sions of the DL-SHOIN(D), SHIF(D) and DL-Lite respectively, he uses the lexicographic entailment and his work is based on Nilsson's probabilistic logic [11].
Reasoning About Belief Uncertainty in π«π³β π³ππππ΅ ππππ Ala Djeddai, Hassina Seridi and Med Tarek Khadir LabGED Laboratory, Computer Science Department, Badji Mokhtar-Annaba University P.O. Box 12, 23000 Annaba, Algeria. {djeddai,seridi,khadir}@labged.net
Abstract. Dealing with uncertainty is a very important issue in description logics π (DLs). In this paper, we present πππ·πΏβ πΏππ‘πππππ a new probabilistic extension of π π·πΏβ πΏππ‘πππππ , known as a very expressive DLβLite, by supporting the belief degree interval in a single axiom or a set of axioms connected with conjunction (by β§) or π π disjunction (by β¨) operators. The πππ·πΏβ πΏππ‘πππππ semantics is based on π·πΏβ πΏππ‘πππππ π features which are a new alternative semantics for π·πΏβ πΏππ‘πππππ having a finite strucπ ture and its number is always finite unlike classical models. πππ·πΏβ πΏππ‘πππππ also supports terminological and assertional probabilistic knowledge and the main reasoning tasks: satisfiability, deciding the probabilistic axiom entailment and computing tight interval for entailment are achieved by solving linear constraints system. A prototype is implemented using OWL API for knowledge base creation, Pellet for reasoning and LpSolve for solving the linear programs. Keywords: uncertainty, description logics, probabilistic reasoning, DLβLite, belief probability.
1
Introduction
Dealing with uncertainty in knowledge representation and reasoning is a very important issue and research direction. Uncertainty arises from various causes such as: automatically extracting and processing data, integration of information from different heterogeneous sources, inconsistency, incompleteness and incorrect information. Ontology merging, user or automatic annotations, ontology alignment and information retrieval are also important sources of uncertainty. An example of uncertain information is given as follows: βRoufaida is a postgraduate student with degree β₯ 0.7β. From the main languages used to represent and reason about knowledge, there are description logics DLs [1] that are designed for crisp and deterministic information. Thus, they are not able to deal with the unknown and therefore must be extended in order to comply with uncertain knowledge. DLs are the formal foundation of the ontology web language OWL which is a W3C standard used for knowledge modeling in the semantic web. To handle uncertainty, many approaches proposed DLs extensions such as, probabilistic approaches when the degree of
uncertainty is interpreted as probability value. Most of them are difficult to use and based on classical models or use graphical models (such as: Bayesian Network) and don't supπ porting the belief in terminological axioms. In this paper, πππ·πΏβ πΏππ‘πππππ a novel probaπ bilistic extension of π·πΏβ πΏππ‘πππππ [2] by using the belief probability is presented. An example of probabilistic axiom is: a given professor is a PhD student with probability in [0.7,0.9]. We choose working with intervals instead of single values because the probabilities can be extracted from different sources and different agents can compute different probabilities so intervals are a good choice for working under uncertainty. The π π πππ·πΏβ πΏππ‘πππππ semantics is based on π·πΏβ πΏππ‘πππππ features which are a new alternative π semantics for π·πΏβ πΏππ‘πππππ . We use features instead of classical models because they have π finite structure and its number is always finite unlike models. πππ·πΏβ πΏππ‘πππππ supports terminological and assertional probabilistic knowledge. Unlike approaches with graphical π models when the model must be fully specified, πππ·πΏβ πΏππ‘πππππ needs only the belief interval in a single axiom or a set of axioms connected with β§ or β¨. Using features with belief is a new contribution compared to the work in [13] which uses probabilistic interpretation on all features but with conditional probability that are interpreted as statistical information and not belief. π Section 2 presents the π·πΏβ πΏππ‘πππππ language and the feature notion. In section 3 the proposed probabilistic extension is presented and the syntax and semantics of π πππ·πΏβ πΏππ‘πππππ knowledge base based on features are explained. Section 4 details the π reasoning tasks supported by πππ·πΏβ πΏππ‘πππππ . The implementation and experimentation are given in section 5. The section 6 is for the related works where the conclusion and future works are presented in the last section.
2
π«π³β π³ππππ΅ ππππ Language and the Feature Notion
π In this section, we start by defining π·πΏβ πΏππ‘πππππ , its syntax and semantics, then the notion of types and features are detailed.
2.1
The π«π³β π³ππππ΅ ππππ Language
Description logics [1] abbreviated by DLs from a family of languages that are used for knowledge representation and reasoning. Their complexity increased with their expressivity. Therefore some researchers propose π·πΏβ πΏππ‘π language [3,4] with very good computational property but less expressivity. It is behind OWL 2 QL which is OWL 2 profile. π Thus, we focus on π·πΏβ πΏππ‘πππππ [2] which is an expressive superset of π·πΏβ πΏππ‘π where the latter is extended with full Booleans and number restrictions on roles. It contains or individual, atomic concepts and atomic roles. General concepts and roles are defined as follows: π
β π|πβ , π΅ β β€ π΄ β₯ π π
, πΆ β π΅ οΏ’πΆ πΆ1 β πΆ2 , where π΄ is an atomic concept, π is an atomic role, π
is a general role and π β₯ 1. π΅ is called basic concept and πΆ is a gen-
eral concept. We abbreviateοΏ’β€, β₯ 1 π
,οΏ’(οΏ’πΆ1 β οΏ’πΆ2 ) andοΏ’(β₯ π + 1 π
) respectively by β₯, βπ
, πΆ1 β πΆ2 and β€ π π
. A signature is a finite set π = ππΆ βͺ ππ
βͺ ππΌ βͺ ππ where ππΆ is the set of atomic concepts, ππ
is the set of atomic roles, ππΌ is the set of individual names and ππ is the set of π natural numbers used in ππΆ (1 is always in ππ ). A π·πΏβ πΏππ‘πππππ ππ΅ππ₯ π is a finite set of concept inclusions on the form πΆ1 β πΆ2 , where πΆ1 and πΆ2 are general concepts. A π π·πΏβ πΏππ‘πππππ π΄π΅ππ₯ π΄ is a finite set of assertions of the form πΆ(π) (concept membership) or π
(π, π) or οΏ’π
(π, π) (role membership) where π and π are individuals names. The pair π π (π, π΄) forms a π·πΏβ πΏππ‘πππππ knowledge base. The semantics of π·πΏβ πΏππ‘πππππ is given by πΌ πΌ πΌ an interpretation πΌ = (β³ , . ) where β³ is a non empty set called the interpretation domain and .πΌ is an interpretation function that associates every individual π with an element ππΌ ββ³πΌ such that ππΌ β ππΌ for every pair π, π β ππΌ , every atomic concept π΄ with a subset (unary relation) π΄πΌ of β³πΌ and every atomic role π with a subset (binary relation) ππΌ of β³πΌ Γβ³πΌ . The interpretation πΌ is extended to general concepts and roles. Given an interpretation πΌ , we write πΌ satifies πΆ1 β πΆ2 denoted by πΌ β¨ πΆ1 β πΆ2 if πΆ1 πΌ β πΆ2 πΌ , πΌ β¨ πΆ(π) if ππΌ β πΆ πΌ , πΌ β¨ π
(π, π) if ππΌ , ππΌ β π
πΌ . πΌ satisfies a ππ΅ππ₯ π if πΌ satisfies every inclusion in π, πΌ satisfies a π΄π΅ππ₯ π΄ if it satisfies each assertion in π΄. Given a knowledge base πΎ = π, π΄ and an interpretation πΌ, πΌ is called a model of πΎ if πΌ satisfies π and π΄. A knowledge base πΎ is satisfiable if it has at least one model. For a concept πΆ, we say that πΎ satisfies πΆ if there is a model of πΎ satisfying πΆ. For concept inclusion or assertion π₯, we say that π₯ is entailed by πΎ and we write πΎ β¨ π₯ if π₯ is satisfied by every model of πΎ. Terminological Box π: 1. πππ·ππ‘π’ππππ‘ β ππ‘π’ππππ‘ 2. πΆππ’ππ π β οΏ’ππ‘π’ππππ‘ β οΏ’ππππππ π ππ 3. βπ‘ππππ β ππππππ π ππ 4. βπ‘ππππβ β πΆππ’ππ π
Assertion Box π΄: 5. ππππππ π ππ(FOFO) 6. πΆππ’ππ π(DESCRIPTION LOGICS) 7. π‘ππππ(FOFO, DESCRIPTION LOGICS)
π Fig. 1. An example of π·πΏβ πΏππ‘πππππ knowledge base
π Example 1. An example of π·πΏβ πΏππ‘πππππ knowledge base πΎ = β©π, π΄βͺ is presented in fig.1 which contains the atomic concepts ππ‘π’ππππ‘, πππ·ππ‘π’ππππ‘, ππππππ π ππ and πΆππ’ππ π. It also contains the atomic role π‘ππππ. The TBox π tells that all PhD students are students (1) and a course is not a professor or a student. It also says that the atomic role π‘ππππ has ππππππ π ππ as domain (3) and πΆππ’ππ π as range (4). The ABox π΄ defining FOFO as a professor teaches DESCRIPTION LOGICS (5) (6) (7). The signature of πΎ is ππΆ βͺ ππ
βͺ ππΌ βͺ ππ where: ππΆ = {ππππππ π ππ, ππ‘π’ππππ‘, πππ·ππ‘π’ππππ‘, πΆππ’ππ π} , ππ
= {π‘ππππ} , ππΌ = {FOFO, DESCRIPTION LOGICS}, ππ = {1}.
2.2
π«π³β π³ππππ΅ ππππ Features
Working with models has some difficulties. Domains have complex (possibly infinite) structures and DL knowledge bases may have infinitely many models. Thus an alternative
π semantics has been proposed called feature [15] which is prposed for the π·πΏβ πΏππ‘πππππ knowledge bases. In contrast to classical models the features have always finite structures and every knowledge base has a finite set of features. According to these motivations, we choose working with features instead of models. Another motivation is that we must consider all possible knowledge base situations and the number of the latters must be finite. The feature notion is based on the type notion proposed in [8]. We begin by presenting types and then the feature is explained. Given a finite signature π, an πβ π‘π¦ππ π is a set of basic concepts over π, such that: β€ β π and for any π, π β ππ with π < π , π
β ππ
βͺ {πβ |π β ππ
} , β₯ π π
β π implies β₯ π π
β π. In what follows, β€ β π is omitted for simplicity and πβ π‘π¦ππ will be specifies as π‘π¦ππ. Type π satisfies basic concept π΅ if π΅ β π, π satisfies οΏ’πΆ if π not satisfies πΆ, and π satisfies πΆ1 β πΆ2 if π satisfies πΆ1 and πΆ2 . The satisfaction relation is denoted by β¨. We say that π satisfies πΆ1 β πΆ2 ( π β¨ πΆ1 β πΆ2 ) if π β¨ οΏ’πΆ1 or π β¨ πΆ2 . Thus π satisfies a ππ΅ππ₯ π if π satisfies every concept inclusion axiom in π. Types are sufficient to capture the semantics of the ππ΅ππ₯, they are not able to capture the semantics of the π΄π΅ππ₯ because they are not suited for individual, thus they must be extended with additional set. The latter is dedicated to the concept and role memberships and is called a πβ π»πππππππ set for the π΄π΅ππ₯, defined in [15] as: Definition 1. An πβ π»πππππππ set π» (or π»πππππππ) is finite set of assertions of the form π΅(π) or π(π, π), where π, π β ππΌ , π β ππ
and π΅ is a basic concept over π, satisfying the following conditions: ο· For each π β ππΌ , β€(π) β π», and β₯ π π
(π) β π» implies β₯ π π
(π) β π» for π, π β ππ with π < π. ο· For each π β ππ
, π(π, ππ ) β π» (π = 1, β¦ , π) implies β₯ π π(π) β π» for any π β ππ such that π β€ π. ο· For each π β ππ
, π(ππ , π) β π» (π = 1, β¦ , π) implies β₯ π πβ(π) β π» for any π β ππ such that π β€ π. For a given individual π, the type π = π΅1 , β¦ π΅π (π β₯ 1) is called the type of π in π», where π΅1 (π), β¦ π΅π (π) are all basic concept assertions associated with π in π» . A π»πππππππ set π» satisfies πΆ(π) if the type of π in π» satisfies πΆ , π» satisfies π(π, π) or πβ (π, π) if π(π, π) β π» (the same is with οΏ’π π, π and οΏ’πβ(π, π) ). π» satisfies an π΄π΅ππ₯ π΄ if π» satisfies every assertions in π΄. The pair β©π, π»βͺ can be used to provide a semantics characterization but it is proved in [15] that using this pair is not sufficient to capture the connection between the ππ΅ππ₯ and the π΄π΅ππ₯. Thus feature which is a set of π types can provide a complete semantics of π·πΏβ πΏππ‘πππππ knowledge bases. The notion of feature is defined in [15] as follows: Definition 2. Given a signature π, an πβ ππππ‘π’ππ (or simply ππππ‘π’ππ) is a pair πΉ = β©Ξ, π»βͺ, where Ξ is a non empty set of πβ π‘π¦πππ and π» an πβ π»πππππππ set, satisfying the following conditions: ο· βπ β βΞ if βπ β β βΞ for each π β ππ
.
ο· For each π β ππΌ we have π β Ξ, where π is the type of π in π». Given a feature πΉ = β©Ξ, π»βͺ, πΉ satisfies πΆ1 β πΆ2 if every type in Ξ satisfies πΆ1 β πΆ2 , πΉ satisfies an assertion πΆ(π) or π(π, π) if π» satisfies this assertion, πΉ satisfies a ππ΅ππ₯ π if πΉ satisfies every inclusion in π, πΉ satisfies an π΄π΅ππ₯ π΄ if πΉ satisfies every assertion in π΄. π Given a π·πΏβ πΏππ‘πππππ knowledge base πΎ = β©π, π΄βͺ and a feature πΉ, πΉ is a model feature of πΎ if πΉ satisfies π and π΄. The set of all model features of πΎ is denoted by ππ (πΎ). For a concept inclusion or assertion π₯, πΎ β¨π π₯ if all model features of πΎ satisfies π₯ [15]. For further reading, the readers are referred to [8] and [15]. Example 2. Given a feature πΉ = β©Ξ, π»βͺ defined over the signature of πΎ such that: Ξ = {π1 , π2 } where: π1 = ππππππ π ππ, βπ‘ππππ and π2 = {βπ‘ππππβ, πΆππ’ππ π} , π» = {ππππππ π ππ FOFO , πΆππ’ππ π(DESCRIPTION LOGICS), π‘ππππ(FOFO, DESCRIPTION LOGICS)}. The feature πΉ = β©Ξ, π»βͺ respects the conditions in definition 2 and the π»πππππππ set respects the definition 1. Every individual π in ππΌ has a type in π», π1 is for FOFO and π2 is for DESCRIPTION LOGICS. πΉ is model feature of πΎ since every type satisfies every inclusion in π and π» satisfies every assertion in π΄. Thus πΉ = β©Ξ, π»βͺ β ππ (πΎ).
3
Probabilistic Extension based on Features Using Belief
π A novel probabilistic extension of π·πΏβ πΏππ‘πππππ based on features is here presented. The π π·πΏβ πΏππ‘πππππ is extended to specify belief interval about its axioms. The extension is π π called πππ·πΏβ πΏππ‘πππππ . In this section, the syntax and semantics of πππ·πΏβ πΏππ‘πππππ probaπ bilistic knowledge bases using π·πΏβ πΏππ‘πππππ features are given.
3.1
Syntax of π·ππ«π³β π³ππππ΅ ππππ Probabilistic Knowledge Bases
π The Axioms in π·πΏβ πΏππ‘πππππ knowledge bases can be annotated by belief degree interval. The following types of probabilistic axioms are supported: 1. Probabilistic terminological axioms (ππ΅ππ₯ axioms): probabilistic concept inclusions (PCI for short) about relationship between concepts. Each one has the form (πΆ β π·)[πΌ,π½ ] which signifies that we have a belief degree in [πΌ, π½] that the concept π· is subsumed by πΆ or πΆ is sub class of π·. 2. Probabilistic π΄π΅ππ₯ axioms: probabilistic assertions about concepts and roles instances: πΆ(π)[πΌ,π½ ] means that we have a belief degree in [πΌ, π½] that the individual π is an instance of the concept πΆ. π
(π, π)[πΌ,π½ ] means that the individual π is related with the individual π by the role π
with a belief degree in [πΌ, π½]. 3. Using conjunction (β§) or disjunction (β¨) are not allowed in DLs, thus the satisfaction of axioms that contain these notations is not defined for features. Therefore we define it as π follow: for a given axiom π₯ = π₯1 β§ π₯2 β¦ β§ π₯π where each π₯π is a π·πΏβ πΏππ‘πππππ axiom, we say that a feature πΉ = β©Ξ, π»βͺ satisfied π₯ if πΉ β¨ π₯π for every π₯π in π₯ and we write
πΉ β¨ π₯. We say that πΉ satisfies π₯ = π₯1 β¨ π₯2 β¦ β¨ π₯π if πΉ satisfies at least one π₯π . Belief π about these axioms is allowed in πππ·πΏβ πΏππ‘πππππ . Thus, the probabilistic axiom π₯ = (π₯1 β§ π₯2 β¦ β§ π₯π )[πΌ,π½ ] means that we have a belief in [πΌ, π½] that all π₯π can be satisfied in the same situation. The probabilistic axiom (π₯1 β¨ π₯2 β¦ β¨ π₯π )[πΌ,π½ ] means that we have a belief in [πΌ, π½] that at least one π₯π can be satisfied. This type of axioms is called probabilistic conjunction and disjunction axioms (ππΆπ·π΄ for short). Using β§ is not allowed with β¨ in the same ππΆπ·π΄ axiom The values πΌ and π½ are in [0,1] where πΌ β€ π½, πΌ is the lower bound and π½ is the upper π bound. In πππ·πΏβ πΏππ‘πππππ , the probabilistic terminological box ππ is a finite set of PCIs. The probabilistic assertions box ππ΄ is a finite set of probabilistic assertions. ππΆπ·π΄ is a set of conjunction and disjunction axioms. A probabilistic knowledge base πΎπ΅ in π πππ·πΏβ πΏππ‘πππππ is defined as πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ, the axioms of πβπ΄ are called certain axioms whereas the axioms in ππβππ΄βππΆπ·π΄ are uncertain axioms. Probabilistic Axiom with [1,1] are considered as certain axiom. Thus they are removed and added to πβπ΄. A single belief value is allowed, thus in this case πΌ = π½ (see axiom 10 in fig.2). π To model a πππ·πΏβ πΏππ‘πππππ probabilistic knowledge, we must have an ontology (π and π π΄) in π·πΏβ πΏππ‘πππππ and then extend it by adding probabilistic axioms. Example 3. We present an example of probabilistic knowledge base π πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ in πππ·πΏβ πΏππ‘πππππ which is an extension of πΎ = (π, π΄) in fig.1 with additional probabilistic axioms. Axioms numbered from 1 to 7 are the same in πΎ. The PTBox ππ tells that the concept professor is a subclass of the concept PhD student with a belief degree in [0.44,0.65] (8). The ππ΄ defining RIDA and KAMEL as respectively a PhD student (9) and a professor (10) with belief degree respectively in [0.55,0.60] and [0.80,0.80]. The ππΆπ·π΄ axiom 11 says that we have a belief degree in [0.45,0.67] that LOTFI is a professor and teacher of DISCRIPTION LOGICS. Axiom 12 tell that ALA is a PhD student or professor with a belief degree in [0.30,0.60]. Probabilistic Terminological Box ππ: Probabilistic Assertion Box ππ΄: 9. πππππ‘π’ππππ‘(RIDA)[0.55,0.60] 8. (ππππππ π ππ β πππ·ππ‘π’ππππ‘)[0.44,0.65] 10. ππππππ π ππ(KAMEL)[0.80,0.80] Probabilistic Conjunction and Disjunction Axioms ππΆπ·π΄: 11. (ππππππ π ππ(LOTFI) β§ π‘ππππ(LOTFI, DESCRIPTION LOGICS))[0.45,0.67] 12. (πππ·ππ‘π’ππππ‘(ALA) β¨ ππππππ π ππ(ALA))[0.30,0.60] π Fig. 2. An example of πππ·πΏβ πΏππ‘πππππ probabilistic knowledge base
3.2
Semantics of π·ππ«π³β π³ππππ΅ ππππ Probabilistic Knowledge Bases
Given a probabilistic axiom, the certain axiom is specified by removing the belief interval. The certain one of (πΆ β π·)[πΌ,π½ ] denoted by πππ( πΆ β π· πΌ ,π½ ) is πΆ β π·. Given a set of PCIs ππ, the certain set of ππ denoted by πππ(ππ) is a set of concept inclusions. Proba-
bilistic assertion and the set of probabilistic assertions ππ΄ are treated in the same manner. Because every axiom in ππΆπ·π΄ has at least two axioms (β¨ or β§ must connects more than an axiom), a set of certain axioms can be extracted from every ππΆπ·π΄ axiom, this set can includes concept inclusions and assertions. Therefore two functions: πππ_π and πππ_π are defined to extract these sets where πππ_π((π₯1 β§ π₯2 β¦ β§ π₯π )πΌ ) is a set contains all πππ(π₯π ) (all π₯π must be satisfied), and πππ_π((π₯1 β¨ π₯2 β¦ β¨ π₯π )πΌ ) is a set contains at least a πππ(π₯π ) (at least one π₯π must be satisfied). Thus πππ(ππΆπ·π΄) is a set contains all sets of certain axioms of all ππΆπ·π΄ axioms. A set of deterministic knowledge bases can be extracted from πΎπ΅, everyone must contains π βͺ π΄ and it can contains selected axioms from πππ(ππ) βͺ πππ(ππ΄) and selected sets of axioms from πππ(ππΆπ·π΄). Axioms from πππ(ππ) βͺ πππ(ππ΄) are added directly. For every selected set from πππ(ππΆπ·π΄), all its certain axioms are added. Each determinisπ tic knowledge base respects the π·πΏβ πΏππ‘πππππ language and every satisfiable knowledge base is called πππ π ππππ ππππ€πππππ πππ π. Definition 3 (ππππππππ πππππππ
ππ πππππ ). Given a probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ. The possible knowledge bases of πΎπ΅, denoted by πππ πΎπ΅ is defined as follows: πππ πΎπ΅ = {πΎ = β©π βͺ π β² , π΄ βͺ π΄β² βͺ|πΎ is satisfiable where π β² (resp. π΄β² ) contains selected axioms from πππ ππ (resp. πππ ππ΄ ) and concept inclusions (resp. assertions) in the selected certain sets from πππ(ππΆπ·π΄)}. In other words, the possible knowledge base πΎ is a possible consistent situation of πΎπ΅. In this situation, the actual world is a model of πΎ. The satisfiability condition is important for preventing inconsistencies and contradictions that may occur between the axioms in πππ(ππ), πππ(ππ΄) and πππ(ππΆπ·π΄). If there are probabilistic axioms that have certain axioms which are inconsistent with π βͺ π΄ then they canβt participated in creating πππ πΎπ΅, thus they are omitted and do not considered in reasoning. Using definition 3, another notion is defined which is π‘ππ πππ π ππππ ππππ‘π’ππ: Definition 4 ππππππππ ππππππππ . Given a probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ, let πππ πΎπ΅ be the set of all possible knowledge bases of πΎπ΅. The possible features related to πΎπ΅ , denoted by πππ β± is defined as follows: πππ β± = {πΉ|πΉ ππ π πππππ ππππ‘π’ππ ππ πΎ π π’ππ π‘πππ‘ πΎ β πππ πΎπ΅}. From the definition, πππ β± contains all model features of all possible knowledge bases in πππ πΎπ΅. Like models, the possible features are used to describe the current πΎπ΅ situation. The signature of probabilistic knowledge base πΎπ΅ is defined as a finite set π = π ππ(π) βͺ π ππ(πππ ππ ) βͺ π ππ(π΄) βͺ π ππ(πππ ππ΄ ) βͺ π ππ(ππππ‘ ππ·πΆπ΄ ). π The semantics of πππ·πΏβ πΏππ‘πππππ probabilistic knowledge bases is given by probabilistic interpretations. A probabilistic interpretation ππ is a probability function on all possible features in πππ β± (ππ: πππ β± β 0,1 ) such the sum of all ππ πΉ is equal to 1. We have a probability distribution over πππ β± and ππ distributed its probability values only on possible features i.e. considering only possible situations of πΎπ΅. The probability of any axiom π₯ is the sum of the probabilities associated with all features that satisfy π₯: ππ π₯ =
πΉβπππ β± πππ πΉβ¨π₯ ππ(πΉ) . A probabilistic interpretation ππ satisfies a PCI (πΆ β π·)[πΌ,π½ ] denoted by ππ β¨ (πΆ β π·)[πΌ,π½ ] if and only if ππ πΆ β π· β [πΌ, π½]. ππ satisfies a set of PCIs ππ denoted by ππ β¨ ππ if ππ satisfies all elements in ππ. The same is with a probabilistic assertion and a set of probabilistic assertions ππ΄. ππ satisfies ππ·πΆπ΄ denoted by ππ β¨ ππ·πΆπ΄ if ππ satisfies all elements in ππΆπ·π΄. ππ satisfies a certain concept inclusion πΆ β π·, denoted by ππ β¨ πΆ β π· if and only if for every feature with ππ πΉ > 0, we have πΉ β¨ πΆ β π·. ππ satisfies a set of concept inclusions π denoted by ππ β¨ π if ππ satisfies all elements in π. The same is with certain assertion and certain set of assertions π΄. Therefore, a probabilistic interpretation ππ is a model of probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ if and only if ππ satisfies π, π΄ , ππ, ππ΄ and ππ·πΆπ΄. πΎπ΅ is satisfiable (or consistent) if there is at least a model of πΎπ΅. Given a probabilistic axiom π₯[πΌ,π½ ] , πΎπ΅ entails π₯[πΌ,π½ ] , denoted by πΎπ΅ β¨ π₯[πΌ,π½ ] if for every ππ such that ππ β¨ πΎπ΅ we have ππ β¨ π₯[πΌ,π½ ] . Before checking the satisfiability of πΎπ΅, β©π, π΄βͺ must be consistent.
Example 4. The πΎπ΅ in fig.2 has the signature π = ππΆ βͺ ππ
βͺ ππΌ βͺ ππ where: ππΆ , ππ
, and ππ are the same of πΎ in fog.1 but ππΌ = {FOFO, ALA, LOTFI , RIDA, KAMEL}. The set πππ πΎπ΅ of πΎπ΅contains for example πΎ1 = β©π βͺ {πππ 8 }, π΄ βͺ {πππ 9 , πππ(10)}βͺ. We observe that πΎ1 π is satisfiable and it respects the expressiveness of π·πΏβ πΏππ‘πππππ .
4
Reasoning and Inferences Tasks
π The main reasoning and inference tasks for πππ·πΏβ πΏππ‘πππππ are the following: ο· Probabilistic Knowledge Base Satisfiability (ππΎπ΅ππ΄π): Given a probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ, decide whether πΎπ΅ is satisfiable. ο· Tightest Belief interval for Logical Entailment (ππ΅πΌπΏπππΈπ): Given a probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ and an axiom π₯, compute the tightest belief interval [πΌ, π½] such thaπ‘ πΎπ΅ β¨ π₯[πΌ,π½ ] . ο· Logical Entailment ( πΏπππΈπ ): Given a probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ and probabilistic axiom π₯[πΌ,π½ ] associated with belief interval [πΌ, π½], decide whether πΎπ΅ β¨ π₯[πΌ,π½ ] or not. The first task is achieved by using theorem 1. The second and the thirds uses theorem 2. Similarly to [9], the reasoning tasks use a system of linear constraints.
Theorem 1. Let πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ a probabilistic KB. This latter is satisfiable if the next linear constraints system over variables ππΉ (πΉ β πππ β±) is solvable: πΉβπππ β±,πΉβ¨π₯ πΉβπππ β±,πΉβ¨π₯ πΉβπππ β±
ππΉ β₯ πΌ (πππ πππ ππππ π₯ πΌ,π½ ππ ππ πππ ππ΄ πππ ππΆπ·π΄) ππΉ β€ π½ (πππ πππ ππππ π₯ πΌ,π½ ππ ππ πππ ππ΄ πππ ππΆπ·π΄)
ππΉ = 1
ππΉ β₯ 0 πππ πππ πΉ β πππ β± Proof. The solver tries to find assignments to every ππΉ such that all constraints are satisfied. Every ππΉ is considered as probability of πΉ β πππ β±. Thus the solver tries to find a probability function on πππ β± that respects the constraints. The two first constraints are to respect the satisfiability of every probabilistic axiom in πΎπ΅ i.e., the sum of probabilities of the features that satisfy π₯ is in [πΌ, π½]. The third one is to respect that the sum of all probabilities associated with all features is 1. The condition of positive probabilities is specified in the last constraint. By respecting the two last constraints, the condition that the probability β [0,1] is also respected. If the solver get a solution i.e., value for every ππΉ then the solution respects the satisfiability conditions and πΎπ΅ has a model ππ that assigns for every πΉ β πππ β± a value ππΉ . Thus πΎπ΅ is satisfiable. We can proof by the same manner that if πΎπ΅ is satifiable then the previous system of linear constraints is solvable. Theorem 2. Given a probabilistic knowledge base πΎπ΅ = β©π, ππ, π΄, ππ΄, ππΆπ·π΄βͺ. Suppose πΎπ΅ is satisfiable. Let π₯ be an axiom. The values πΌ and π½ such that πΎπ΅ β¨ π₯[πΌ,π½ ] are taken over all possible solutions of the system in Theorem 1 as follow: πΌ = πππ πΉβπππ β±,πΉβ¨π₯
ππΉ π π’πππππ‘ π‘π π π¦π π‘ππ ππ πππππππ 1
π½ = πππ₯ πΉβπππ β±,πΉβ¨π₯
ππΉ π π’πππππ‘ π‘π π π¦π π‘ππ ππ πππππππ 1
Proof. Suppose that πΎπ΅ is satisfiable, the system in Theorem 1 has at least one solution i.e., probabilistic interpretation of πΎπ΅. Given an axiom π₯, for every solution i.e., probabilistic interpretation, the solver computes the sum of all values assigned with features that satisfy π₯. To minimization, it keeps πΌ and π½ for the maximization. Thus, in every interpretation ππ we have ππ π₯ β [πΌ, π½] so πΎπ΅ β¨ π₯[πΌ,π½ ] . The interval computed by theorem 2 is called the tightest belief interval i.e., πΌ (resp., π½) is the min (resp., max) of ππ(π₯) subject to all models ππ of πΎπ΅. Theorem 2 can be used to decide if a given probabilistic axiom is entailed by a satisfiable probabilistic πΎπ΅. Thus πΎπ΅ β¨ π₯[πΌ,π½ ] if the tightest belief interval that π₯ is entailed by πΎπ΅ is in [πΌ, π½]. Example 5. From the example 3, if πΎπ΅ is satisfiable using Theorem 1, then we can use theorem 2 to compute ππ΅πΌπΏπππΈπ of some given axioms such as: computing the tightest belief interval [πΌ, π½] such that πΎπ΅ β¨ πππ·ππ‘π’ππππ‘(KAMEL)[πΌ,π½ ] , finding the ππ΅πΌπΏπππΈπof πππ·ππ‘π’ππππ‘ β ππππππ π ππ. We can also decide whether πππ·ππ‘π’ππππ‘ LOTFI [0.4,0.6] is a logical consequence of πΎπ΅.
5
Implementation and Experimentation
π A prototype of πππ·πΏβ πΏππ‘πππππ is implemented in Java with Eclipse, using Pellet [18] for reasoning, owlapi [19] for knowledge base creation, and LpSolve 5.5 [20] for solving the
π linear programs. π·πΏβ πΏππ‘πππππ supportes only concept inclusion, concept and role assertions. In owlapi the previous axioms are respectively created by OWL. π π’ππΆπππ π ππ, OWL. ππππ π π΄π π πππ‘πππ , OWL. πππππππ‘π¦π΄π π πππ‘πππ . For example πππ·ππ‘π’ππππ‘ β ππ‘π’ππππ‘ is represented by OWL.π π’ππΆπππ π ππ(πππ·ππ‘π’ππππ‘, ππ‘π’ππππ‘). Therefore π, π΄, ππ, ππ΄ and ππΆπ·π΄ are created using owlapi where every probabilistic axiom is associated with its belief interval. During the building of the possible knowledge bases, Pellet reasoner is used to test the satisfiability of the latters. For a given probabilistic πΎπ΅, we can generate 2 ππ + ππ΄ + ππΆπ·π΄ knowledge base, thus the one in fig.2 has 25 = 32 knowledge bases where all of the latπ ters are consistent so πππ πΎπ΅ = 32 (tested by the πππ·πΏβ πΏππ‘πππππ prototype). We conππ + ππ΄ + ππΆπ·π΄ clude that in worst cases, we have πππ πΎπ΅ β€ 2 . For every πΎ in πππ πΎπ΅, using its signature π ππ(πΎ), we compute all types which satisfy its TBox, everyone is a set of basic concepts represented using owlapi by a set of OWLClassExpression. From the ABox of πΎ, all basic concept and basic role assertions are extracted using Pellet. Thus these assertions are used create the π»πππππππ set π» of πΎ according to the definition 1 (π» is created in owlapi as owl ontology). All possible combinations of the types associated with πΎ are generated. For every individual in π ππ(πΎ), its type in π» is extracted and added to every combination. The latter is tested if it forms with π» a feature of πΎ using. Every element in πππ πΎπ΅ is treated with the same manner and all features are added to πππ πΉ.
Semantics based on features
Terminological probabilistic knowledge Probabilistic Assertions Conjunction axioms Disjunction axioms Reasoning tasks
Implementation and evaluation
π πππ·πΏβ πΏππ‘πππππ Supported. Probabilistic interpretation on only possible features. Supports the belief in concept inclusions. Supported supported Supported Strong conclusions by computing the tightest belief interval. Deciding entailment is supported. The reasoning tasks use linear programming for efficient computation. A prototype is implemented and evaluated
Probabilistic DL-Lite in [13] Supported. Probabilistic interpretation on all features of a given signature. Supports the conditional constraints. Supported Not supported Not supported Weak conclusions. It uses an inference rules that get only lower bounds. In most cases, the upper bound is 1. The inferences cover only special cases. The tightest interval is not supported No implementation and evaluation
π Table 1. Comparison between πππ·πΏβ πΏππ‘πππππ and the work in [13]
One feature can model more than one knowledge base, thus the number of all possible features can be reduced. Using the prototype, we found that the probabilistic KB in fig.2 has 1548 possible features and this proved that the number of features is finite. The linear
program Lp in theorem 1 is created using LpSolve, the variables number is πππ πΉ because every πΉ β πππ πΉ is associated with one variable ππΉ . Using ππ , ππ΄ , ππΆπ·π΄ and πππ πΉ, two constraints are created for every axiom π₯, one for its interval lower bound and one for the upper bound. The coefficient of every variable ππΉ in these constraints can be 1 (πΉ satisfies π₯) or 0 (πΉ does not satisfy π₯). The Lp for the KB in fig.2 has 1548 variables and 12 constraints (10 for the probabilistic axioms). Example 6. Using the prototype and the probabilistic KB in fig.2, we have found that: πΎπ΅ β¨ πππ·ππ‘π’ππππ‘(KAMEL)[0.24,0.65] , (πππ·ππ‘π’ππππ‘ β ππππππ π ππ)[0.0,0.45] and πππ·ππ‘π’ππππ‘ LOTFI [0.40,0.60] is not entailed by KB because the tightest belief interval of πππ·ππ‘π’ππππ‘ LOTFI is [0.45,0.67]. For comparison, our work and the one in [13] are used (see table 1) because the only π probabilistic extension of π·πΏβ πΏππ‘πππππ which based on features is in [13]. The goal of comparison is to understand the points of difference between the two works. We have not presented this section in detail because of the limitation in paper length.
6
Related Works
π Closest to our work, we have [13] where the π·πΏβ πΏππ‘πππππ is extended to use conditional constraints on concepts i.e., statistical information about concepts, it supports probabilistic terminological knowledge and probabilistic assertions about concepts and roles. The probabilistic interpretation is defined on the set of all features of a given signature, unlike our work which uses only the possible features instead of all features. In [13], no reasoning tasks and implementation are presented and the belief in concept inclusion is not supported. Its inferences are weak and consider only special cases. PrDLs [14] is a probabilistic description logic which supports the belief in DL axioms. From the probabilistic knowledge, similarly to our approach, the last reference extracts a set of certain knowledge bases but using a discrete probability distribution on all possible worlds. The author in [9] who proposed P-SHOIN(D), P-SHIQ(D) and P-DL-Lite that are probabilistic extensions of the DL-SHOIN(D), SHIF(D) and DL-Lite respectively, he uses the lexicographic entailment and his work is based on Nilssonβs probabilistic logic [11]. He defines a probabilistic interpretation on possible objects (everyone contains concepts that are free of probabilistic individuals), one for terminological probabilistic knowledge and one for every probabilistic individual. PRONTO reasoner [6] is based on the works in [9]. Unlike our work, [9] does not allow probabilistic role assertions and it uses a separation between probabilistic interpretations for the individuals and this makes difficulties to draw conclusions about relations between individuals. Another work is in [22] where probabilistic DLs based on the DL-ALC are presented. The work does not allow probabilistic terminological knowledge and its semantics is subjective by considering the probabilities as believe degrees. The authors use probability distributions on possible worlds where everyone is associated with a FOL interpretation. The quasi model is defined to check the con-
sistency of the probabilistic KB. This model shares some similarities with the feature because the former is a pair of two sets of types one contains ABox types and the other contains types for the individual. Contrary to this model the feature includes TBox and ABox types and this is important to capture the KB semantics. The authors in [22] use linear constraint systems that help to construct the worlds. In contrast, our work uses only one linear constraint system after the features construction. The work in [16] allows for epistemic and statistical probabilistic annotations in DL axioms by transforming the annotated axioms to predicate logics. The authors consider the epistemic probability as belief degree. The BUNDLE [17] is a reasoner for the work in [16]. Other set of probabilistic DLs use graphical models such as Bayesian network BN as underlying probabilistic formalisms, some of these works are [7] and [5]. P-CLASSIC [7] is a probabilistic DL based on BN that supports terminological probabilistic knowledge about concepts and roles but does not allow assertional knowledge about concepts and roles. The work in [5] is an extension of DL-Lite that uses BN towards tractable probabilistic DL. For further reading about probabilistic uncertainty in semantic web, readers are referred to [10], [12] and [21].
7
Conclusion and Future Works
π π In this paper, πππ·πΏβ πΏππ‘πππππ a novel probabilistic extension for π·πΏβ πΏππ‘πππππ knowledge π bases is presented. The proposed work allows belief interval in a single π·πΏβ πΏππ‘πππππ π axiom or a set of π·πΏβ πΏππ‘πππππ axioms that are connected with β§ or β¨. Its semantics is π based on π·πΏβ πΏππ‘πππππ features. Both terminological and assertion probabilistic knowledge are supported. Using this work, meaningful conclusions can be drawn from the probabilistic knowledge. Analysing the computational complexity of our work and implementing efficient reasoner are the main future work directions. Another future work consists in using specific application domains such as: medical.
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