},\bullet^{\mathcal{I}(w)})$ , where $\triangle^{\mathcal{I}}$ is a domain of interpretation with non-empty set and $\bullet^{\mathcal{I}(w)}$ is a fuzzy interpretation function that maps each concept, role, and individual into a membership function between 0 and 1.

A fuzzy $\mathcal{ALC}$ knowledge base $\mathcal{K}$ consists of a fuzzy TBox $\mathcal{T}$ and a fuzzy ABox $\mathcal{A}$ . A fuzzy TBox $\mathcal{T}$ is a finite set of fuzzy General Concept Inclusions (fuzzy GCIs) axioms of the form $\langle C\sqsubseteq D\bowtie k\rangle$ with $\bowtie\in\{\geqslant,>,<,\leqslant\},k\in$ [0, 1]. A fuzzy ABox $\mathcal{A}$ is a finite set of fuzzy assertions, $\langle C(a)\bowtie k\rangle$ , $\langle R(a,b)\bowtie k\rangle$ . In knowledge base $\mathcal{K}$ , we call fuzzy GCIs and fuzzy ABox fuzzy $\mathcal{ALC}$ axioms.

3. Fuzzy spatio-temporal representation models

Spatio-temporal objects and their relationships are two most important attributes of the spatio-temporal representation model. However, in practical applications, spatio-temporal objects are not always crisp, but with fuzzy and uncertain in nature [21, 23]. Therefore, how to represent fuzzy spatio-temporal model (objects and topological relations) has become an important research issue. In the following, we first review some models of fuzzy spatio-temporal objects. On this basis, we then summarize fuzzy spatio-temporal topological relationships model in the literature.

3.1 Fuzzy spatio-temporal objects model

To represent the fuzziness and uncertainty of spatio-temporal objects, much work has been carried out toward a formal representation of fuzzy spatio-temporal objects [62, 41, 12]. Discrete models and continuous models are two widely used spatio-temporal object models. In the discrete model, time is considered as the third geometric dimension and the model represents the discrete change of spatial objects over time. In the continuous model, the continuous change of spatial object over time can be represented. Table 3 summarizes some main works on modeling of fuzzy spatio-temporal objects. Also, some mainly discussed issues can be found in Table 3.

Table 3
Some relevant works on modeling of fuzzy spatio-temporal objects

Fuzzy spatio-temporal object models	Corresponding references	Mainly discussed issues
Discrete models	[62]	Fuzzy points
	[11]	Fuzzy regions
	[68]	Fuzzy points
Continuous model	[41]	Fuzzy regions
	[21]	Fuzzy lines, fuzzy regions
	[23]	Fuzzy points, fuzzy lines, fuzzy regions

Several efforts have been made to extend spatio-temporal object models with fuzzy set theory. For example, [62] constructed a moving point object model based on fuzzy techniques. The cause of this fuzzy model is the sampling error and measurement error. In other words, fuzziness can be found in the positions of a spatial point at some temporal points. It should be noted that, [62] only discussed the spatio-temporal uncertainty of moving fuzzy points, but the study of the change of fuzzy lines and fuzzy regions over time is lacking. [41] proposed a moving fuzzy “Egg/Yolk” model, which only deals with the representation of fuzzy regions and does not take into account the representation of fuzzy points and fuzzy lines. In [68], the authors modeled the fuzzy spatial and temporal information based on triangular fuzzy numbers, and got 117 spatio-temporal topological relationships by combining 13 temporal relationships with 9 kinds of spatial relationships. However, the described models only deal with fuzzy point and do not represent fuzzy line and fuzzy region. In [12], the authors proposed a spatio-temporal object model based on rough set. In this model, a rough spatial region is divided into three different parts, i.e., lower bound, boundary region, and outside region. Based on fuzzy topological space, a method for modeling fuzzy spatio-temporal objects was proposed by [11], which defines a land change as a fuzzy spatio-temporal object. In this model, every object consists of a maximum of five parts which includes interior, boundary of the boundary, interior of the boundary, boundary of the interior, and exterior. Thus, based on the object model, a traditional 3 $\times$ 3-intersection matrix model can also be extended into 5 $\times$ 5-intersection one. The land use status and land cover situation can be analyzed through the change of the five parts between time starting point and time ending point The approach in [11] can only help to model a discrete change of fuzzy region over time, but have no effect on the continuous changes. In addition, the temporal variation of fuzzy point and fuzzy line is missing. That is, the method mentioned in [12] focuses on the discrete changes of fuzzy regions over time rather than continuous changes.

[21, 23] considered fuzzy spatio-temporal objects as moving objects, which model the evolution of fuzzy spatial objects over time. The motion of a moving object can be defined formally by a continuous function $F$ from fuzzy time $t$ to fuzzy spatial data types $S$ :

$\displaystyle F:t\rightarrow S$

where fuzzy spatial data types $S$ consist of fuzzy points, fuzzy lines, and fuzzy regions, $t$ denotes fuzzy time with a degree of membership [0, 1]. Thus, the moving objects model describe the continuous evolution of fuzzy spatial objects. Based on the formal definition of motion, the authors defined 3 moving objects as follows:

•

mfpoint (moving fuzzy point): the position information ( $x$ , $y$ coordinates) of a fuzzy point changes over time,

•

mfline (moving fuzzy line): the position and route information of a fuzzy line changes over time,

•

mfregion (moving fuzzy region): the position information of a fuzzy region changes over time such as a hurricane.

3.2 Fuzzy spatio-temporal topological relations model

Since fuzzy topological relation is the most important property of fuzzy spatio-temporal objects, many researchers put forward the modeling of fuzzy topological relation on the basis of fuzzy spatio-temporal objects. These approaches are divided into five categories:

•
fuzzy set theory based method [68, 8, 21, 23];
•
rough set-based method [12];
•
point set topology method [11];
•
action-based method [41, 50];
•
dimensional-extended method [10, 8].

Table 4 summaries some main works on topological relations models between all fuzzy spatio-temporal objects. In Table 4, we use $f p$ , $f l$ , and $f r$ to denote fuzzy point, fuzzy line, and fuzzy region, respectively. The whole table includes six kinds of fuzzy spatio-temporal objects. For example, $f p$ vs $f l$ denotes “between fuzzy point and fuzzy line”.

Table 4
Some topological relation models between fuzzy spatio-temporal objects

References Methods Fuzzy spatio-temporal objects

Fuzzy Rough Point set Action-based Dimension $f p$ vs $f l$ vs $f r$ vs $f p$ vs $f p$ vs $f l$ vs

set-based set-based topology extension $f p$ $f l$ $f r$ $f l$ $f r$ $f r$

[41] $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$

[10] $\times$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$

[12] $\times$ $\surd$ $\times$ $\times$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$

[11] $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$

[68] $\surd$ $\times$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\times$ $\times$

[50] $\times$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\times$ $\times$ $\times$ $\times$ $\times$

[9] $\times$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\surd$ $\times$ $\times$ $\times$

[8] $\surd$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\times$ $\surd$ $\times$ $\times$ $\times$

[21] $\surd$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\times$ $\surd$

[23] $\surd$ $\times$ $\times$ $\times$ $\times$ $\surd$ $\times$ $\surd$ $\surd$ $\times$ $\times$

In the following, we report these approaches of modeling fuzzy topological relations between fuzzy spatio-temporal objects.

Fuzzy set theory based method. [8] proposed a fuzzy spatio-temporal XML model, and presented a fuzzy extension of spatio-temporal topological relationship between fuzzy spatio-temporal data by using fuzzy set theory. A total of five kinds of fuzzy topological relations between fuzzy regions are obtained, including FRRequal, FRRcontain, FRRoverlap, FRRmeet, and FRRdisjoint. In [68], the authors proposed a spatio-temporal logical module for supporting 117 kinds of spatio-temporal topological relationships, which are based on a combination of 13 temporal relationships and nine spatial relationships between fuzzy points. Based on a 9-intersection matrix model, [23] presented a formal model of fuzzy spatio-temporal topological relations and identified three kinds of topological relations between two moving fuzzy points, six topological relations between a moving fuzzy point and a moving fuzzy region, and fifty topological relations between two moving fuzzy regions. At the same time, [21] identified a total of eighteen basic topological relations between a moving fuzzy line segment and a moving fuzzy region.

Rough set-based method. [12] constructed a spatio-temporal topological relation model between regions on the basis of rough set theory. In this model, lower bound, boundary and outside regions can be defined as a rough set’s core concepts. Point set topology method. To represent topological relationships between different land covers, [11] defined a new fuzzy spatio-temporal model which extends a classical 9-intersection spatial topological relation model to 25-intersection model. The land use and land cover were analyzed by comparing the changes of different regions between fuzzy regions over time.

Action-based method. [41] extended Muller’s spatio-temporal theory to fuzzy Egg-Yolk region, and proposed an action-based spatio-temporal representation model, which represents a spatial evolution of fuzzy Egg-Yolk region over time. The authors further constructed some action classes such as POSSIBLE-HIT, POSSIBLE-CROSS, and POSSIBLE-REACH, etc. [50] proposed a spatio-temporal relationship model based on fuzzy logic. In order to qualitatively explain a real-time behavior of entities (for example, animals), this model formalizes four spatio-temporal relationships, namely, IsMoving, IsGoingAlong, IsGoingAway, IsComingCloseTo, and IsGoingAlong. [8] defined a fuzzy spatio-temporal position in Euclidean space by using the minimum boundary matrix (BMR), and further represented six kinds of spatio-temporal topological relations between based on 9-intersection model predicate.

Dimensional-extended method. [10] extended 2D Egg-Yolk model into 3D space, defined a gradual uncertainty region, and further obtained a total of 46 spatio-temporal topological relationships between uncertain regions. On the basis of the idea proposed by [10, 9] also extended the two-dimensional Egg-Yolk model, analyzed all the types of topological relations in 3D space, and obtained 92 kinds of topological relationships between fuzzy regions. The corresponding graph representation and intersection matrix are given for each topological relation.
4. Fuzzy spatial description logics

References	Methods	Fuzzy spatio-temporal objects
	Fuzzy	Rough	Point set	Action-based	Dimension	$f p$ vs	$f l$ vs	$f r$ vs	$f p$ vs	$f p$ vs	$f l$ vs
	set-based	set-based	topology		extension	$f p$	$f l$	$f r$	$f l$	$f r$	$f r$
[41]	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$
[10]	$\times$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$
[12]	$\times$	$\surd$	$\times$	$\times$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$
[11]	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$
[68]	$\surd$	$\times$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$	$\times$	$\times$
[50]	$\times$	$\times$	$\times$	$\surd$	$\times$	$\surd$	$\times$	$\times$	$\times$	$\times$	$\times$
[9]	$\times$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\surd$	$\times$	$\times$	$\times$
[8]	$\surd$	$\times$	$\times$	$\surd$	$\times$	$\surd$	$\times$	$\surd$	$\times$	$\times$	$\times$
[21]	$\surd$	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\times$	$\surd$
[23]	$\surd$	$\times$	$\times$	$\times$	$\times$	$\surd$	$\times$	$\surd$	$\surd$	$\times$	$\times$

Fuzzy spatial description logics are based on crisp spatial description logics. In order to understand the research effort on fuzzy spatial DLs, we first briefly review crisp spatial DLs. Then, we summarize the related research of fuzzy spatial DLs. Table 5 summaries the current research progresses of some fuzzy spatial description logics.

Table 5
The current research progresses of some fuzzy spatial description logics

Type	Name	Syntax and semantics	Tableau algorithm	Complexity analysis	Reasoner
Spatial DLs	$\mathcal{ALCRP}(S_{2})$	[32]	[32]	[53]	$\times$
	$\mathcal{ALCRP}^{3}(D_{\textit{cCOA}})$	[45]	$\times$	[45]	$\times$
	$\mathcal{ALC}(S)$	[53]	$\times$	[53]	$\times$
	$\mathcal{ALCRP}(D_{\textit{rcc}8})$	[58]	$\times$	$\times$	$\times$
	$\mathcal{ALC}(C)$	[55]	[55]	[55]	[55]
	$\mathcal{ALC}(\textit{CDC})$	[25]	$\times$	$\times$	$\times$
	$\mathcal{ALC}(\textit{CDCRCC})$	[25]	$\times$	$\times$	$\times$
	$\mathcal{ALCI}_{\textit{RCC}}$	[82]	$\times$	$\times$	$\times$
	$\mathcal{ALC}_{s}$	[80]	[80]	[80]	$\times$
Fuzzy spatial DLs	fuzzy- $\mathcal{ALC}(D)$	[77]	[77]	$\times$	$\times$
	$\mathcal{ALC}_{F}$	[39]	[40]	$\times$	$\times$

4.1 Crisp spatial description logics

Description logic is a logical foundation of knowledge representation and reasoning. To represent and reason different types of knowledge in the Semantic Web, the study on the extension of different requirements of description logics has become an important research direction in the Semantic Web, which has attracted extensive attention from researchers at home and abroad. Thus, for the spatial knowledge representation and reasoning in the Semantic Web, a lot of research work is devoted to extending the spatial requirements of classic description logics in the last two decades. Currently, various spatial extensions forms of description logics can be distinguished, including some spatial extensions of description logics based on:

•
spatial concrete domain [32, 45, 53, 58];
•
spatial constraint system [55, 25];
•
spatial relation combination table [82, 80].

For each type of extension, we summarize some components of spatial description logics and do some comparisons and analyses to highlight similarities and differences in the existing researches as follows.
4.1.1 Extension of DLs based on spatial concrete domain

To support the reasoning of spatial knowledge, some spatial concrete domain techniques have attracted much attention and many extension approaches of DLs based on spatial concrete domain have been proposed. The initial idea combining the classic description logic $\mathcal{ALC}$ and a concrete domain was presented by [6]. They proposed a description logic $\mathcal{ALC(D)}$ that supports the representation of concrete domain. In $\mathcal{ALC(D)}$ , the concrete domain $\mathcal{D}$ is comprised of a set of concrete objects and a set of predicates defined over these objects. $\mathcal{ALC(D)}$ also divides the set of logical objects into two types of sets, namely, abstract and concrete objects. In fact, abstract objects and concrete objects are related by features, while relations between two concrete objects can be described by a set of predicates. In terms of reasoning, $\mathcal{ALC(D)}$ combines inference algorithms of description logics with the reasoning algorithms for the concrete domain $\mathcal{D}$ . Baader and Hanschke also pointed out that the only admissible concrete domain can be extended to the description logics so as to realize the reasoning of knowledge. Also, the reasoning algorithm of $\mathcal{ALC(D)}$ is implemented in a reasoner called Taxon system [34].

Based on $\mathcal{ALC(D)}$ , [32] presented a description logic $\mathcal{ALCRP(S)}_{2}$ with an admissible spatial concrete domain $\mathcal{S}_{2}$ and a role-forming predicate operators, which supports representation of spatial relationships. The admissible spatial domain $\mathcal{S}_{2}$ provides a spatial predicate that can qualitatively represent the RCC-8 topological relationships between spatial objects. Similar to $\mathcal{ALC(D)}$ , $\mathcal{ALCRP(S)}_{2}$ provides two kinds of logical levels: (i) abstract logical level, using abstract domain to represent spatial knowledge; (ii) concrete logical level, using spatial concrete domain to support the reasoning of spatial relationships. From the perspective of complexity, [53, 54] proved that $\mathcal{ALCRP(S)}_{2}$ -concept satisfiability is in NEXPTIME. Subsequently, based on $\mathcal{ALCRP(S)}_{2}$ , [45] proposed a spatial description logic $\mathcal{ALCRP}^{3}(\mathcal{D}_{c\mathcal{COA}})$ to represent and reason about ternary qualitative spatial relationships between 2D objects. The spatial concrete domain $\mathcal{D}_{c\mathcal{COA}}$ is comprised of predicates which describe spatial point, cardinal directions (e.g., West-East-South-North), and relative orientation (e.g., left, straight and right). In terms of decidability, the consistency problem of $\mathcal{ALCRP}^{3}(\mathcal{D}_{c\mathcal{COA}})$ ABoxes is proved to be undecidable. However, if some restrictedness criteria is added to $\mathcal{ALCRP}^{3}(\mathcal{D}_{c\mathcal{COA}})$ concept and role terms, then the restricted $\mathcal{ALCRP}^{3}(\mathcal{D}_{c\mathcal{COA}})$ ABox is decidable. The biggest difference between description logics $\mathcal{ALCRP(S)}_{2}$ and $\mathcal{ALCRP}^{3}(\mathcal{D}_{c\mathcal{COA}})$ is that the former can only represent binary spatial topological relationship between spatial regions, while the latter can only represent ternary spatial relations between 2D spatial objects.

On the basis of a standard two-dimensional topological space, [54] defined a spatial concrete domain $\mathcal{S}$ consisting of spatial regions and RCC-8 [65] topological relation predicates and proved that $\mathcal{S}$ is admissible and the satisfiability problem for a finite conjunctions of topological relation predicates is in NP. Then, the spatial concrete domain $\mathcal{S}$ is introduced into description logic $\mathcal{ALC(D)}$ and a spatial description logic $\mathcal{ALC(S)}$ is proposed. Also, [54] showed that the satisfiability problem of $\mathcal{ALC(S)}$ -concept is in PSPACE-complete. According to the requirements of multimedia information retrieval, [58] presented a spatial description logic $\mathcal{ALCRP}(\mathcal{D}_{\textit{rcc}8})$ which extends the DL $\mathcal{ALCRP(S)}_{2}$ with a new spatial concrete domain $\mathcal{D}_{\textit{rcc}8}$ . The most special feature of $\mathcal{D}_{\textit{rcc}8}$ is that it can represent the spatial topological relations between spatial lines and spatial regions. It should be noted that these spatial topologies are based on the 9-intersection model, rather than the Region Connection Calculus. For the reasoning problem, [58] did not give a decision reasoning process.

4.1.2 Extension of DLs based on spatial constraint system

Spatial constraint system, in fact, is a special concrete domain, which focuses on binary jointly exhaustive and pairwise disjoint spatial predicates, such as RCC-8. Many approaches have been proposed for extending DLs with spatial constraint system. [55] proposed a spatial description logic $\mathcal{ALC(C)}$ which extends basic DL $\mathcal{ALC}$ with a spatial constraint system $\mathcal{C}$ . In $\mathcal{ALC(C)}$ , the constraint system $\mathcal{C}$ is based on RCC-8 spatial topological relations and the authors also prove that the decidable $\mathcal{C}$ is $\omega$ -admissible. $\mathcal{ALC(C)}$ allows the definition of spatial concept through some spatial operators occurring in $\mathcal{C}$ . That is, the instances of abstract concepts layer can only be connected to the spatial regions of the constraint system $\mathcal{C}$ by spatial operators. As far as inference is concerned, the authors presented a tableau-based algorithm for deciding satisfiability of $\mathcal{ALC(C)}$ -concepts with respect to TBoxes and showed that the satisfiability problem of $\mathcal{ALC(C)}$ is EXPTIME-complete. For the empirical evaluation of tableau algorithm, the authors provided an idea that the algorithms of $\mathcal{ALC(C)}$ can be integrated into existing DL reasoners such as FaCT and RACER [55].

Based on the combination of $\mathcal{ALC(C)}$ with Cardinal Direction Calculus (short for $\mathcal{CDC}$ ), [25] presented a framework of spatial description logic $\mathcal{ALC(CDC)}$ . The constraint system $\mathcal{CDC}$ focuses on eight cardinal direction relations in a two-dimensional space, i.e., north, east, west, south, northwest, northeast, southwest, southeast. The $\mathcal{CDC}$ is also proved to be a $\omega$ -admissible constraint system. Similar to $\mathcal{ALC(C)}$ , satisfiability of the DL $\mathcal{ALC(CDC)}$ is also decidable. In $\mathcal{ALC(CDC)}$ , the computational complexity can not be provided. Using the same method, [25] also integrated the constraint system $C D R C C$ of topological and directional relations proposed by [51] into $\mathcal{ALC(C)}$ to form a spatial description logic $\mathcal{ALC}(\textit{CDCRCC})$ , which can be used to reason the spatial information combined by topological and directional relations. This description logic has also been proved to be decidable.

4.1.3 Extension of DLs based on spatial relation combination table

In order to infer spatial knowledge in the Semantic Web, many researches are devoted to extending DLs by spatial relation composition table. [82] introduced RCC composition tables into DLs, and proposed a family of DLs $\mathcal{ALCI_{RCC}}$ to support spatial reasoning. In $\mathcal{ALCI_{RCC}}$ , a spatial relationship is simply seen as a role from a spatial relation combination table in an abstract concept level, rather than a spatial predicate in a concrete domain. The decidability of $\mathcal{ALCI_{RCC}}$ -concept satisfiability had also been proved, for example, the DLs $\mathcal{ALCI_{RCC}}_{1}$ , $\mathcal{ALCI_{RCC}}_{2}$ , $\mathcal{ALCI_{RCC}}_{3}$ are decidable. However, for $\mathcal{ALCI_{RCC}}_{5}$ and $\mathcal{ALCI_{RCC}}_{8}$ , the decidability of their reasoning problems is unknown. In the context of geospatial semantic Web, [80] presented a spatial DL called $\mathcal{ALC}_{s}$ . In contrast to $\mathcal{ALCI_{RCC}}$ , $\mathcal{ALC}_{s}$ considers RCC8-based spatial relationships as concepts in abstract levels rather than roles. In addition, by using $\mathcal{ALC}_{s}$ , the constraint satisfaction problem of RCC-8 can be determined within EXPTIME-complete.

4.2 Fuzzy spatial description logics

In real applications, spatial knowledge is not always crisp but with the nature of fuzziness and imprecision [23, 21]. However, the classic DLs can not deal with the representation and reasoning of fuzzy spatial knowledge. Thus, with the demand of fuzzy information processing in the areas of Geographical Information Systems (GIS) and spatial databases, fuzzy spatial DLs emerge. Although much work is currently devoted to the study of fuzzy DLs, there are few fuzzy DLs that can represent and reason about uncertain and fuzzy spatial relations. In order to represent a large amount of fuzzy spatial knowledge existing in the real application domains, [77] proposed fuzzy spatial DL called fuzzy $\mathcal{ALC(D)}$ to reason fuzzy spatial topological relations. The specific ideas are roughly divided into two steps: (i) defining a fuzzy spatial concrete domain $\mathcal{D}$ composed of spatial region and fuzzy RCC topological relations; (ii) introducing the concrete domain $\mathcal{D}$ into fuzzy DL $\mathcal{ALCF(D)}$ [76]. In addition, [77] provided a tableau-based reasoning algorithm to determine the Best Satisfiability Degree and the Best Entailment Degree of fuzzy $\mathcal{ALC(D)}$ knowledge base. It should be noted that the author did not show whether the concrete domain $\mathcal{D}$ is admissible, and does not prove the correctness of the reasoning algorithm based on tableau, i.e., termination, soundness, and completeness.

To qualitatively represent the two-way and uncertain spatial relationships described in medical images, [39] introduced a bipolar fuzzy domain $\mathcal{F}$ into the classical DL $\mathcal{ALC(D)}$ [6], and presented a fuzzy DL $\mathcal{ALC(F)}$ . In $\mathcal{ALC(F)}$ , the bipolar fuzzy concrete domain regards spatial relationship as a relative spatial relationship pair, such as (Close To, Very Far From), (Right of, Top of), and (Top, Bottom). This DL satisfies the spatial relationship representation in medical images, with negative spatial information indicating the degree of anatomical deviation, and positive spatial information indicating the real position of the image. Subsequently, [40] further gave an inference decision algorithm of $\mathcal{ALC(F)}$ , to determine the consistency problem of $\mathcal{ALC(F)}$ ABox. However, $\mathcal{ALC(F)}$ is limited to the representation and reasoning of the typical fuzzy bidirectional spatial relationship, but cannot reason fuzzy RCC topological relationships that can be found in real-world applications. Moreover, $\mathcal{ALC(F)}$ can only deal with fuzzy spatial information, but do not apply to general fuzzy knowledge, such as fuzzy concepts and fuzzy roles.

5. Fuzzy temporal description logics

Crisp temporal description logics are the basis of fuzzy temporal description logics. Hence, we first briefly review the related work of crisp temporal description logics. The detailed survey on crisp temporal description logics can be found in [56]. Then, we focus on a summary of the related research of fuzzy temporal description logics.

5.1 Crisp temporal description logics

To represent and reason temporal knowledge that exists widely in the Semantic Web, temporal extension of DLs has attracted wide attention. Currently, various forms of temporal requirement extension of DLs have been proposed, including the extensions of DLs based on:

•
propositional temporal logics [69, 83, 84, 5, 81];
•
actions [2, 20, 19, 38, 78, 52];
•
temporal concrete domains[53].

To facilitate the comparison of the existing research works on some temporal description logics studied in this paper, we summarize these works and make some comparisons and analyses in Table 6.

Table 6
The current research progresses of some temporal description logics

Type Name Syntax and Tableau Complexity Reasoner

semantics algorithm analysis

Propositional temporal logic extension of DLs $\mathcal{ALCT}$ [69] [69] [69] $\times$

$\mathcal{CIQ_{US}}$ [83] [83] [83] $\times$

$\mathcal{TL_{ALC}}$ [84] [84] [84] [31]

$\mathcal{ALC}$ -LTL [5] [5] [5] $\times$

$\mathcal{ALC}\textit{CTL}$ [81] [81] [81] $\times$

Action extension of DLs $\mathcal{TL}$ - $\mathcal{F}$ [2] $\times$ [2] $\times$

$\mathcal{TL}$ - $\mathcal{FU}$ [2] $\times$ [2] $\times$

$\mathcal{D}$ - $\mathcal{ALCO}$ [20] [20] $\times$ $\times$

$\textit{DDL}(\mathcal{ALCQIO})$ [19] [19] $\times$ $\times$

$\mathcal{TL}$ - $\mathcal{SI}$ [38] [38] $\times$ $\times$

$\mathcal{DLTL_{DL}}$ [78] $\times$ $\times$ $\times$

T- $\mathcal{ALC}$ [52] [52] [35] $\times$

Concrete domain extension of DLs $\mathcal{ALC}(P)$ [53] [53] [53] [53]

$\mathcal{ALC}(I)$ [53] [53] [53] [53]

5.1.1 Extension of DLs based on propositional temporal logic

Type	Name	Syntax and	Tableau	Complexity	Reasoner
		semantics	algorithm	analysis
Propositional temporal logic extension of DLs	$\mathcal{ALCT}$	[69]	[69]	[69]	$\times$
	$\mathcal{CIQ_{US}}$	[83]	[83]	[83]	$\times$
	$\mathcal{TL_{ALC}}$	[84]	[84]	[84]	[31]
	$\mathcal{ALC}$ -LTL	[5]	[5]	[5]	$\times$
	$\mathcal{ALC}\textit{CTL}$	[81]	[81]	[81]	$\times$
Action extension of DLs	$\mathcal{TL}$ - $\mathcal{F}$	[2]	$\times$	[2]	$\times$
	$\mathcal{TL}$ - $\mathcal{FU}$	[2]	$\times$	[2]	$\times$
	$\mathcal{D}$ - $\mathcal{ALCO}$	[20]	[20]	$\times$	$\times$
	$\textit{DDL}(\mathcal{ALCQIO})$	[19]	[19]	$\times$	$\times$
	$\mathcal{TL}$ - $\mathcal{SI}$	[38]	[38]	$\times$	$\times$
	$\mathcal{DLTL_{DL}}$	[78]	$\times$	$\times$	$\times$
	T- $\mathcal{ALC}$	[52]	[52]	[35]	$\times$
Concrete domain extension of DLs	$\mathcal{ALC}(P)$	[53]	[53]	[53]	[53]
	$\mathcal{ALC}(I)$	[53]	[53]	[53]	[53]

Extension of DLs based on Propositional temporal logic is to combine the temporal logic based on time-points or time-intervals with the classical DLs to achieve the representation and inference of temporal knowledge. At present, a lot of work has been done on this research. [71] first extended temporal logic based on time-interval into DLs and addressed a temporal description logic, in which temporal logic operators based on time-interval includes temporal quantifier at, existential temporal quantifier sometime and complete temporal quantifier alltime. It should be noted that these temporal logic operators can be applied to the construction of concepts and roles. Subsequently, [69] embeded point-based tense operators into DLs $\mathcal{ALC}$ and presented a temporal description logic called $\mathcal{ALCT}$ . The temporal concept operators consist of existential future operator ( $\lozenge$ ), universal future operator ( $\square$ ), next instant operator ( $\circ$ ), and until ( $\mathcal{U}$ ). From a syntactic point of view, these temporal concept operators can be applied to the construction of concepts and roles. From semantic perspective, a concept can be interpreted as a collection of objects at a certain time point, and a role can be interpreted as a relationship of roles between two objects at a certain time point. For example, given a time point $t$ , a concept $\square C$ represents all instances belonging to concept $C$ for each time point $t_{1}\geqslant t$ . Also, [69] proved that the concept satisfiability problem of $\mathcal{ALCT}$ has the same level of complexity as the concept satisfiability problem of $\mathcal{ALC}$ , i.e., PSPACE-complete. [83] extended two temporal operators $\mathcal{U}$ (Until) and $\mathcal{S}$ (Since) into a DL $\mathcal{CIQ}$ , and proposed a temporal DL $\mathcal{CIQ_{US}}$ . In $\mathcal{CIQ_{US}}$ , the two operators can be used not only to concepts, but also to formulas. $\mathcal{CIQ_{US}}$ can be interpreted as a sequence of time points of a standard DL $\mathcal{CIQ}$ , reflecting the evolution of concepts and formulas over time. At the same time, [83] first introduced a quasimodel technology to prove the satisfiability problem of $\mathcal{CIQ_{US}}$ formula. [84] presented a temporal extension of DL $\mathcal{ALC}$ called $\mathcal{TL_{ALC}}$ , where the temporal operator is only $\mathcal{U}$ (Until). For the interpretation domain, $\mathcal{TL_{ALC}}$ restrict the original $\mathcal{ALC}$ constant interpretation domain to an extended interpretation domain. For the realization of the reasoning problem, the tableau calculus is an elaborate combination of the tableau algorithm for $\mathcal{ALC}$ [70], tableau calculus for linear temporal logic (LTL) [47], and the method of quasimodels. It should be noted that the authors use quasimodels to bridge a gap between the tableau algorithm model and the intended models for $\mathcal{TL_{ALC}}$ . In particular, it is known that the complexity of the tableau calculus is still PSPACE. [31] implemented a reasoner called TMT to reason the temporal description logic $\mathcal{TL_{ALC}}$ .

To reduce the reasoning complexity of temporal description logic, [5] also extended the DL $\mathcal{ALC}$ , and proposed a temporal description logic based on linear temporal logic LTL called $\mathcal{ALC}$ - $\mathcal{LTL}$ . In this language, temporal operators can only be allowed to occur in front of formula, but not inside concept constructions. To satisfy different inference needs, the authors considered the satisfiability problem of the $\mathcal{ALC}$ - $\mathcal{LTL}$ formula with general cases, rigid concepts, rigid roles, and rigid concepts. The so-called rigid concepts mean that the semantic interpretation of concepts do not change over time. The general cases refer to the cases where rigid concepts and rigid roles are not considered. At the same time, it is proved that the satisfiability problems with these three cases are decidable, and the complexity of $\mathcal{ALC}$ - $\mathcal{LTL}$ can be reduced. For example, the complexity decreases to NExpTime-complete with rigid concepts, and the complexity is ExpTime-complete with the general cases.

[81] combined DL $\mathcal{ALC}$ with computational tree logic CLT [27] and presented a decidable temporal DL $\mathcal{ALCCTL}$ . The authors also gave an inference algorithm to generate counterexamples and proved the completeness and soundness of this algorithm.

5.1.2 Extension of DLs based on action

In temporal knowledge, time is usually accompanied by actions or events. [2] pointed out that action is a temporal evolution between possible worlds. For the purpose of representation and reasoning about actions in temporal knowledge, many researchers are devoted to the work on action extension of description logics. [2] combined an interval-based temporal logic $\mathcal{TL}$ and feature description logic $\mathcal{F}$ , and constructed a temporal description logic $\mathcal{TL}$ - $\mathcal{F}$ to represent and reason about actions. The main purpose of $\mathcal{TL}$ - $\mathcal{F}$ is to study the temporal relationship between actions, i.e., reasoning about actions by inferring temporal relationships. For the reasoning problem, the authors presented a sound and complete algorithm for deciding concept inclusion problem (also called subsumption relationship between concepts) and proved that this reasoning problem is an NP-complete. To reason about more complex actions, the authors further proposed two more expressive temporal description logics $\mathcal{TL}$ - $\mathcal{FU}$ and $\mathcal{TL}$ - $\mathcal{ALCF}$ . $\mathcal{TL}$ - $\mathcal{FU}$ adds disjunctive operation of temporal concepts and non-temporal concepts on the basis of $\mathcal{TL}$ - $\mathcal{F}$ , while $\mathcal{TL}$ - $\mathcal{ALCF}$ replaces the feature description logic in $\mathcal{TL}$ - $\mathcal{F}$ with a complete description logic $\mathcal{ALCF}$ . It should be noted that these two temporal description logics are still decidable.

Currently, there are much temporal knowledge containing action in the area of Semantic Web. With the aim of solving the problem of reasoning about action in the semantic Web, [4] combined description logics (such as $\mathcal{ALC}$ and $\mathcal{ALCQIO}$ ) with a typical action theory, and proposed a decidable framework for supporting action reasoning. In this framework, all description logic concepts are only used to represent different states of possible worlds, pre-conditions, and post-conditions of action. By embracing actions into DL $\mathcal{ALCO}$ , a dynamic description logic called $\mathcal{D}$ - $\mathcal{ALCO}$ was presented by [20] in which an action can be obtained by atomic actions and operators (* and ;). For example, actions $\pi,\pi^{\prime}$ of $\mathcal{D}$ - $\mathcal{ALCO}$ can be defined inductively by the following syntax rules:

$\displaystyle\pi,\pi^{\prime}\longrightarrow(p,e)|\varphi?|\pi\cup\pi^{\prime}% |\pi;\pi^{\prime}|\pi^{*}$

where, $(p,e)$ is an atomic action, $\varphi$ is a $\mathcal{D}$ - $\mathcal{ALCO}$ formula, and the four operators $\varphi?,\pi\cup\pi^{\prime},\pi;\pi^{\prime},\pi^{*}$ are called test action, choice action, sequential action and iterated action, respectively. Reasoning problems about actions can be reduced to the reasoning of $\mathcal{D}$ - $\mathcal{ALCO}$ formulas, such as actions subsumption problems. For reasoning about action and temporal properties of dynamic domains in the Semantic Web, [19] further proposed a family of dynamic description logics, such as $\textit{DDL}(\mathcal{ALCO})$ and $\textit{DDL}(\mathcal{ALCQIO})$ . The detailed introduction can be found in [19].

Based on the idea of temporal description logic $\mathcal{TL}$ - $\mathcal{F}$ , [38] constructed a temporal description logic $\mathcal{TL}$ - $\mathcal{SI}$ by extending $\mathcal{SI}$ with a interval-based temporal logic $\mathcal{TL}$ . The authors provided a tableau-based algorithm for deciding the $\mathcal{TL}$ - $\mathcal{SI}$ concepts satisfiability and the subsumption relationship between temporal concepts. At the same time, for reasoning about Semantic Web service, a dynamic linear time temporal description logic $\mathcal{DLTL_{DL}}$ is proposed by [78]. The $\mathcal{DLTL_{DL}}$ is a combination of dynamic linear time temporal logic and description logic and thus suitable for reasoning about temporal changes of web service concepts, instances, and roles.

In order to enhance expressive ability of action, [52] extended the description logic $\mathcal{ALC}$ and presented a temporal description logic T- $\mathcal{ALC}$ . In the T- $\mathcal{ALC}$ , a temporal constraint is applied only to assertion axiom, not to concept axiom. Moreover, the temporal constraint is also a constraint based on time interval, rather than on temporal operators. For example, during a period, an individual belongs to a concept, while an individual does not belong to the whole concept in the whole period. For the reasoning problem of actions consistency, the authors gave a tableau-based reasoning algorithm and proved the completeness and soundness of the algorithm. To introduce action execution into knowledge base, [35] studied a description logic knowledge and action base (short for KAB), discussed the semantic structure of action execution in KAB, and proved that he satisfiability problem of KAB is decidable in the case of simple temporal properties. The authors further showed that the complexity of determining the satisfiability problem of KAB is PTIME-complete in the case of acyclic KAB.

5.1.3 Extension of DLs based on temporal concrete domain

Similar to the spatial reasoning method, some work on extension of DLs based on temporal concrete domain have been proposed for reasoning temporal knowledge. In general, these approaches can be divided into two categories: time point-based temporal concrete domain P and time interval-based temporal concrete domain I. P is a concrete domain defined on a linear time point and I is a concrete domain defined on 13 Allen time interval relationships [1]. In [49], the authors proved that the P-satisfiability problem is decidable and further proved that P is admissible. Similarly, in [59], the authors proved that I is also admissible, and the I-satisfiability problem is NP-complete. [53] extended the description logic $\mathcal{ALC}$ with P, and proposed a temporal description logic $\mathcal{ALC}(P)$ to support time-point reasoning, and further gave a decision algorithm based on tableau’s reasoning problem. At the same time, [53] proved that the $\mathcal{ALC}(P)$ ABox consistency problem is EXPTIME-complete in the case of general TBox. Subsequently, [53] combined the description logic $\mathcal{ALC}$ with I, and presented a temporal description logic $\mathcal{ALC}(I)$ for reasoning time interval relationships. Also, the authors proved that the $\mathcal{ALC}(I)$ satisfiability problem is decidable. The problem of $\mathcal{ALC}(I)$ satisfiability is proved to have the same level complexity as $\mathcal{ALC}(P)$ . A detailed review of temporal description logics can be found in [84, 56].

5.2 Fuzzy temporal description logics

In real-world applications, most temporal knowledge exists in vague and fuzzy forms. Some temporal information and its relationships are fuzzy and vague. In order that temporal logics can deal with imprecise/uncertain temporal knowledge and its applications, there have been many efforts in the past to extend temporal logics with fuzzy set theory. Some different fuzzy extensions of temporal logics had been proposed, such as, fuzzy linear temporal logic (FLTL), fuzzy branch temporal logic (FBTL) [57], fuzzy metric temporal logic based on Łukasiewicz [28], and fuzzy temporal timing logic (FTL) [29]. Although there have been several kinds of fuzzy extensions of temporal logics, the work is not suitable for dealing with imprecise and uncertain temporal information that is found in the Semantic Web. Aiming at handling fuzzy temporal knowledge widely existed in the Semantic Web, less research on fuzzy extension of temporal description logics has been done. For example, [37] introduced imprecise temporal knowledge represented by fuzzy sets into a description logic $\mathcal{ALC}$ , and proposed a fuzzy temporal description logic FTDL that allows reasoning vague temporal concepts.

6. Fuzzy spatio-temporal description logics

Description logic is a logical foundation of ontologies in the Semantic Web. In this section, we concentrate on a summary of fuzzy spatio-temporal ontologies and fuzzy spatio-temporal description logics.

6.1 Fuzzy spatio-temporal ontologies

In real-world applications, many application domains involve crisp/fuzzy temporal and spatial knowledge. With the rapid development of the Semantic Web, a large number of crisp/fuzzy spatio-temporal knowledge and its related applications are incorporated into the Semantic Web. How to construct crisp/fuzzy spatio-temporal ontologies has become an important problem to enable the Semantic Web, and many methods and tools were developed to help people to construct crisp/fuzzy spatio-temporal ontologies.

Currently, some researchers have proposed several approaches for constructing crisp spatio- temporal ontologies [16, 18, 66]. The ideas of these approaches can be roughly divided into three types as follows:

•
adding the support of temporal elements based on existing spatial ontologies, for example, ;
•
adding spatial support based on existing temporal ontologies;
•
constructing spatio-temporal unified ontologies representation model using formal methods

To represent both qualitative temporal and spatial information, [14] constructed a framework for settling spatio-temporal information in OWL 2.0 called SOWL, which can reason cone-shaped directional relations and RCC-8 topological relations. [18] presented a knowledge representation method to representing spatio-temporal geographic data and to grounding spatio-temporal geographic ontologies upon the data. Also, the authors provided a useful and natural approach to establish a gap between the spatio-temporal data and ontologies. [66] constructed a spatio-temporal geographic ontology model which is made up of three parts: geographical object model, spatial model, and temporal model. It should be noted that the above ontology languages are not suitable for dealing with imprecise and fuzzy information that is found in spatio-temporal domain.

For reasoning fuzzy spatio-temporal knowledge that exists widely in the Semantic Web, some efforts on fuzzy spatio-temporal extensions of ontologies have been investigated. In [79], a novel framework of multimedia ontologies based on fuzzy spatio-temporal relations are presented in order to reason media property.
6.2 Fuzzy spatio-temporal description logics

Since description logic is the logical basis of ontology representation and reasoning in the Semantic Web, fuzzy spatio-temporal description logic naturally becomes the logical basis of fuzzy spatio-temporal ontology. Currently, the study of fuzzy spatio-temporal description logic is mostly limited to two separate levels: fuzzy spatial description logic, and fuzzy temporal description logic. The aim of fuzzy spatial description logic is to achieve the reasoning of fuzzy spatial knowledge, while fuzzy temporal description logic can represent and reason fuzzy temporal knowledge. The research on fuzzy spatio-temporal description logic which can represent both fuzzy spatial knowledge and fuzzy temporal knowledge is still in a developing stage. Based on the idea of temporal description logics, [22] proposed a fuzzy spatio-temporal description logic called $f$ - $\mathcal{ALC(D)}$ -LTL extending fuzzy spatial description logic $f$ - $\mathcal{ALC(D)}$ with linear temporal logic (LTL). This description logic can reason fuzzy RCC relationships which change over time and reflect temporal evolutions of fuzzy spatial relations. For deciding satisfiability of $f$ - $\mathcal{ALC(D)}$ -LTL formula, a tableau-based algorithm was also presented. The biggest feature of $f$ - $\mathcal{ALC(D)}$ -LTL is to apply Hintikka model to settle the infinite expansions of interpretation domains, which ensures the correctness of the tableau algorithm.

7. Summaries and discussions

Representing and reasoning fuzzy spatio-temporal knowledge with description logics have been an important topic in the areas of Knowledge Engineering and Artificial Intelligence. On one hand, there is an increasing demand for reasoning a large amount of fuzzy spatio-temporal knowledge in the context of the Semantic Web, and on the other hand, description logic, as the logical foundation of knowledge representation and reasoning, becomes an effective method to solve the problem of fuzzy spatio-temporal knowledge representation and reasoning. The researches of fuzzy spatio-temporal extension of description logics provide a feasible solution for the realization of the automatic reasoning of fuzzy spatio-temporal knowledge.

This paper provides an up-to-date review of the current state of the art in fuzzy spatio-temporal representation and reasoning with description logics. In the end of this paper, we briefly summarize the main content of this paper again to have a general understanding of the field. First, for the problem of the formal representation of fuzzy spatio-temporal knowledge in the real spatio-temporal application, many proposals for modeling fuzzy spatio-temporal knowledge with different formalisms such as fuzzy set theory, rough set, point set topology, action-based, and dimensional-extended were proposed, and thus several different formal model of fuzzy spatio-temporal objects and their topological relations were presented as shown in Section 3. Furthermore, as we have known that fuzzy spatial description logic is a special crisp spatial description logic, and thus we first review various forms of spatial extensions of crisp description logics to represent and reason crisp spatial knowledge and then summary some fuzzy spatial extensions of description logics in Section 4. Also, for representing crisp/fuzzy temporal knowledge, several kinds of crisp temporal description logics (such as $\mathcal{TL_{ALC}}$ , $\mathcal{TL}$ - $\mathcal{SI}$ ) and fuzzy temporal description logics (such as FTDL) were addressed as mentioned in Section 4. Finally, in order that description logics can represent crisp/fuzzy spatio-temporal knowledge, several approaches for constructing crips/fuzzy spatio-temporal ontologies were proposed and how to develop fuzzy spatio-temporal description logic was addressed as mentioned in Section 5.

After surveying most of the proposals of fuzzy spatio-temporal extensions to description logics, it has been widely approved that extending description logics to make it able to have the ability to reason fuzzy spatio-temporal knowledge is an important research problem. To achieve knowledge representation and reasoning in the fuzzy spatio-temporal domain, some extension proposals (fuzzy spatio-temporal representation model, fuzzy spatial description logics, fuzzy temporal description logics, and fuzzy spatio-temporal description logics) have been proposed. However, the researches on fuzzy spatio-temporal extensions of description logics are still in their relative infancy and still the full potential of fuzzy spatio-temporal description logics have not been fully explored. Hence, we are currently exploring several interesting research directions. The following issues may be a good starting point for finding some appropriate approaches and techniques in the Semantic Web.

•

Representation of fuzzy spatio-temporal models needs to be further investigated to meet the demands of complex fuzzy spatio-temporal objects. Due to the irregularity of fuzzy spatial objects, such as spatial regional objects with holes, it is necessary to further study the representation model of complex fuzzy spatio-temporal objects (fuzzy spatio-temporal regions with holes) and their topological relations. Moreover, there needs to consider some topological relationships between fuzzy lines.

•

The complexity and optimization techniques of reasoning algorithms of fuzzy spatial description logics need to be studied deeply. Actually, the complexity and optimization of reasoning algorithm is always a very important issue in case of the Semantic Web. Although there have been several reasoning algorithms of crisp spatial description logics as introduced in Section 4.1, most of them cannot support fuzzy spatial relations reasoning.

•

Fuzzy extensions of temporal description logics are a major open issue in order to handle fuzzy spatial information that exists in practice applications. As introduced in Section 5.2, although there have been some reports on fuzzy extensions of temporal logics, there is little research on fuzzy extensions of temporal description logics. Thus, some theoretical frameworks and reasoner of fuzzy temporal description logics should be further exploited.

•

Fuzzy spatio-temporal description logic will increasingly receive attention in the future work. There are very few reports on fuzzy spatio-temporal description logic. Only an effort can be found in [22], where the description logic can support the change of fuzzy RCC topological relations over crisp time. Three issues about fuzzy spatio-temporal extension of description logics are crucial and interesting. The first issue is an extension of fuzzy spatial description logics with branch temporal logic. The second issue is an extension of fuzzy spatial description logic with fuzzy temporal logics. The third issue is an extension of crisp spatial description logics with fuzzy temporal logics.

•

Implementation of fuzzy spatio-temporal description logic reasoners may be needed to support the inference of fuzzy spatio-temporal knowledge. Currently, most efforts of fuzzy spatial description logics and fuzzy temporal description logics mainly focus on setting up theoretical frameworks (syntax, semantics, knowledge base, and tableau algorithms). The implementation of effective fuzzy spatio-temporal description logic reasoner is still rarely investigated in the literature. Thus, to meet the requirement of spatio-temporal applications, several efficient fuzzy spatio-temporal reasoners and their empirical evaluations should be implemented.

Footnotes

Acknowledgments

This work is supported by National Key R&D Program of China (Grand No. 2018YFB1003201), National Natural Science Foundation of China (Grand No. 61672296, Grand No. 61602261), Major Natural Science Research Projects in Colleges and Universities of Jiangsu Province (Grand No. 18KJA520008), and NUPTSF (Grand No. NY217133).

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Representing and reasoning fuzzy spatio- temporal knowledge with description logics: A survey

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Fuzzy set theory

Table 1 Four popular families of fuzzy logics ( a , b ∈ [ 0 , 1 ] )

3. Fuzzy spatio-temporal representation models

3.1 Fuzzy spatio-temporal objects model

Table 3 Some relevant works on modeling of fuzzy spatio-temporal objects

Table 5 The current research progresses of some fuzzy spatial description logics

4.1.2 Extension of DLs based on spatial constraint system

4.1.3 Extension of DLs based on spatial relation combination table

4.2 Fuzzy spatial description logics

5. Fuzzy temporal description logics

5.1 Crisp temporal description logics

5.1.2 Extension of DLs based on action

5.1.3 Extension of DLs based on temporal concrete domain

5.2 Fuzzy temporal description logics

6. Fuzzy spatio-temporal description logics

6.1 Fuzzy spatio-temporal ontologies

7. Summaries and discussions

Footnotes

Acknowledgments

References

Table 1
Four popular families of fuzzy logics ( $a,b\in[0,1]$ )

Table 3
Some relevant works on modeling of fuzzy spatio-temporal objects

Table 5
The current research progresses of some fuzzy spatial description logics