Survey of Temporal Knowledge Representation. (Second Exam)

Transcription

1 Survey of Temporal Knowledge Representation (Second Exam) Sami Al-Dhaheri The Graduate Center, CUNY Department of Computer Science June 20, 2016 Abstract Knowledge Representation (KR) is a subfield within Artificial Intelligence that aims to represent, store, and retrieve knowledge in symbolic form that is easily manipulated using a computer. The vision of Semantic Web has recently increased because of the interest of using and a applying the Knowledge Representation methodology in both academia and industry. Knowledge Representation formalism are often named as one of the main tools that can support the Semantic Web. Knowledge Representation has many forms, including logic-based and non-logic based formalisms. This survey will only be concerned with logic-based KR. The review will present various methods authors have used in applying logic-based KR. We will present each methodology with respect to its formalism, and we will present the reasoning in Description logic. The survey will also discuss possible temporal extensions of Description Logic, RDF and OWL. 1 Introduction Knowledge Representation and Reasoning is a scientific domain within the area of Artificial Intelligence in Computer Science. It aims to encode human knowledge in a computerized fashion using formal models that allow the computer to take human knowledge and represent it in a meaningful manner [1]. In this context, computers are able to represent implicit knowledge found within the encoded body of human knowledge provided to the system, thus resulting in a behavior that is known as Intelligent one. Computer Scientists have devised systems to represent knowledge and make inferences based on the encoded knowledge. Such

2 systems have been referred to as Knowledge Base Systems; their main components are a knowledge base and a reasoning engine. Mathematicians were the first to explore and apply logic-based Knowledge Representation before the computers revolution. However, the field acquired more recognition during the 1970s with the rise of the field of Computer Science in the academy and industry. Two forms of Knowledge Representation emerged: logic based and non-logic based. With the introduction of the World Wide Web in 1989, Knowledge Representation became of crucial importance to achieve its goal of representing implicit knowledge found within the wide array of information created and shared by people [2]. Knowledge Representation has played a significant role in the development of Semantic Web. One of the areas KR has been active in within Semantic Web is Ontologies. Ontologies provide vocabulary for annotations of semantics in application domains. The first and foremost significant aim of Knowledge Representation is creating new forms of knowledge and reasoning by using existing ones. Computers can be taught to use encoded human languages to extract implicit trends, dimensions, semantics and other forms of knowledge. The formal specification of things is also accessible to machines. Practically, a standard web ontology language is needed and have been developed for the above applications. Knowledge Representation becomes useful in many fields including medical decisions support, financial decision making, institutional research frameworks and others. The developed Knowledge Based Systems can be used to not only extract knowledge, but also instruct to make decisions based on such knowledge. Therefore, the wide application of Semantic Web renders Knowledge Representation as one of the most worthy topics of research and development. Science not only strives to explain variance in a given variable, it also tries to develop an understanding of a particular phenomenon. One element scientists consider in their analyses is the effect of time. As in social sciences, time matters and witnesses many outcomes. It is also of utmost significance to Computer Scientists, especially in Semantic Web. For instance, the description of a concept in ontology often reference to temporal patterns. e.g., the definition of salary in employee ontology that changes over time. Similarly, time plays a critical role when using Description logics (DL) to represent and reason about conceptual representation of temporal knowledge. Therefore, models used in Knowledge Representation includes time as an important element. We can represent static knowledge" that means not changing. Besides that, "temporality" is an important issue imposed by the flows of time. The issue comes when the represented knowledge changes over time. Which is called dynamic knowledge. Dynamic knowledge is maintained by updating, deleting, or putting "outdated flag" on the "old" knowledge. In the past two decades, temporal databases have been extensively researched. They cover several dozens of temporal data models that are mainly extension of the relational model. Temporal

3 query languages are proposed. Issues of representing time have also been tackled. In general, a knowledge representation formalism can be extended to cover the temporal part of the knowledge. There are two basic strategies: one is to formally extend the model. For instance, extend the Description Logics to Temporal Description Logics. The other is that the temporal extension is added on top of the formalism. Ontology versioning is an example. The reification approach is somewhat between these two extremities [3]. In this survey we review the recent research and developments in KR. We examine various logic-based knowledge representation approaches with the aim of deciding upon a better approach for temporal knowledge representation. Our focus is on temporal aspect of knowledge representation within semantic web. It s worth mentioning that much of this survey is concerned with logic-based knowledge representation. In this survey, we will explore important issues in knowledge representation formalisms, including their formal semantics when a syntax is referred, and reasoning and inference services provided. Temporal extensions will be presented next. The survey paper is organized as follows: in section 2, different logic-based knowledge representation formalisms are introduced. Section 3 discusses various nonlogic-base knowledge representation technologies associated with the Semantic Web.Section 4 covers the temporal extension to Description logics, Temporal RDF and Temporal OWL. Section 5 provides a brief comparisons between the formalisms. Section 6 general discussion. Finally, in Section 7, we draw conclusion. 2 Logic-based Knowledge Representation Formalisms One may classify knowledge by its nature: declarative or procedural. Declarative knowledge represents simple facts, it includes the truth of propositions or statements about the world. For instance, humans can eat at any given time of the day. Declarative knowledge can be easily retrieved from a knowledge base using a simple operation, a look up process. The retrieval of declarative knowledge could be the solution for a problem a user has, or it could be part of a larger solution. Conversely, procedural knowledge involves knowing HOW to do something. It details a set of processes as opposed to facts. Procedural knowledge outlines a step by step process for reaching a particular end or solution to a problem facing the user. For instance, operating a vehicle on the road necessitates procedural knowledge allowing a driver to fulfill the end of driving a car on a certain road. Unlike declarative knowledge, simple look up operations are inadequate to retrieve procedural knowledge. To retrieve procedural knowledge one may resort to the use and implementation of algorithms. Moreover, procedural knowledge involves implicit learning. Researchers developed a number of knowledge representation formalisms within the semantic web context. The decision of choosing a particular formalism heavily depends on the need and

4 nature of the problem faced by the user. Semantic networks, production rules and formal logic provide the main knowledge representation formalisms in semantic web. RDF graphs can be utilized to specify models of semantic networks [4]. Logic is used to recognize a precise semantic interpretation for both of the other forms. By providing formal semantics for knowledge representation languages. A Knowledge representation formalism help in representing the relevant knowledge of a specific domain as facts and rules in such a way that make it efficiently retrievable. Despite this useful application of KR, one may be interested in retrieving knowledge that is not explicit in the knowledge base. One may be interested not only in facts or rules, but also in reasoning based on a combination of declarative, as well as procedural knowledge. In retrieving explicit knowledge from a knowledge base, one may need extra programs in order to specify what is wanted from the machine. In the case of implicit knowledge representation, this process becomes more complex. The process of asking the machine to produce implicit knowledge based on explicit knowledge in a KB is referred to as reasoning or inferences. Two main categories of inference problems are: subsumption and instance problems. The former is to check whether one concept is more specific than the other. The latter is to check if a given individual is an instance of a particular class [5]. In this survey, whenever reasoning and/or inference are mentioned, they correspond to similar concepts stated above. A knowledge-based system is based on acquired knowledge of a specific domain represented using some formalism. Basic components of a knowledge-based system are: knowledge base (KB) and an inference engine/reasoning mechanism that simulates the problem solving process. The knowledge acquisition is also a core component. Reference [6] provides a survey of fundamental knowledge acquisition models and techniques. The acquisition of knowledge is not a trivial task. Each Knowledge Representation formalism has an expressive power and a complexity for retrieving and reasoning about a specific knowledge. An efficient knowledge based system that is a component of knowledge base and an inference engine should have high expressive power and low complexity. The knowledge representation formalism in this survey will be outlined as follows. In Section 2.1 we give a brief introduction about first-order logic, which has an advantages over the varieties of other logics due to its expressivity, however it has a decidability problem. In Section 2.2 we introduce modal logic. Modal logic is appropriate for small knowledge-based system, knowledge representation is more restricted in modal logic. Section 2.3 focuses on description logics, an extension of the semantic network and frame formalism. Section 2.4 concentrates on temporal logics. 2.1 First-Order Logic First-Order Logic (FOL) is the most commonly studied and implemented logic, it is one of the widely used Knowledge Representation formalisms. Computer scientists, especially those

5 working within the umbrellas of Artificial Intelligence picked up the most applied form of logic FOL [7]. It comes out of propositional logic that postulates the existence of real facts that could be obtained using predicates. This formalism mostly concerns declarative knowledge. Nevertheless, it could be used to represent complex facts or concepts by combing two simple declarative facts by logical connectives such as AND, OR. With the first order logic, problem solving could be achieved by applying logic inference to facts and sentences. The mostly involved operation would be "look-up" operations, followed by some reasoning algorithm such as Tableau algorithm or resolution proof methods, which are the two main methods of reasoning in FOL. One of the greatest concern when applying FOL is in determining the validity of formulae involving sentences. In particular, we are interested in whether a set of sentences S logically entails a sentence R. The set S of sentences used as a basis for calculating entailment can be thought of as a knowledge base (KB). This pertains to whether the specified mathematical formulae are doing what they are intending to do. The application of FOL is intended to produce meaningful representation of knowledge based on a knowledge base, body of facts. FOL usually rely on half a dozen symbols to draw a given reasoning based on a given knowledge base. Depending on the choice of predicates, highly expressive knowledge representations can be constructed using these elements.nevertheless, FOL does not perform greatly when considering its complexity. FOL faces a challenge when it comes to its reasoning algorithm, when we represent a lot of domain knowledge and the knowledge-base gets large, FOL will eventuality become undecidable. Therefore, FOL is not the most preferred formalism when it comes to the current practice of knowledge representation. We first define a syntax for the first-order logic and then we define its meaning (semantics). We incorporate the syntax and semantic as outlined in [8]. A vocabulary of a first-order logic is comprised of the following. A set of unique predicate symbols of arity n. These predicate symbols are often denoted using capitalized letters, or more commonly using P, and Q. Usually (but not always) the vocabulary also contains: A set of unique constant symbols. These symbols are often denoted using lower case, or more generally using a, and b. A set of unique function symbols of arity m. These symbols are often denoted using lower case letters, or more generally using f, and g. Depending on the choice of vocabulary, we can construct a first-order language over that vocabulary by using terms and formulas, the two syntactic atomic expressions in FOL. Both terms and formulas are built by the following rules:

6 An infinite set of variable symbols, often denoted using x, y and z. The logical connectives (negation), (conjunction), (disjunction) and (implication). The variable-binding quantifiers (universal) and (existential). Left and right parenthesis and the comma. We now define formulas, more precisely WFFs, or well-defined formulas: An atomic formula is a WFF. If ϕ and ψ are WFFs, then so are ϕ, ϕ ψ, ϕ ψ and ϕ ψ. If ϕ is a WFF and x is a variable, then x.ϕ and x.ϕ are WFFs. Nothing else is a WFF. A formula is said to be a FOL sentence if all the variables in the formula are within the scope of some quantifier, i.e., if all the variables in the formula are bound. Formulas can be seen as descriptions within FOL, while sentences can be seen as representing knowledge. A language here can be seen as a string of symbols put together according to the precise rules given by the syntax of the language. These strings do not have any intrinsic meaning or content assigned to them; a meaning is assigned to them through a preassigned notion of meaning in a precise sense. A model or an interpretation is an associated semantics consisting of additional information that allows us to assign a truth value to any such string (meaning sentence here; in the case of FOL, the truth value is either true or false; in other logics there are other options.) This assignment is done recursively, from the assignment of world/domain elements to constants to the assignment of meaning to n-ary predicates for all values of n. Notice that the truth value assigned to a sentence is dependent on our choice of interpretation. In order to define a semantics or meaning for first-order logic, the author in [18] introduced the notion of a model. A model for a given vocabulary can be thought of as a concrete example where axioms (elements of the syntax) are verified or realized, in a precise sense (given by an interpretation function F; please see below) together with a specification of the objects we are referring-to, i.e., the world of discourse, e.g., a model for the axioms of group theory is a group. We can also see a model as a possible world in which the axioms are verified with respect to some pre-specified meaning (again, given by the function F). Formally, a model for a given first-order logic language is a pair (D, F) specifying a non-empty domain D and an interpretation function F. The domain contains the kinds of things we want to talk about, e.g. group elements, individuals, organizations, events, places, or objects; notice that different models may assume different domains (and different functions F). The interpretation function assigns a meaning to each symbol in the vocabulary in such a way that

7 we can use Tarski s truth definition to unambiguously decide and determine the truth of a sentence. Basically, the function F assign an interpretation for each symbol in the vocabulary. Each of the constant symbols a,b, etc. are interpreted as an element of the domain and each predicate is assigned a meaning, e.g., the 2-ary predicate expression B(a,b), defined in the domain of all people, can be interpreted as meaning Andrew is the brother of Bob, by first assigning elements (Andrew, Bob respectively) to each of the constants a,b and then assigning, through our function F, the meaning Is a brother of, to the 2-ary predicate B. Then through our truth function we can determine whether, in our world our sentence is true. Collection S of sentences in which these sentences are true is said to be a model for such set of sentences. For example F(Student) is some subset of D, which we can specify as the set of students within the domain. Another example is F(Enrolled) which is some subset of D1 D2, which we can specify as being the set of individual student of the domain D1 who are enrolled in a set of individual courses of the domain D2. Each function symbol f of arity m is interpreted as an m-ary function over the domain. For example F(BrotherOf) is some function D D, which we can specify as being the function which maps an individual to his or her brother. Note again that a given vocabulary can be mapped to the same (or even different) domain via a different interpretation function, henceforth it is probable to create multiple models for a given vocabulary.there are also considerations here such as the Compactness theorem in Logic, whereby the existence of a model of a given infinite cardinality implies the existence of models of all different infinite cardinalities. The main limitations of knowledge representation using First Order Logic comes from the complexity of its reasoning algorithm. In [9] Drew wrote a critique of pure reason and he argued that most reasoning about the world is not deductive. He also mentioned that theorem prover or proving things in FOL generally happens from vague intuition which is far from human experts reasoning mode. Thus, it might not be appropriate to use logical deduction as a main mechanism for inference in logic based knowledge system. Another limitation First Order Logic reasoning suffers from is the lack of rigorous reasoning presented by the binary true or false formulae used in FOL inference. This result in no space for deciding about uncertainty which is a common in real life problems. For instance in clinical diagnosis, if a patient had a breast cancer and she would like to be clear about the cause of such illness, the doctor may observe her and analyze her history of her medical records. Finally, the doctor could come to a conclusion that the possible reason for her illness results from 80% of excessive stress and 20% of environmental factors. This conclusion can t be easily handled by first order logic. Variations of other logic such as multi-value logic and fuzzy logic were developed to handle such deficiency of first order logic [9]. Third limitation of knowledge representation using First Order Logic comes from the monotonicity property of FOL. Some researchers argued that monotonicity is a deficiency of

8 FOL in both representing and reasoning about knowledge. To illustrate the problem let s suppose we have a simple domain defined by a set of axioms. Any additional axioms must be consistent with the original axioms, otherwise the system stops working. The following demonstrates the argument: x[bird(x)] flies(x) bird(duck) There would be a conflict and the system would break down if we add flies(duck) to the set of axioms. This deficiency can somewhat be augmented by using other instruments such as temporal logic if the inconsistency is introduced because the known fact changes over time. 2.2 Modal Logic Model logics are designed to reach the objective reality of a specific phenomenon in the world. While First-order logic FOL semantics only allow us to talk about truths within the range {T, F} (true or false), Modal logics are designed for reasoning about different modes of truth. For example, they allow us to specify what is necessarily true, known to be true, or believed to be true [10]. These diverse modes usually denoted to as modalities include possibility, necessity, knowledge, and belief. The most important mode among these are what must be true (necessity) and what may be true (possibility). As discussed in [10], the interpretation of such modes gives rise to different variations of modal logics. For example, alethic modal logic deals with the two basic common modes necessity and possibility which are interpreted as necessary (possible) truth. We say we are in epistemic logic if the necessity or the possibility is interpreted as stating that which is known (not known) to be true. Finally, if the mode has a time dimension, such that if necessity or possibility is interpreted as stating that which always has been or to which will always be (possibly) true, we have temporal logic. Note that much of the following section is derived from [11, 12].Modal logic is often formulated by augmenting any existing logic such as propositional or even non-classical logic with logical operators referring to modalities. The following example illustrates the formation of modal logic. There are two modal operators of the first-order modal logic that are syntactically represent a formula in modal logic. These modal operators are the diamond symbol for the possibility and box symbol for necessity. The author in [12] define a well-formed formula (or WFF, or simply a formula ) of the logic as follows. An atomic formula is a WFF. If ϕ and ψ are WFFs, then ϕ, ϕ ψ, ϕ ψ, ϕ ψ, ϕ and ϕ are WFFs. If ϕ is a WFF and x is a variable, then x.ϕ and x.ϕ are WFFs.

9 Nothing else is a WFF. Given ϕ as a formula, then the modal logic formula ϕ interpreted as ϕ is possibly true, whereas ϕ interpreted as ϕ is necessarily true. With such a logic we can, for example, represent the following sentences. It is possible that the Professor is late. It is possible that DB course will be offered in the Fall. It is not possible that: every Student is Smart, Tom is a student, and Tom is not Smart. It is necessary that Alice pass the exam or Alice does not pass the exam. Other modal logics usually have different modal operators. For example in epistemic logic, we need to represent what is known (not known) to be true. The two basic epistemic logic modal operators are K and C. The modal operator K represent it is known that, and the modal operator C represent it is common knowledge that. A formula such as KTom x.late(x) represents Tom knows that someone is late, whereas x.ktomlate(x) represents Tom knows someone who is late. A formula such as CSmart(John) represents the common knowledge that John is smart. By common knowledge we mean that everyone knows that John is smart, everyone knows that everyone knows that John is smart, everyone knows that everyone knows that everyone knows that John is smart, and so on. We will discuss the modal temporal logic in section 2.4. Attributable to [13], the semantics of modal logic is defined as the following. Any modal logic can be assigned a possible world semantics. Essentially, a possible world is any world which is considered possible. This includes not only our own, real world, but any imaginary world whose characteristics or history is different. Here we will supply a semantics for our example first order modal logic. Given a modal logic language vocabulary of unique constant and predicate symbols, a model M for that vocabulary consists of 4 elements (W, R, D, F) which comprises the following: the first element of the model M is a non-empty set W of possible worlds. The second element of the model M is an accessibility relation R W W between two worlds, where the relation R(w,w ) represents that world w has access to world w, or that w is accessible from w. The third element of the model M is a domain D of the kinds of individuals, places or objects we want to talk about, and which is common to all worlds. And the fourth element of the model M is an interpretation function F which assigns a semantic value in D to each symbol in the vocabulary at each world of W. One assumption the author made in their example modal logic is that the constants are rigid. By this it means the interpretation of a constant symbol is the same at every world. Hence if we interpret F(Professor) as being an individual called the Professor in some world, then F(Professor) is interpreted as the same individual in any other world. Note however that the

10 interpretation of a predicate symbol at some world may differ from its interpretation at some other world. The domain of individuals in Modal logic may change from state to state (or world to world). Also if we ignore the common domain requirement, modal logic even may allow for things to exist in one world but not another. 2.3 Description Logics Description logics which is earlier called terminological logics firstly developed for a purpose of providing a formal foundation for semantic networks. Semantic networks were first proposed by Charles Peirce s under the name existential graphs in Semantic networks represent semantic relations between concepts. It is a directed graph consisting of vertices, and edges. Vertices represent concepts; concepts could be an objects, individuals, or abstract classes; and the edges represent semantic relations between these concepts. Semantic networks have long been used in philosophy, psychology and linguistics. However It has been introduced as a knowledge representation formalism in early 1960s [16]. One of the features of the Semantic Networks is the inferential dependencies which are type of semantic information that relates two or more concepts, such as IS-A, TYPE-OF edges. Due to their hierarchical nature and ability to represent class relationships through links, semantic networks can easily model the inheritance relationship. Another important feature of semantic networks is that the representation tends to cluster information relating to an object around that object. This feature not only has useful consequence for computational efficiency, but is also claimed to be the way human memory works. Semantic Networks can also be used to represent events and natural language sentences [17]. Description Logics (DLs) are a well-known family of logic-based knowledge representation formalisms that represent the conceptual knowledge as well as the instances of knowledge of an application domain. The representation of such knowledge start by first defining the relevant concepts of the domain, and then defining properties of objects and individuals of these concepts occurring in the domain. It has numerous applications in the semantic web field. For instance, DLs is a standard for the W3C's web ontology language OWL. Other relevant applications of DLs include the representation of and reasoning about conceptual database models. Strictly speaking, extended entity-relationship (EER) model and UML class diagrams can be embedded into DLs, and DL reasoners can be used to verify their consistency and to derive implicit consequences of the model [14]. Description logics focuses on representing the terminological knowledge of an application domain. The central ingredients to such a representation are concepts, roles and individuals. In DL a knowledge base is a finite set of TBox (terminological box) and the ABox (assertional box).

11 In general, the TBox contains sentences describing concept hierarchies (i.e., relations between concepts) while the ABox contains ground sentences stating where in the hierarchy individuals belong (i.e., relations between individuals and concepts) [14]. For example, the statement every manager is employee belongs in the TBox, while the statement Bob is an employee belongs in the ABox. In this section, we briefly introduce the formal framework of DLs which will serve as the basic representation language for the non-temporal knowledge. There are different description languages in the DL family. The description of ontologies and knowledge in description logics uses constructs that have semantics given in predicate logic. However, due to historical reasons, different notation is used. Typically, the notation is much closer to notations in semantic networks and frame based systems. The presentation of Description logic formal framework will strictly follow the ALC (Attributive Concept Language) notation introduced by Schmidt-Schauss and Smolka [15]. Attributed to [15] the syntax and semantic of DL defines as the following, DL concepts are built from a countably infinite set NC of concept names that relates atomic and complex concepts, a countably infinite set NR of role names, and countably infinite set NI of individual names by applying the available concept constructors. In the basic standard propositionally closed description logic ALC, these constructors are A,, T, C, C П D, R.T, R.C Where A is the atomic concept and C, D are complex concepts that range over NC.R represents a role and ranges over NR. T is the universal concept, and as usual we use as an abbreviation for T.Complex concepts are formed by constructors such as C П D.We use C D as an abbreviation for ( C П D). We also use R.C as an abbreviation for the negation of existential R. C A concept denotes the set of all individuals satisfying the properties specified in that concept. For example, the concept Person represents the set of people, whereas Student П Male represents the set of male student. A concept such as Student represents all the individuals which are not students. The intersection and complement of atomic concepts or complex concept are often denoted as concept conjunction and negation respectively. Concerning the and restrictors, if the concept C denotes some class of individuals, then the restriction R.C represents those individuals who are R-related only to the individuals of that class. For example, teaches.phd represents those professors who only teach PhD students, whereas teaches. represents those professors who never teach. The restriction R.T represents all individuals that are R-related to at least one other individual. For example, ownscar.t represents all individuals who own at least one car.

12 TBoxes represent knowledge at the conceptual level that can be formed by sentences. A sentence may include concept subsumption C D or concept equality C D. They are formally build from atomic concepts and atomic roles or complex concepts with constructors like the above ALC constructors. On the other hand, ABoxes represent knowledge at the instance level, It is formally built by asserting each concept C(a) and role r(a,b) with individuals, where a and b are individuals. The semantic of DLs is based on the notion of an interpretation I of an expressions, which consists of two parts I = < Δ I,. I > where Δ I is a set of non-empty individual domain of interpretation Δ I, and an interpretation function. I that maps each ALC concept name C NC into a subset of C I Δ I and every role name r NR to a relation r I Δ I x Δ I and every individual name a NI to an element in the domain a I Δ I.The interpretation function. I can be defined for a complex concept as follows T I = Δ I I = ɸ ( C) I = Δ I \ C I (C П D) I = C I D I = {a ϵ Δ I a ϵ C I a ϵ D I } ( r.c) I = { a ϵ Δ I b.(a,b) ϵ R I b ϵ C I } We say that C I (r I ) is the extension of the concept C ( role name r ) in the interpretation I. If a C I then we say that a is an instance of C in an interpretation I. The core inference task over concept expressions in DL is subsumption. The meaning of subsumption is to check whether the concept described by D is more general than another concept denoted by C, written as C D. A special case of subsumption is satisfiability which is the problem of checking whether a concept expression does not necessarily denote an empty concept [18]. Other basic reasoning tasks are: instance checking C(a) is to check if an individual a is an instance of the concept c; equivalence checking A B is to check if both A B and B A are true; and retrieval is the reasoning task of retrieving a set of individuals that instantiate the concept C Expressive Description Logics In the previous section we discussed the basic ALC (Attributive Concept Language) description logic which is an extension of the basic AL (Attributive Language) logic. In order to increase the expressive power of Description logic several extension to the ALC were proposed by adding new constructs. We ll now give a brief introduction to the more expressive description logics, for a detailed discussed we refer the reader to [16]. These languages feature complex roles such

13 as R1 R2 (intersection), R1 R2 (union), R (complement), R1 R2 (composition), R+ (transitive closure) and R 1 (inverse). An example of conjunction include hasfather hasmother which can be used to define hasparent. Another example is the composition hashusband hasbrother which can be used to define hasbrotherinlaw, the transitive closure of haschild which is the role hasdescendant, and the inverse of haschild which is the role hasparent. Additional concepts occurring in expressive description languages is the cardinality of a role participating in a concept include n R.C, and n R.C (qualified number restriction) and {a1,..., an} (set concepts, for n 1). Examples include 2ownsCar.BMW which expresses the set of individuals who own at least two BMW cars and {John, Tom, Mary, Jennifer} which can be used to define the concept of students at some class. In order to avoid Description Logic languages with long names, (S) was introduced as an abbreviation for an ALC language with transitive roles. The extension (H) allows role hierarchies, i.e. role inclusions R1 R2, and the extension (R) incorporates role intersection. (I) allows for inverse roles, whereas (O) allows for set concepts. The extension (Q) incorporates qualified number restriction. The extension (D) allows the integration of an arbitrary concrete domain within a description logic language. An example of an expressive description language is SHOIN(D). Table 2 below show how the AL description logic can be further extended by adding new constructs. The name of the logic is then formed from the string AL[U][E][N][C],so for example the logic ALEN is the attributive language logic extended with full existential quantification and number restrictions. Some of the combinations are not unique from the semantic point of view - for example, union and existential quantification can be expressed using negation Reasoning with DL Table 1: AL logic extensions There are different reasoning tasks in Description Logics, According to [18] we can categorize the type of reasoning tasks as two categories: either a reasoning for a terminology (TBox) or a reasoning for an assertions (ABox). Typical reasoning tasks for a terminology (TBox) are (1)

14 satisfiability which determine whether a particular concept is satisfiable, i.e. non-contradictory; and (2) Subsumption which determine whether one concept is more general than another, i.e. whether the first description subsumes the second. Given a terminology and the set of all possible models M for that terminology (whereby the models have differing domains and interpretation functions), we say that a concept C is satisfiable if it is satisfied in at least one model of M. Concept C is unsatisfiable if it is satisfied in no model of M. A concept C1 is said to be subsumed by a concept C2 if C1 C2 is satisfied in every model of M. Two concepts C1 and C2 are said to be equivalent if C1 C2 is satisfied in every model of M. From their definitions, satisfiability and equivalence can be reformulated in terms of subsumption. C is unsatisfiable iff C is subsumed by, C. C1 and C2 are equivalent iff C1 C2 and C2 C1. Moreover, if the type of Description Logic allow full concept negation and conjunction, then the subsumption and equivalence can be reformulated in terms of satisfiability. C1 is subsumed by C2 iff C1 П C2 is unsatisfiable. C1 and C2 are equivalent iff both C1 П C2 and C1 П C2 are unsatisfiable. A structural subsumption algorithms can compute the subsumption of concepts if the description logic is simple. These structural subsumption algorithms perform greatly well if the DL has less expressive power which do not allow negation at all, such algorithms compare the syntactic structure of concept descriptions. We refer the reader to [19] for an in-depth discussion. On the other hand, in order to determine the satisfiability of a more expressive description logic which allow concept negation and conjunction, we use tableau-based algorithms [19]. We will shortly give an example of a tableau-based algorithm for the ALC language. The second category of the reasoning tasks involves an assertions (ABox). Typical reasoning tasks for the ABox are (1) Consistency which determine whether a particular assertion is consistent, non-contradictory, i.e. whether it is satisfied in some model; and (2) whether one particular individual is described by some concept. We say that an ABox A is consistent with respect to a TBox T if there is at least one model which is both a model for T and a model for A. We simply say A is consistent if it is consistent with respect to the empty TBox. An example is the set of assertions {Mother(Mary), Father (John)} which is consistent with respect to the empty TBox, but inconsistent with respect to a TBox defining the disjoint concepts Mother and Father. Given an ABox A and the set of all possible models M for A, we say an assertion α is entailed by A if α is satisfied in every model of M. Hence Mother(Mary) is entailed by an ABox if the Mary is a mother in every model for that ABox. Derived from their definitions, consistency and entailment are related as follows. α is entailed by A iff A { α} is inconsistent.

15 Consequently, concept satisfiability in the TBox can be reduced to consistency in the ABox, because for every concept C and arbitrarily chosen individual name a, we have the following. C is satisfiable iff {C(a)} is consistent. Basically, the tableau-based algorithm for ALC works as the following. Let s assume we have a concept C, the algorithm works on constructing a model M (D, F) such that C is satisfied in M, i.e. F(C). The algorithm start by arbitrarily chooses an individual x where x F(C). Starting with the ABox A {C(x)}, consistency-preserving tableau rules are applied next until the ABox is complete, i.e. until no more rules can be applied. A is consistent (which means that C is satisfiable) if the subsequent ABox does not have a contradiction, otherwise it is inconsistent. Note that a contradiction or a clash in an ABox A is such that either { (a)} A, or {C(a), C(a)} A for some individual name a and concept C. The example presented here is incorporated from [20], suppose we want to determine whether C0 ( ownscar.bmw) П ( ownscar. BMW) is unsatisfiable. We choose an arbitrary individual x and start with ABox A0 {(( ownscar.bmw) П ( ownscar. BMW))(x)} We then apply the following tableau rule. П -rule Condition: Ai contains (C1 П C2)(x), but not both C1(x) and C2(x). Action: Ai+1 Ai {C1(x), C2(x)}. After applying this rule, we have A1 A0 {( ownscar.bmw)(x), ( ownscar. BMW)(x)} We next apply the tableau rule below. -rule Condition: Ai contains ( R.C)(x), but there is no z such that. C(z), R(x, z) are in Ai. Action: Ai+1 Ai {C(y), R(x, y)} where y does not occur in Ai. Thus we have A2 A1 {( ownscar.bmw)(x), BMW(y), ownscar(x, y)} The universal rule is as follows. -rule

16 Condition: Ai contains ( R.C)(x) and R(x, y), but it does not contain C(y). Action: Ai+1 Ai {C(y)}. After application of this rule we have A3 A2 { BMW(y), BMW(y)} The final ABox A3 contains a contradiction or clash. This conclude that the concept ( ownscar.bmw) П ( ownscar. BMW) is unsatisfiable and the original ABox A0 is inconsistent. 2.4 Temporal Logic The representation and reasoning about time and temporal information within a logical framework has been done using temporal logic. There are a rich variety of temporal models and logics introduced and studied over the past 50 years including Interval temporal logic (ITL), Linear temporal logic (LTL), Computational tree logic (CTL), Property specification language (PSL), Halpern-Shoham Temporal logic and others [52]. In this subsection we will briefly refer to the temporal logic that use modal-logic operator approach that was introduced around 1960 by Arthur Prior under the name of Tense Logic and subsequently developed further by many logicians and computer scientists [52]. Other than representing and reasoning about temporal information, temporal logic could also be used as a tool for the specification, formal analysis, and verification of the executions of computer programs and systems. Attributed to [41], the syntax of temporal logic has two kinds of operators: logical operators and modal operators. Logical operators are usual truth-functional operators ( ). There are two type modal operators used in Linear Temporal Logic and Computation Tree Logic; Binary operator and Unary operator are defined as follows.

17 Halpern and Shoham [41] introduced temporal logic HS which could be seen as the combination of a propositional modal logic with the interval-based temporal logic. The syntax of HS wellformed formulae are built by extending the propositional calculus with the modal temporal operators corresponding to the Allen interval relations [1] in the table2 below are: before (b), meets (m), during (d), overlaps (o), starts (s), finishes (f), equal (=),after (a), met-by (mi), contains (di), overlapped-by (oi), started-by (si), finished-by (fi). Table 2: Allen relations between pairs of intervals and Halpern-Shoham modal operators on them.

18 The semantics of the logic is based on a Kripke structure whose domain is a set of intervals, and its logical formulae are interpreted as sets of intervals. We say that a formula holds at an interval if it is evaluated as true at that interval. The modal operators relate the reference interval now with other intervals. As an example incorporated from [41], the notion of Mortal can be expressed in HS temporal logic as: LivingBeing <met-by> LivingBeing Which defines a LivingBeing as the one who will not be alive in some interval met by the current interval. In [41], the semantic of Halpern-Shoham Temporal logic is defined as the following. A linear and unbounded temporal structure T = (P, <) is assumed, where P is a set of time points and < is a strict partial order on P. The interval set of a structure T is defined as the set T *< of all closed intervals [t1, t2] = {x P t1 <= x <=t2} in T. An interpretation I = <T * <,. I > consists of a set T *< (the interval set of the selected temporal structure T), and a function. I which maps each primitive proposition into a set of closed intervals where it is true. Halpern and Shoham improve the expressive power of the temporal logic and prove many interesting complexity results concerning the validity and satisfiability issues. However, this natural and seemingly simple logic turned out to be highly undecidable [Halpern and Shoham 1991]. The explanation of the heavy computational behavior of HS temporal logic is that it can be viewed as a two dimensional modal logic interpreted over products of liner Kripke frames which provide a good playground for simulating Turing machines [Artale, et al 2015]. 3 The Semantic Web With the expansion of the World Wide Web in the late eighties, Tim Berners-Lee identified two purposes for internet use by people: information and knowledge sharing and information analysis. The internet allowed people to share a multitude of information stored in the web with each other across time and space. Internet use also witnessed a great deal of analysis with the purpose of generating meaningful knowledge that caused a need for tools that can help people analyze and manage the information they share in a meaningful way. This vision has been referred to as semantic web [2]. The World Wide Web Consortium has initiated an effort to increase the depth of information sharing, analysis and automation using various standards and technology which has become known as the semantic web intuitive. Lukasiewicz [21] outlined the Semantic Web intuitive goals as 1) add machine-readable meaning to web pages using ontologies, thus assigning a precise definition to shared terms. (2) Increase the use of automated reasoning within the context of knowledge representation, and

19 3) generate automated cooperation among machine systems to produce meaningful information based on web content. It is clear that semantic web involves the use of knowledge representation and reasoning to a great extent. Below, we outline few semantic web techniques that heavily utilize knowledge representation. These techniques include a wide range of languages such as XML, the resource description framework RDF, the XML and RDF schema languages, the RDF query language called SPARQL; and the Web ontology language abbreviated as OWL. These languages and technologies allow computers to represent human knowledge on the web in an automated, reliable and accessible manner. XML stand for extensible Markup Language - is a general purpose computer language that uses a set of explicit tags to produce documents interpretable by humans as well as machines. The purpose of XML is to facilitate the sharing of data structures across information systems [22]. XML is an extensible language, which means that users may develop their own tags. Tags allow users to specify, identify or share a data set for a particular use. For example the pair of tags <b>...</b> in HTML specifies that the enclosed text is displayed in bold font, whereas the pair <message>...</message> in XML labels the enclosed text as a message. A XML schema - written in a schema language like DTD (Document Type Definition) - specifies a vocabulary for the kinds of tags which can be included in a XML document and the valid arrangements of those tags. A XML parser checks whether a document conforms to its schema. XML Schema type System has two commonly cited drawbacks. First, XML Schema user-defined datatypes are not accessible by RDF(S) and OWL causing an inconvenience for users. Second, XML limits datatype constraints that RDF(S) and OWL could provide because XML Schema does not support n-ary datatypes. XML schema has multiple varieties. More information regarding these can be found at [23] RDF Stands for Resource Description Framework - is a language with the purpose of representing and sharing information about resources, web resources in particular [24]. The language solves several practical problems, such as creating metadata for web resources. Note that this metadata mechanism supplies the necessary platform for processing information among various agents making it easier for more users to use the resources data. Another important issue solved by RDF is allowing users to specify statements about any resources. This leads to an open world semantics The resource in RDF data models is represented as a Uniform Resource Identifier (URI) reference. URIs are generic types of Uniform Resource Locator URLs. URLs are string characters specifying the resource in web pages, whereas a URI is an identifier for the resource without having to indicate the location of the resource. URIs can be used within a RDF statement to refer to essentially any specified resource. RDF statements and models can be graphically represented showing three main ingredients as a triple, subjects, objects and predicates. A

20 predicate is a directed arc emanating from a subject pointing to an object. URI reference nodes are depicted as ellipses, whereas constant values are depicted as rectangles [24]. The meaning of an RDF graph is the conjunction of the statements corresponding to all the triple it contains. However, some of the limitations of RDF is that it does not provide means to express negation (NOT) or disjunction (OR). A formal definition of RDF triples are given in [25]: _ an infinite set U containing RDF URI references, _ an infinite set B = Nj ; j ϵ N (called blank nodes), _ an infinite set L (RDF literals). A triple(s; p; o) ϵ (U B) x U x (U B L) is called RDF triple. The symbols s; p; o are Subject, Property, and Object respectively. An RDF, or RDF graph is a set of triples. A subgraph is a subset of an RDF graph. According to the above definition, objects of the graph can be pieces of text instead of resources (URIs); those pieces of text are called literals. This is syntactically useful in RDF documents (being able to write directly some text rather than storing it in another resource). It has to be emphasized that only "object" can assume literals. In RDF graphs, a Blank node represents a special node that has no URI or intrinsic reference. Blank nodes are used in n-ray RDF statements or graphs specifying relationships. Blank nodes breaks up n-ray relationships into binary relationships. Blank nodes are also used to represent a statement or graph that does not have an explicit URI reference. RDF s chief purpose is to make knowledge accessible, machine-readable and processable by computers and other software applications. To achieve this aim RDF utilizes XML. A particular type of notation called RDF/XML is used to specify RDF graphs as XML files. This process of converting RDF into XML is called serialization. Serialization has many versions such as Turtle, N3 and N-triples. We refer the reader to [24] for more information regarding RDF and its serialization notations. To draft a statement using RDF, W3C has recommended RDF vocabularies guiding users in creating RDF graphs. Part of this vocabularies are certain types known as reserved words, which are used to write RDF statements. Some URI references possess unique meanings. For instance, the URI reference: " conventionally represent a namespace whose prefix is rdf:. URI: " represent another namespace whose prefix is rdfs:. The set of reserved words in RDF vocabulary description language, called RDF schema (RDFS) is described in [26]. The vocabularies allow the user to specify the properties, as well as the relationships among resources. Also, vocabularies allows defining classes of references. A class represents resources that share a set of properties. Elements of a class are called instances of that class. In schemas, new resources can be defined as specialization of an existing ones, thus inferring implicit triples is possible. Schemas also