Extending Nonmonotonic Description Logic withTemporal Aspects

Ouarda BettazRIIMA Laboratory, Computer Science Department FEI

Ecole Normale Supérieure de Kouba, Algeriaouarda.bettaz.nehab@gmail.com

Narhimene BoustiaRIIMA Laboratory, Computer Science Department FEI

Saad Dahlab University of Blida, Algerianboustia@gmail.com

Aicha MokhtariRIIMA Laboratory, Computer Science Department FEI

USTHB, Algeriaaissani_mokhtari@yahoo.fr

Abstract—This paper is about extending nonmonotonic de-scription logic with temporal aspects; this attempt permitsactually to represent both default and temporal features inconcepts definition. The introduction of defaults in the definitionof concepts in previous researches has allowed to go beyond thestrict limitations on their description and permitted consequentlyto fully define them; by providing both necessary and sufficientconditions for their representation. This allowed improving theclassification process. Contrary to the use of strict knowledge thatprovides only necessary conditions leaving the concepts partiallydefined. The nonmonotonic language that allows using defaultsin the definition of concepts is AL augmented with default andexception connectors that allow respectively representing defaultand exception properties in concepts definition. However wefrequently need to add the temporal aspect to the nonmonotonicfeature as it is the case in causal reasoning, planning process, andaction theory. In our case, we will use it in the field of accesscontrol. Our aim in this paper is to extend this logic furtherwith temporal connectives to grant the possibility to representtemporal properties of concepts and that by referring to temporaldescription logic.

Keywords-component; Nonmonotonic description logic;Default knowledge; Temporal description logic; Temporalnonmonotonic description logic.

I. INTRODUCTION

Description logics (DLs) are good formalisms for knowl-edge base representation [8]. However classical forms ofdescription logics do not permit to represent neither default norexception facts about concepts, for example: they do not allowrepresenting the fact that all birds fly by default, but a penguinis a bird that exceptionally doesn’t fly. The impossibility ofrepresenting this kind of information leaves the knowledgebase partially defined which subsequently affects the inferenceprocess. The solution to represent such kind of knowledge isto rely on nonmonotonic reasoning that is based on the use ofdefault description logic.Many approaches were proposed in the literature: Quantz andRoyer [17], Padgham and Nebel [15], Padgham and Zhang[16], Baader and Hollunder [7]. The problem with theseapproaches is that they use a limited form of default reasoning;

where concepts are defined only by using strict propertieswhile default knowledge is represented using incidental rules,considering the fact that most of concepts can’t be justdefined by the use of strict properties, that will leave theknowledge base inevitably partially defined, consequently theclassification process won’t be complete.The approach that overcomes this problem was proposed byCoupey and Fouqueré [13]. In fact they developed a new non-monotonic description logic named ALδε that permitted theintroduction of the notion of default and exception in conceptsdefinition, it was elaborated by adding to the description logicAL [8] two connectives: (δ) to represent default facts and (ε)to represent exception facts.This language was improved by the addition of connectorsfrom C-classic which permitted to augment its expressivityand thus make it usable from a practical point of view. Thisnew language was called JClassicδε [10], [11]. Using thisdescription logic we can define the concept Tree as havingby default branches and always having a trunk and roots:Tree v δWith_branches uWith_trunk uWith_rootsNow if we want to define the concept Scion as being a treethat is by default one year old and exceptionally has brancheswe will write it this way:Scion v δOne_year_old u Tree uWith_branchesε.In this example the concept Scion that is subsumed by theconcept Tree will only inherit the properties With_trunk andWith_roots, but not the property With_branches since thisproperty is an exception for the concept Scion.What we tried to do in this work is to develop a temporalnonmonotonic description logic and that by adding to the non-monotonic description logic JClassicδε temporal connectives.The purpose of doing so is to represent temporal conceptswhile having default knowledge.Differing from the existing temporal description logics wheretemporal components are added to classical description logics.This will permit to better manage the time aspect in a varietyof domains such as reasoning about actions and plans andenhance natural languages comprehension, ...etc, it will also

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allow us to improve access control and more specifically forthe model we are working with, the so called DL−OrBACδεmodel inspired by OrBAC to which we attributed a part fromthe temporal nonmonotonic language that we developed in thispaper [9].For this purpose we referred to the use of temporal descriptionlogic (TDL) [2], [3], [4], [5], [8], [12]. In the research fieldon temporal description logic two approaches for modelingthe notion of time were considered: the modal temporal logicand the reified temporal logic [14]. In our work we willemploy modal temporal logic. In which the connectors �and ♦ represents respectively the notion of (always in thefuture) and (sometimes in the future). The flow of time canbe taken from two different angles, we can actually considertime as a set of points (instances) or as a set of intervals.In [2], [3], [8], [12], the different approaches on temporallogic based on points and intervals have been widely spread.For the purpose of our work we are interested in the use ofinterval based approach to define the specific interval at whicha concept is valid. Concerning this approach many studieswere undertaken. Artale and Franconi [2], [3], [4] put intoevidence a TDL inspired by Schmiedel’s [18] approach, thatthey restricted by discarding the � operator for decidabilitymatters. Example [6]:♦ (X Y) (Y starts X).(Student@ X Bachelor_student @Y).In this example, we have two intervals X and Y, where Xand Y start at the same time but Y is ended before X. So thedescribed persons are students during the interval X and theyare specifically Bachelor students for the initial sub-interval Yof X. The temporal part that we will be using for extendingthe nonmonotonic description logic is the one used in the TDLdefined by Artale and Franconi [2], [3], [4].The rest of the paper is organized as follows. In the secondsection we introduce temporal nonmonotonic description logicby providing its syntax, in the third section we expose itsalgebraic framework with both descriptive and structural pointsof view we also define the subsumption algorithms. We endthis paper by a conclusion and a future work presented insection four.

II. TEMPORAL NONMONOTONIC DESCRIPTIONLOGIC

In this section we introduce our new temporal nonmono-tonic description logic that we elaborated by the combinationof the nonmonotonic description logic JClassicδε [10], [11]and the temporal part [2], [3], [4]. The result of this process isthe temporal nonmonotonic description logic that we namedT − JClassicδε.

A. Syntax of T − JClassicδεT − JClassicδε is defined using a set of atomic concepts

P, a set of atomic roles R, the constants > (Top) and ⊥(Bottom), a set of individuals I called "classic individuals"and the following syntactical rule given in Table 1:

C and D are concepts, δ (Default) and ε (Exception)are unary connectives, u is a binary conjunction, ∀enables

universal quantification on role values, u is real number,n is an integer, Ii are "classic individuals", X and Y aretemporal variables (intervals), # is a reference interval,T and S denote Allen’s interval relations [1], Tc denotestemporal constraints, @ is an operator that specifies theinterval at which a concept applies, C[Y]@X renames inthe defined concept C the variable Y with X; it’s a way toperform a co-reference between two temporal variables, ♦(i)is an existential quantifier on temporal intervals; it actuallyintroduces temporal intervals and relates them throughtemporal constraints based on Allen’s relations.

Using this description logic we can define the concept Birdas being an animal that flies by default and that is mortal:Bird ≡ Animal u δF ly uMortalWhile Mortal is defined using temporal connectives as aliving being at a certain period (interval) and not leavingbeing anymore after that period:Mortal ≡ LivingBeing u (♦(i) (after i #) )(¬ LivingBeing@ i))

If we consider the concept of Penguin as a bird thatexceptionally does’t fly:Penguin ≡ Bird u FlyεIn this case the concept Penguin inherits from the conceptBird the property Animal and the temporally defined propertyMortal but not the property Fly since it’s an excepted propertyin the definition of Penguin. The specificity of our logic isthe use of an algebraic-based semantics unlike the classicalpractice where the semantics is based rather on a first orderlogic interpretation. Following this algebraic semantics, theconcepts are written in a particular form called normal form.

III. ALGEBRAIC FRAMEWORK

We endow T − JClassicδε with an algebraic framework,it allows actually characterizing and formalizing the differenttypes of subsumption namely descriptive and structural.

A. Descriptive point of view

From a descriptive point of view the calculation ofsubsumption consists of the comparison of terms throughan equational system called EQ which defines formally themain properties of connectors and determines the equivalenceclasses of terms.

EQ: an equational system for T − JClassic+δε

∀A,B,C ∈ T − JClassic+δε:01: (A uB) u C = A u (B u C)02: A uB = B uA03: A uA = A04: > uA = A05: ⊥ uA = ⊥06: (∀R : A) u (∀R : B) = ∀R : (A uB)07: ∀R : > = >08: (δA)ε = Aε

C,D → > Universal concept| ⊥ Bottom concept| P Atomic concept| C uD concept conjunction| ¬P negation of primitive concept| ∀r : C C is a value restriction on all roles R| R AT-LEAST n cardinality for R (minimum)| R AT-MOST n cardinality for R (maximum)| δC default concept| Cε exception to the concept| ClcsD concept disjunction| C@X Qualifier| C[Y ]@X Substitutive qualifier| ♦(X)Tc.C Existential quantifier| Tc→ (X(T )Y )|(X(T )#)|(#(T )Y ) Temporal constraint| Tc→ Tc|TcTc Temporal constraint| T, S → T, S| Disjunctiont| T, S → starts|finishes|met− by|... Allen’s relations| X,Y → x|y|z|... Temporal variables| X → X|XX

TABLE ISYNTAX OF T − JClassicδε

09: δ(A uB) = (δA) u (δB)10: A u δA = A11: Aε u δA = Aε

12: δδA = δA13: ♦ x (x a #).C@x = ♦ xy (y mi #)(x mi y).C@x14: ♦ x (x d #).C@x= ♦ xy(y s #)(x f y).C@x15: ♦ x(x o #).C@x =♦ xy(y s #)(x fi y).C@x16: C@X uD@X = (C uD)@X17: (C@X1)@X2 = (C@X1)18: (C@X1 uD)@X2 = C@X1 uD@X2

19: C u ♦(X)Tc.D = ♦(X)Tc.(C uD)

The first twelve axioms correspond to JClassicδεconnectives [10], [11] and the axioms from thirteen tonineteen correspond to our temporal connectives.The axioms express that the conjunction of concepts:01: is associative.02: commutative.03: idempotent.04: the most general concept in the hierarchy top (>) is theneutral element of the conjunction.05: the most specific concept bottom (⊥) is the absorbentelement.06: the connector ∀R : A is distributive over the conjunction.07: represents a false restriction on roles.08: an exception to the default concept is the same as anexception to the underlying concept.09: a default on a conjunction of concepts is similar to theconjunction of two defaults.10 and 11: express the fact that both A and Aε are subsumedby δA.12: Allows redundant default chains to be removed.13, 14 and 15 express the fact that among the thirteenrelations represented by Allen only three are really needed:starts (s), finishes (f) and met-by (mi) since all the others:after (a), during (d) , overlaps (o) and there inverses can be

represented using these three ones.16: the @ operator is distributive over the conjunction.17: the concept C is valid at the first interval it is related to.18: the @ operator is distributive over the conjunction butstill C is valid at the first interval it is related to.19: if there is a conjunction between the concepts C andD on which the existential temporal quantifier applies thenthe existential temporal quantifier ♦ will apply on theirconjunction.

Descriptive Subsumption:We denote vd for descriptive subsumption. vd is a partial

order relation on terms. Equality (modulo the axioms of EQ)between two terms is denoted =EQ. =EQ is a congruencerelation which partitions the set of terms, i.e., =EQ allowsto form equivalence classes between terms. We define thedescriptive subsumption using the congruence relation andconjunction of concepts as follow:

Definition 1:(Descriptive Subsumption)Let C and D two terms of T − JClassicδε, C vd D, i.e.,

D subsume descriptively C, iff C uD =EQ C.

From an algorithmic point of view, terms are not eas-ily manipulated through subsumption. We adopt a structuralpoint of view closer to the algorithmic aspect of computingsubsumption. This allows us to first formalize calculationof subsumption in the implementation of T − JClassicδεand secondly to endow T − JClassicδε with an intensionalsemantics.

B. Structural point of view

We present in this section the structural point of viewfor the subsumption in T − JClassicδε. This provides avery closer vision to the algorithmic aspect and a formalframework to validate the algorithmic approach. For this

purpose we define CLδε−t an intentional semantic forT − JClassicδε. Elements from CLδε−t are the canonicalintentional representation of terms of T − JClassicδε thatallows representing the properties of concepts by using anormal form. These elements are formed of a pair of 7-uples,both having the same structure, the first is used to representthe strict properties and the second for the default properties,the six first fields of the 7-uples are the same as for thenormal form defined for JClassicδε [10], [11], we introducedthe 7th field to represent temporal aspect. The structure ofthe elements of CLδε−t is as follows:

Definition 2:An element of CLδε−t corresponding to a term T of

T − JClassicδε is a pair 〈tθ, tδ〉, where tθ is the strict partof T and tδ the default part of T, tθ and tδ are 7-uples (dom,min, max, π, r, ε, t) defined as follows:

- dom: is a set of individuals if the definition of T containsa property ONE-OF otherwise the special symbol UNIV.- min(resp. max): its either a real if T contains a propertyMIN (resp MAX), or the special symbol MIN-R (resp.MAX-R) otherwise.- π: is the set of primitive concepts contained in T.- r: contains the following elements:〈R, fillers, least,most, c〉 where: R: is the role name,fillers: is a set of individuals if T contains a property RFILLS or ∅ otherwise, least (resp. most): is an integer if Tcontains the property R AT-LEAST (resp. R AT-MOST) or ∅(resp. NOLIMIT) otherwise, c: is the normal form of C if Tcontains the property ∀R.C.- ε: is a set of seven elements (dom, min, max, π, r, ε, t).- t: is the set of temporal concepts contained in T.

Notation: the complete structure is noted:〈(tθdom, tθmax, tθmin, tθπ, tθr, tθε, tθt),(tδdom, tδmax, tδmin, tδπ, tδr, tδε, tδt).

Example: 〈(Univ,Min−R,Max−R,B, ∅, ∅, D),(Univ,Min−R,Max−R,B,C, ∅, ∅, D) is a description of:A ≡ B u δC u ♦ (i) D @ i

Structural Subsumption:Two terms C and D of T − JClassicδε are structurally

equivalent iff their normal forms are equal. We denote vs forstructural subsumption. vs is a partial order relation.

The structural equality of two terms of T − JClassicδεis noted =CL. =CL is a congruence relation as =EQ indescriptive subsumption.

We define the structural subsumption using the congruencerelation and conjunction of concepts as follow:

Definition 3: (Structural Subsumption)Let C and D two terms of T − JClassicδε, C vs D; i.e.,

D subsume structurally C, iff C uD =CL C.To infer new knowledge in this system, the subsumption

relation is used. In the next section, we outline the subsump-tion algorithm t− subδε used for T − JClassicδε. The other

inference operation in concern is inheritance that we will notexposed here for space issues, it is however available in [9].

C. Subsumption algorithm t− subδεt − subδε is composed of two stages. The first is a

normalization of descriptions. The second is a syntacticcomparison between normal forms. Let C and D be twoterms of T − JClassicδε. To answer the question "Is Csubsumed by D?", we apply the next procedure. The normalform of C and C u D are calculated with the procedureof normalization. If the two normal forms are equal, thealgorithm returns "Yes" otherwise it returns "No".

- Procedure of normalization of description. Thisprocedure uses the semantics functions union-uple and ⊗which compute respectively the union of two 7-uples (⊕) andtwo normal forms. The normalization permits to calculatethe normal form of a concept C from its given descriptiondenoted by fn(C) in algorithm 1.

- Procedure of comparison of normal forms. Wefirst define the procedure Compar which checks equalitybetween two 7- uples t1 and t2 . t1 resp t2 have theform (ti.dom, ti.min, ti.max, ti.π, ti.r, ti.ε, ti.t) with i=1 (respi=2). Compar calls the procedure Compar-roles which allowschecking equality of sets which denote roles. This procedureis denoted Compar(fn(C1), fn(C2), rep) in algorithm1. Thesubsumption algorithm t− subδε is defined bellow1:

Algorithm 1 Algorithm t-SubδεRequire: C and D two description of concepts of T −JClassicδε

Ensure: Response “Yes” or “No” to question “Is C subsumedby D?”{Compute normal forms}fn(C) ← Normalization(C)fn(C u D) ← Normalization(C u D){Treatment of bottom}if fn(C)=b0 then

Response ← “Yes”else

if fn(C u D)=b0 thenResponse ← “No”

else{Comparison of the obtained normal forms}Compar(fn(C)θ, fn(Cu D)θ, rep1)if rep1=”Yes” then

Compar(fn(C)δ , fn(Cu D)δ , rep1)Response ← rep2

elseResponse ← “No”

end ifend if

end if

1The Normalization (c) and Compar(fn(c1), fn(c2, rep) procedures are notcited here for space issue, however there signification were defined previously.

As we mentioned in the introduction the major goal behinddeveloping this temporal nonmonotonic DL is to improveaccess control, in particular for DL − OrBACδε model forwhich the above subsumption algorithm will serve for theclassification process [9]. In this model we can express apermission at a given period of time, for example we canexpress the fact that in the Hospital H1, Physicians have theright to consult any medical record during Working hours:Permission(P1)@Working_hours v

PermissionAv.Activity(Consult) uPermissionR.Role(Physician) uPermissionV.V iew(Med_db) uPermissionOr.Organization(H1)

If we want to express the fact that Cardiologists have theright to access the system during Working hours or On daySunday, we will just write the permission concerning theperiod On day Sunday; if we take into consideration thatCardiologist is subsumed by Physician then the permissionfor Working hours will be inherited.

Permission(P2)@On_day_Sunday vPermissionAv.Activity(Consult) uPermissionR.Role(Cardiologist) uPermissionV.V iew(Med_db) uPermissionOr.Organization(H1).

IV. CONCLUSION AND FUTURE WORK

In this paper, we presented a new notion of descriptionlogic namely the temporal nonmonotonic description logicthat we formed by the combination of two other logics: thenonmonotonic description logic JClassicδε and the temporaldescription logic.The purpose behind developing this idea is to grant theopportunity to represent at the same time temporal and defaultknowledge. The implementation of the corresponding reasoneris in progress.This work is a preamble for our initial objective: extendan access control model; in fact a first work consisted ofintroducing normal and exceptional context for the purpose ofpermission management, our goal is to introduce the temporaland or the spatial context.

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