Tableau-based Revision for Expressive Description Logics with Individuals

Transcription

1 Tableau-based Revision for Expressive Description Logics with Individuals Thinh Dong, Chan Le Duc, Myriam Lamolle LIASD - IUT de Montreuil, Université Paris 8, France Abstract Our understanding of an application domain evolves over time If we consider an ontology as a set of semantic constraints which describes our understanding of an application domain, ontologies must be revised Indeed, adding a new semantic constraint to an ontology may make it inconsistent To restore consistency with minimal loss of semantics, we would change other semantic constraints of the ontology such that the resulting ontology is semantically as close as possible to the initial ontology To be able to say that an ontology is semantically close to another one, it is needed to define a distance over finite structures representing models, called completion graphs, which characterize the semantics of an ontology In this paper we present a tableau algorithm for building such completion graphs of an ontology expressed in the description logic SHIQ with individuals Based on the distance defined over completion graphs, we introduce a revision operation applied to a SHIQ ontology with a set of new semantic constraints This revision operation computes the completion graphs that a revised ontology should admit However, there does not always exist an ontology expressible in SHIQ from which a tableau algorithm generates exactly a given set of completion graphs This leads us to introduce the notion of upper approximation ontology from which a tableau algorithm can generate the smallest set of completion graphs including a given set of completion graphs This notion allows us to design an algorithm for constructing a revised ontology from an initial ontology with a set of new semantic constraints We also implement the proposed algorithms with optimizations and report some experimental results to show that a model-based approach to revision of expressive ontologies is practicable Keywords: Ontology, Revision, Tableau Algorithm, Description Logics 1 Introduction Formalisms based on Description Logics (DLs) such as OWL are widely used to represent ontologies encapsulated in semantics-based applications An interesting feature of ontologies expressed in DLs (called DL ontologies) is to support automated inference services which allow ontology designers to detect eventual errors and allow users to entail new knowledge from ontologies with help of a reasoner However, ontologies are not static but evolve over time When changing ontologies, we are confronted with the problem of dealing with inconsistencies since new knowledge may contradict what exists in the ontology If we consider an ontology as a set of statements that describe our understanding about an application domain, ontologies must be revised to take into account new knowledge about the domain The problem of revising a DL ontology is closely related to the problem of belief revision which has been widely discussed in the literature Among early works on belief revision, Alchourrón, Gärdenfors and Makinson (AGM) [1] introduced intuitive and plausible constraints (namely AGM addresses: dong@iutuniv-paris8fr (Thinh Dong), leduc@iutuniv-paris8fr (Chan Le Duc), lamolle@iutuniv-paris8fr (Myriam Lamolle) postulates) which should be satisfied by any rational belief revision operator Existing belief revision approaches can be classified into syntax-based and model-based (semantic) approaches [11] Syntax-based (or formula-based) approaches manipulate directly syntactical entities such as formulas occurring in a knowledge base (KB) To take into account a new formula in preserving consistency, these approaches try to identify other formulas which should be removed A main advantage of syntax-based approaches is to allow for distinguishing between the relevance of different formulas [22] For instance, one can affect a lower priority to formulas that can change and a higher priority to those that would be protected The main issues are that (i) the procedures resulting from these approaches heavily depend on the syntax of knowledge bases that is for two different KBs X and Y that have the same meaning, ie Cn(X) = Cn(Y ) where Cn(X) denotes all logical consequences of X, revision may lead to two different results; and (ii) formulation of the principle of minimal change would refer to possibly infinite sets such as Cn(X), that is not necessarily computable Despite these issues, there have been some syntax-based belief revision operations developed for revising a DL ontology composed of TBox and ABox The former consists of axioms expressing subsumption relation- Preprint submitted to Elsevier September 15, 2017

2 ships between concepts (or roles) while the latter consists of assertions expressing memberships of individuals in a concept or role Qi and colleagues [24] have reformulated the AGM postulates for DL ontologies and proposed two revision operators by weakening assertion axioms and GCIs However, none of their operators satisfies all the AGM postulates Ribeiro and Wassermann [26] investigated ontology contraction with Levy identity (ie removing an axiom) The absence of axiom negation in DLs has led the authors to study ontology semi-revision with two different constructions for revising a knowledge base K by an axiom α The first one ensures that the revised KB is always consistent but α is not necessarily entailed from the resulting KB, ie, success of the revision does not hold In the second construction, success always holds but consistency of the resulting KB is not guaranteed Therefore, the postulates about success and consistency are not simultaneously guaranteed 2 Contrary to syntax-based approaches, semantic approaches investigate and manipulate models of ontologies rather than their syntactical entities The main issues in adapting semantic approaches to DL ontologies are how to define a distance between models and how to compute a revised ontology from the models selected according to the defined distance In addition, other problems may arise from dealing with models of DL ontologies First, DL ontologies may have infinitely many models which make impossible to construct directly a revised ontology from models Second, models of a DL ontology have usually (possibly infinite) complex structures, which may require a complex definition of distance between two models Third, there may not exist an ontology that admits exactly a given set of models [14, 21] Despite these problems, there have been several attempts to adapt classical model-based revision approaches to DL ontologies Qi and Du [23] adapted the well-known Dalal s revision operator [8] for revising terminologies in DL Their operators that do not depend on a specific DL are defined by using sets of models of KBs They also showed that their operators satisfy the AGM postulates but no algorithm has been proposed for computing revised ontology Wang and colleagues [34] have adapted Satoh s approach [27] for propositional KB to introduce a revision operator over DL-Lite ontologies The construction of that operator relies on a finite structure, namely feature, which is used to characterize possibly infinite models of DL-Lite ontologies Although the introduced revision operator does not ensure fully the principle of minimal change, it allows the authors to develop and implement an algorithm for revising DL-Lite ontologies A so-called type semantics [36, 35, 37] generalized from the notion of features has been developed later to provide a method for dealing with revision and contraction on DL- Lite ontologies Since the type semantics is founded on propositional models rather than first-order structures, it is not straightforward to extend this semantics to a more expressive DL In order to find a good tradeoff between the two kinds of approaches, a hybrid one has attempted to integrate semantic aspects within syntax-based approaches Calvanese and colleagues [6] have proposed a revision algorithm that, given a KB K and new knowledge N, returns a subset of the deductive closure cl(k) such that it is a maximal compatible set of axioms for K and N However, the revision result is non-deterministic in general since there may exist several subsets of cl(k) each of which is a maximal compatible set of axioms with respect to K and N To obtain determinism, it is needed to restrict revision operation on ABox as pointed out by the authors [4, 5] Another result that establishes relationships between syntax and model-based approaches was presented by Qi and colleagues [25] They have proved that a syntax-based revision operator can be used to approximate the modelbased revision operators in DL-Lite R proposed by Kharlamov and colleagues [21] In this paper, we propose a new model-based approach for revising ontologies in SHIQ with individuals A preliminary result of the present work for revising ontologies in SHIQ without individuals was published at LPAR forum [10] We base the construction of our revision procedure on the following points : (i) using completion graphs generated by a novel tableau algorithm to characterize the semantics of a SHIQ ontology A completion graph for an ontology O consists of nodes and edges which are respectively labelled by sets of concepts and roles from the signature of O in such a way that the semantics of each axiom from O is satisfied in each node and edge This algorithm must build a set of completion graphs, denoted FM(O), for an ontology O by considering all intrinsic non-deterministic cases instead of building one completion graph as existing tableau algorithms do; (ii) defining a distance over a set of completion graphs for addressing the principle of minimal change Given an ontology O containing new axioms which should be taken into account when revising, this distance can help to choose completion graphs from FM(O ) that are semantically closest to those in FM(O) A revised ontology of O by O should admit the chosen completion graphs as models; (iii) introducing the notion of approximation ontology to overcome inexpressibility issue Our revision procedure returns an approximation ontology that is expressible in SHIQ and smallest in terms of semantics To illustrate the idea behind the construction, we consider the following running example Example 1 Given an ontology UNI presented in Table 1 Assume that researchers and experts do not supervise any student We add to UNI the following axioms which express these semantic constraints: (δ 1 ) : Researcher supervises( Student) (δ 2 ) : Expert supervises( Student)

3 α 1 : Professor Researcher Expert α 2 : Professor supervisesstudent α 3 : Professor ( 2 teachescourse) β : Professor(Alex) Professors are researchers or experts A professor supervises at least a student A professor teaches at least two courses Alex is a professor Table 1: Ontology UNI However, the presence of δ 1 and δ 2 will make UNI inconsistent, and this requires a revision to maintain consistency of UNI One of the ways of revision is to remove a number of axioms from UNI Intuitively, we can eliminate the axiom α 1 or α 2 to maintain consistency of UNI In this case, if we remove α 2 from UNI, we will lose all knowledge from α 2 However, we can replace α 2 by knowledge A professor except Alex supervises at least a student and we will lose less knowledge of α 2 In other words, the goal should be to build a new ontology O which is compatible with the axioms from UNI such that O is semantically as close as possible to UNI We can check that the completion graphs F 1, F 2 in Figure 1 yield models of UNI In this case, each node equipped with a label represents an individual and each edge represents a relation between two individuals Similarly, we can check that the two completion graphs in Figure 2 yield models of {δ 1, δ 2 } If we define a distance between completion graphs based on structural similarity, it would be plausible to say that F 1 is closer to F 1 and F 2 than F 2 Therefore, a revised ontology O should admit F 1 rather than F 2 Indeed, this intuition will be confirmed in Section 5 where we present a procedure for computing the revised ontology The present paper is organized as follows Section 2 describes the DL SHIQ In Section 3, we present a novel tableau algorithm for building a set of completion forests which represents all models of a SHIQ ontology Section 4 introduces a distance over a set of completion forests This distance provides a means to equip a set of completion forests with a total pre-order This is a crucial point allowing for defining a revision operation which satisfies all revision postulates reformulated for DL ontologies Based on the defined revision operation, we introduce in Section 5 the notion of upper approximation which allows us to propose a procedure for computing a revised ontology expressible in SHIQ from a set of completion forests Section 6 describes some techniques for optimizing our procedure We also describe an implementation of our algorithm and report some experimental results in Section 7 Finally, Section 8 consists of a discussion on the obtained results and future work 3 2 Preliminaries We begin by presenting the syntax and the semantics of SHIQ Let R be a non-empty set of role names and R + R be a set of transitive role names We use R I = {R R R} to denote a set of inverse roles Each element of R R I is called a SHIQ-role To simplify notations for nested inverse roles, we define a function Inv(S) = R if S = R; and Inv(S) = R if S = R where R R A role inclusion axiom is of the form R S for two (possibly inverse) SHIQ-roles R and S A role hierarchy R is a finite set of role inclusion axioms A sub-role relation is defined as the transitive-reflexive closure of on R + = R {Inv(R) Inv(S) R S R} We define a function Trans(R) which returns true iff R is a transitive role More precisely, Trans(R) = true iff R R + or Inv(R) R + A role R is called simple with respect to (wrt) R + iff R / R + and, for any R R, R is also a simple role An interpretation I = ( I, I) consists of a non-empty set I (domain) and a function I which maps each role name to a subset of I I such that R I = { x, y I I y, x R I } for all R R, and x, z S I, z, y S I implies x, y S I for each S R + An interpretation I is a model of R, written I = R, if R I S I for each R S R Let C be a non-empty set of concept names The set of SHIQ-concepts is inductively defined as the smallest set containing all C in C,, C D, C D, C, RC, RC, ( n SC) and ( n SC) where n is a positive integer, C and D are SHIQ-concepts, R is a SHIQ-role and S is a simple role wrt a role hierarchy We write for The interpretation function I of an interpretation I = ( I, I) maps each concept name to a subset of I such that I = I, (C D) I = C I D I, (C D) I = C I D I, ( C) I = I \C I, ( RC) I = {x I y I, x, y R I y C I }, ( RC) I = {x I y I, x, y R I y C I }, ( n SC) I = {x I {y C I x, y S I } n}, ( n SC) I = {x I {y C I x, y S I } n} where S stands for the cardinality of a set S An axiom C D is called a general concept inclusion (GCI) where C, D are (possibly complex) SHIQ-concepts, and a finite set of GCIs is called a terminology T An interpretation I satisfies a GCI C D, written I = (C D), if C I D I An interpretation I is a model of T, written I = T, if I satisfies each GCI in T Let I be a set of individual names An assertion is of the form C(a), R(a, b), or a = b for a, b I, a SHIQ-role

4 {{Alex}, Professor, Researcher, Expert, supervisesstudent, 2 teachescourse} x {{Alex}, Professor, Researcher, Expert, supervisesstudent, 2 teachescourse} x F 1 : supervises y teaches z teaches w F 2 : supervises y teaches z teaches w {Student, Professor} {Course, Professor} {Course, Professor} {Student, Professor} {Course, Professor} {Course, Professor} Figure 1: Completion graphs yielding models of UNI {{Alex}, Professor, Researcher, Expert, supervisesstudent, 2 teachescourse} x {{Alex}, Professor, Researcher, Expert, supervises( Student), 2 teachescourse} x F 1: supervises y teaches z teaches w F 2: teaches z teaches w {Student} {Course} {Course} {Course} {Course} Figure 2: Completion graphs yielding models of {δ 1, δ 2 } R and a SHIQ-concept C An ABox consists of a finite set of assertions For an interpretation I = ( I, I), an element x I is called an instance of a concept C iff x C I For ABoxes, the interpretation function I of I maps each individual a I to some element a I I An interpretation I satisfies an assertion C(a) (resp R(a, b), and a = b) iff a I C I (resp a I, b I R I, and a I b I ) I satisfies an ABox A if it satisfies each assertion in A Such an interpretation is called a model of A, denoted by I = A We use O = (T, R, A) to denote a SHIQ ontology, where T is a SHIQ terminology, R is a SHIQ role hierarchy, and A is an ABox An ontology O = (T, R, A) is said to be consistent if there is a model I of T, R and A, ie, I = T, I = R and I = A Additionally, we use Mod(O) to denote all the models, and S(O) = R C I to denote the signature of an ontology O where C is the set of concept names occurring in O, R is the set of role names occurring in O, and I is the set of individual names occurring in A For the ease of construction, we assume all concepts to be in negation normal form (NNF), ie, negation occurs only in front of concept names Any SHIQ-concept can be transformed to an equivalent one in NNF by using De Morgan s laws and the duality between existential and universal restrictions, and between at-most and at-least number restrictions of the form nrc and nrc respectively [18] For a concept C, we use nnf(c) and C to denote respectively the NNF of C and C The function nnf(c) can be computed in polynomial time in the size of C [2] In the remaining of this section, we introduce some notations which will be used in the next sections 4 Definition 1 (Subconcepts) Let O = (T, R, A) be a SHIQ ontology with S(O) = R C I A set sub(o) is inductively defined as follows: sub(o) = sub(t ) sub(a) { C C sub(t ) sub(a)} sub(t ) = sub(nnf( C D)) C D T sub(a) = {sub(nnf(c)) C(a) A} {C, C} if C C sub(e) sub(f ) if C {E F, E F } {C} { R E R R } sub(e) if C = RE sub(c) = {C} { R E R R } sub(e) if C = RE {C} { nr E R R } sub(e) if C =( nre) {C} sub(e) if C = ( nre) Note that sub(o) contains no disjunctions or conjunctions since they are replaced with their disjuncts and conjuncts In addition, the size of sub(o) is bounded by a polynomial function in the size of O since sub(o) is built from subconcepts of concepts occurring in O and from the concepts obtained by substituting roles in sub-concepts for other roles In this paper, we distinguish intrinsic non-determinism from non-intrinsic non-determinism Intrinsic nondeterminism arises mainly from disjunctions and numbering restrictions appearing on ontologies while nonintrinsic non-determinism may be added to completion graphs when the tableau algorithm deals with GCI An amount of this kind of non-determinism can be removed by absorption techniques which will be presented in Section 62

5 To characterize the semantics of an ontology we need to explore all intrinsic non-determinism when constructing a completion graph for the ontology For this reason, we introduce a function Flat(C) which makes explicit all disjunctions at top-level of a concept C (ie those that do not appear in the filler of a universal, existential, numbering restrictions occurring in C) Definition 2 (Flattening) Let C be a SHIQ concept We define a function Flat(C) which returns a set of subsets of sub(c) as follows: 1 If C is a concept name or C is an existential, universal, number restriction, we define Flat(C) = {{C}}; 2 If C = E F, we define Flat(C) = Flat(E) Flat(F ); 3 If C = E F, we define Flat(C) = {W W W Flat(E), W Flat(F )} Applying the item 3 in Definition 2 to a concept C may make Flat(C) increase exponentially For instance, Flat((A 1 B 1 ) (A 2 B 2 )) = {W W W {{A 1 }, {B 1 }}, W {{A 2 }, {B 2 }}} = {{A 1, A 2 }, {A 1, B 2 }, {B 1, A 2 }, {B 1, B 2 }} More general, if C = (A 1 B1) (A n B n ), Flat(C) contains 2 n elements We formulate and prove some useful properties of the function Flat in the following lemma Lemma 1 Let C be a SHIQ concept It holds that: i each element X Flat(C) is a subset of sub(c) and does not contain any conjunction and disjunction at top-level; ii if C is neither conjunction nor disjunction, then Flat(C) contains a unique subset of sub(c) that includes only C; iii if C is a conjunction and there is a conjunct D of C such that D is neither conjunction nor disjunction, then D must appear in all elements of Flat(C); iv if C is a disjunction and each disjunct of C is neither conjunction nor disjunction, then each element of Flat(C) contains exactly a disjunct of C Proof i By the items 2 and 3 of Definition 2, each element X Flat(C) does not contain any conjunction and disjunction at top-level According to Definition 1 and Definition 2, each element X Flat(C) is a subset of sub(c) ii Assume that C is neither conjunction nor disjunction This means that C is atomic or C = RD, or C = RD, or C = ( nrd), or C = ( nrd) By the item 1 of Definition 2, we have Flat(C) = {{C}} iii Assume that C = D 1 D 2 D n and there is a D i (1 i n) which is neither conjunction nor disjunction Without loss of generality, assume i = 1 and rewrite C = D 1 X with X = D 2 D n By 5 applying the item 3 in Definition 2 to C, we obtain Flat(C) = {{D 1 } W W Flat(X)} This implies that D 1 appears in all elements of Flat(C) iv Assume that C = D 1 D 2 D n and each disjunct D i (1 i n) of C is neither conjunction nor disjunction By the item 2 of Definition 2, we have Flat(C) = Flat(D 1 ) Flat(D n ) = {{D 1 }} {{D n }} = {{D 1 },, {D n }} Therefore, each element of Flat(C) contains exactly a disjunct of C 3 Tableau-based characterization of the ontology semantics A model-based approach [1] to revision of an ontology would need to change semantic constraints from the ontology by manipulating a set of its models However, the set of all models of an ontology may be infinite and there exists a SHIQ ontology which admits only infinite models This leads us to propose a method which allows one to characterize the semantics of a SHIQ ontology by using a finite set of completion graphs In this section we introduce a tableau-based algorithm for generating such a finite set of completion graphs for a SHIQ ontology Horrocks and colleagues [19] have proposed a tableau algorithm for checking consistency of a SHIQ ontology and have shown that there always exists a finite completion forest iff the ontology is consistent This algorithm attempts to construct a completion forest, returns YES if it succeeds in building such a completion forest and NO if it fails after considering all possibly non-deterministic cases To be able to characterize the semantics of an ontology, we need rather a set of completion forests which describes different models resulting from non-deterministic logical constructors than one completion forest For this purpose, we adapt the tableau algorithm by Horrocks and colleagues [19] in such a way that it would explore all intrinsic nondeterministic cases Definition 3 (Completion forest) Let O = (T, R, A) be a SHIQ ontology A completion forest F for O is a tuple F = (G, T x 1,, T x n ) where G = (V, E, L) is a directed graph with V a set of root nodes, E a set of edges connecting root nodes and L a labelling function which associates to each node x V a set L(x) sub(o) and to each edge x, ŷ E a set L( x, ŷ ) R R I A node ŷ V is called an R-neighbor of x V if R L( x, ŷ ) or Inv(R) L( ŷ, x ) If it is clear from the context, we use L( x) and L( x, ŷ ) instead of L( x) and L( x, ŷ ) Each T x i = (V i, E i, L i ) (1 i n) is a tree rooted by x i belonging to G (ie x i V), V i a set of nodes, E i a set of edges, and L i a labelling function which associates to each node x V i a set L i (x) sub(o)

6 and to each edge x, y E i a set L i ( x, y ) R R I If it is clear from the context, we use L(x) and L( x, y ) instead of L i (x) and L i ( x, y ) If two nodes x, y V i (of some tree T x i = (V i, E i, L i )) connected by an edge x, y E i, then y is called a successor of x, and x is called a predecessor of y; ancestor is the transitive closure of predecessor A node y is called an R-successor of x if, for some role R with R R, R L( x, y ); x is called an R- predecessor of y, if y is an R-successor of x A node y is called an R-neighbor of x if y is an R-successor or x is an Inv(R)-successor of y A node x V i is called blocked by a node y V i if x is not a root node and it has ancestors x, y and y such that (i) y is not a root node, (ii) x is a successor of x and y is a successor of y, (iii) L(x) = L(y), L(x ) = L(y ), and (iv) L( x, x ) = L( y, y ) Furthermore, there are an inequality relation = and an equality relation = defined over nodes in F In addition, F is said to contain a clash if (i) there is some node x in F such that either {A, A} L(x) for some concept name A C, or (ii) ( nsc) L(x) and there are (n + 1) S- neighbors y 1,, y n+1 of x with y i = y j and X L(y i ) for some X Flat(C) and all 1 i < j (n + 1) In order to build a completion forest, a standard tableau algorithm [19] uses expansion rules each of which corresponds to a logical constructor appearing in ontologies In particular, there are rules whose behavior is nondeterministic such as -, -, ch-rule The -rule adds C or D to L(x) if C D L(x) and {C, D} L(x) = The -rule introduces new non-determinism via concepts of the form C D that is added to L(x) for each node x if an axiom C D belongs to the ontology The ch-rule deals with non-determinism related to each R-neighbor y of a node x with ( nrc) L(x) Based on this standard tableau algorithm, we design a new one which uses the expansion rules in Figure 3 There are two main differences between the rules in Figure 3 and those presented in a standard tableau algorithm: (i) the absence of conjunction and disjunction rules According to Lemma 1, applying the function Flat to a concept freshly added to the label of a node removes all conjunctions and disjunctions at top-level from that concept; (ii) the presence of the sat-rule (sat stands for saturate) A choice of a subset S sub(o) in the sat-rule must include all flattened concepts from GCI axioms (such as those added by the -rule [19]) In addition, the sat-rule adds to each node label either C or C for each C sub(o) These behaviors may lead to an exponential blow-up but they are needed for constructing an approximation ontology from a set of completion forests and a set of sub-concepts (Section 5) An optimization of this rule will be proposed and discussed in Section 6 For an input ontology O = {T, R, A}, our tableau al- 6 gorithm starts by initializing a completion forest F with only root nodes and edges between them This part of F represents individuals and assertions defined in A The algorithm applies the rules from Figure 3 to each node until none of the rules is applicable to each node In this case, F is called complete If F contains no clash it is called clashfree More precisely, F contains a root node x i for each individual a i I occurring in A, and an edge between two any root nodes x i, x j if A contains an assertion R(a i, a j ) for some role R The label of these root nodes and the edges between them with the relations = and = is initialized as follows: L( x i ) := {{a i } X k C(a i ) A, X k X, X Flat(C)}, L( x i, x j ) := {R R(a i, a j ) A}, x i = x j iff a i = a j A, and the =-relation is initialized to be empty Applications of generating rules ( and -rules) can create fresh nodes whose label is entirely filled by the sat-rule, or partially filled by -, + -rules After applying the sat-rule to a fresh node, its label is not changed since this rule saturates its label by adding either X Flat(C) or X Flat( C) for each concept C sub(o) This explains why - and ch-rules in a standard tableau algorithm become useless when allowing for the sat-rule In addition, the function Flat makes - and -rules unnecessary Finally, the r -rule is used for identifying two root nodes which cannot be handled by the -rule This situation is illustrated in the following example Example 2 Let consider an ontology which consists of the following assertions: ( 1RC)(x), C(y), C(z), D(z), SD(z), R(x, y), R(x, z) We construct a completion forest by initializing the root nodes x, y and z, and adding to their label the concepts imposed by the assertions ( 1RC)(x), C(y), C(z), D(z) and SD(z) We then create the edges between x and y and between x and z with the label R imposed by the assertions R(x, y) and R(x, z) Due to the presence of the concept ( SD) in the label of z, the -rule is applied to generate a fresh node z 1 which is an S-successor of z and has a concept D in its label When applying the -rule for merging z to y, we must set L( x, z ) = and thus the relation R between x and z will not be preserved This implies that the assertion R(x, z) will no longer be satisfied For this reason, the r -rule introduces some additional steps to deal with this issue Firstly, this rule adds L(z) to L(y) as what the -rule does, and changes the edges (and their respective labels) between z and its neighbors to the edges (and respective labels) between y and these neighbors Secondly, L(z) and all edges going to/from z are removed from the forest This removal does not cause any problem because all neighbors of z became neighbors of y in the previous step Finally, the identification of z and y is saved in the = relation This identification is needed to construct a model from a complete and clash-free completion forest

7 -rule: if 1 SC L(x), x is not blocked, and 2 x has no S-neighbor y st X L(y) for some X Flat(C) then create a new node y with L( x, y ) {S} and L(y) X for some X Flat(C) -rule: if 1 SC L(x), and 2 there is an S-neighbor y of x st X L(y) for all X Flat(C) then L(y) L(y) X for some X Flat(C) + -rule: if 1 SC L(x), 2 there is an R with Trans(R) st R S, and 3 there is an R-neighbor y of x st RC / L(y) then L(y) L(y) { RC} -rule: if 1 ( nsc) L(x), x is not blocked, and 2 x has no n S-neighbors y 1,, y n such that X L(y i ) for some X Flat(C) and y i = y j for 0 i < j n then create n new nodes y 1,, y n with L( x, y i ) {S}, L(y i ) X for some X Flat(C) and y i = y j for 1 i < j n -rule: if 1 ( nsc) L(x), 2 x has n+1 S-neighbors y 0,, y n st X L(y i ) for some X Flat(C), 3 there are two S-neighbors y, z of x with X 1 L(y), X 2 L(z) for some X 1, X 2 Flat(C), y is not an ancestor of z, and not y = z then (i) L(z) L(z) L(y) and L( x, y ) (ii) if z is an ancestor of x then L( z, x ) L( z, x ) {Inv(R) R L( x, y )}; else L( x, z ) L( x, z ) L( x, y ), and (iii) add u = z for all u such that u = y r -rule: if 1 ( nsc) L(x), 2 x has n + 1 S-neighbors y 0,, y n st X L(y i ) for some X Flat(C), 3 y i = y j does not hold for some 0 i < j n where y i, y j are root nodes, then (i) L(y i ) L(y i ) L(y j ), (ii) for all edges y j, w, if the edge y i, w does not exist, create it with L( y i, w ) = ; set L( y i, w ) L( y i, w ) L( y j, w ), (iii) for all edges w, y j, if the edge w, y i does not exist, create it with L( w, y i ) = ; set L( w, y i ) L( w, y i ) L( w, y j ), (iv) set L(y j ) and remove all edges to/from y j, (v) set u = y i for all u with u = y j, (vi) set y j = yi sat-rule: if sat-rule has never been applied to x then choose a subset S sub(o) st L(x) X Flat(nnf( C D)),C D T and set L(x) S S where S = { C C sub(o) \ S} Figure 3: Expansion rules for SHIQ X S, 7

8 y R {C} x { 1RC} R z {C, D, SD} S z 1 {D} r -rule x { 1RC} R y {C, D, SD} z S z 1 {D} Figure 4: Effect of the r-rule Another specific issue which would lead to non-termination when reasoning with an ABox is the so-called yo-yo effect [3] This issue occurs when merging two neighbors y and z of a node x (due to a concept nrc L(x)) does not change the neighborhood structure of x which triggers merging To solve this issue, Horrocks and colleagues [19] have introduced the -rule which adds L(z) into L(y) and sets L( x, z ) = instead of merging z into y This empty edge isolates x from z and all successors of z which were responsible for reproducing the neighborhood structure of x Lemma 2 (Soundness and completeness) Let O = (T, R, A) be a SHIQ ontology 1 The tableau algorithm with the rules in Figure 3 terminates 2 If the tableau algorithm with the rules in Figure 3 yields a complete and clash-free completion forest from the input ontology O then O is consistent; 3 If O is consistent then the tableau algorithm with the rules in Figure 3 can yield a complete and clash-free completion forest from the input ontology O Proof sketch Termination is a consequence of the blocking condition and monotonicity of the construction of a completion forest F, that is, the algorithm never removes anything from F except for removing edges by the r - rule, which is never restored To prove soundness, we devise a model from a completion forest F by unraveling This process generates descendants of blocked nodes by replicating descendants of blocking nodes The obtained model is of the forest-like structure that may be infinite To prove completeness, we track the construction of a completion forest by using a function π which associates to each node x of the completion forest an individual of the model Whenever a non-deterministic expansion rule is applied to a node x, we consider the two following cases : (i) it makes a good choice if L(x) L(π(x)); (ii) it makes a bad choice if L(x) L(π(x)) In this case, we show that the algorithm can always go back to make a good choice and succeed in building a complete and clashfree completion forest A complete proof can be found in Appendix The tableau algorithm can build a completion forest whose depth is bounded by an exponential function in the size of O due to the blocking condition This implies that 8 the size of a completion forest is bounded by a doubly exponential function in the size of O Since the algorithm creates a completion forest in a non-deterministic way, it runs in non-deterministic doubly exponential time This complexity is far from being optimal in the worst case since SHIQ is ExpTime-complete [32] Notation 1 (Forest-like model) Let O be a SHIQ ontology A complete and clash-free completion forest F that is built by running the tableau algorithm with the expansion rules in Figure 3 over O is called a forest-like model According to soundness of the tableau algorithm (Lemma 2), one can devise by unraveling a model from a complete and clash-free completion forest F, denoted I(F) Given an axiom C D, define I(F) = (C D) if C I(F) D I(F) Given an assertion C(a) (or R(a, b)), define I(F) = C(a) (resp I(F) = (R(a, b))) if a I(F) C I(F) (resp a I(F), b I(F) R I(F) ) Conversely, according to completeness of the tableau algorithm (Lemma 2), it can build a complete and clash-free completion forest F from a model I Mod(O), denoted F(I) Contrary to standard tableau algorithms which terminate when a forest-like model is found, our algorithm has to consider all non-deterministic cases and build all forestlike models for an ontology O Notation 2 (FM(O, sub)) We use FM(O) to denote the set of all forest-like models built by running the tableau algorithm over a SHIQ ontology O Given a set of concepts sub, we define FM(O, sub) to be the set of all forest-like models built by running the tableau algorithm on O such that the sat-rule operates on sub(o) sub (ie it chooses a subset S sub(o) sub) In particular, given an ontology O the set FM(O, sub(o )) can be built by running the tableau algorithm over O with the sat-rule operating on sub(o) sub(o ) In addition, we need to import to O roles which occur in role assertions from O This leads to add to the label of each root edge x, ŷ of each forest F FM(O, sub(o )) a subset of roles S R R O where R O is the set of all roles occurring in O with their inverse This new behavior can be formalized as follows: sat R -rule : for each root edge x, ŷ E with G = (V, E, L), F = (G, T x 1,, T x n ) and F FM(O, sub(o )), we set L( x, ŷ ) L( x, ŷ ) S R for some S R R O

9 The construction of FM(O, sub(o )) allows one to import the signature of an ontology O to O when building completion forests for O Note that we do not import any new semantic constraint from O when building FM(O, sub(o )) What we really perform in this construction is to import into O concepts written in the signature of O This importation may extend FM(O) with new completion forests but never changes consistency of O The effect of the sat R -rule is illustrated in the following example Example 3 Consider an ontology O which consists of the following axiom and assertions: A RA, A(a), A(b), R(a, b) Let α be an assertion R( A)(a) By applying the tableau algorithm over O to construct the set FM(O), we obtain only the following forest F 1 : F 1 x {{a}, A, RA} R y {{b}, A, RA} By applying the tableau algorithm over O with the sat-rule operating on sub(o) sub(o ) = {A, A, RA, R( A), R( A), RA} and the sat R - rule operating on R α = {R, R } to construct the set FM(O, sub(α)), we obtain the two following forests F 2 and F 3 : F 2 x {{a}, A, RA, RA} F 3 R y R {{b}, A, RA, RA} x {{a}, A, RA, RA} R, R y z {A, RA, RA} {{b}, A, RA, RA} Therefore, we obtain FM(O) = {F 1 } but FM(O, sub(α)) = {F 2, F 3 } We now use the notations recently introduced to formulate and prove the following properties on FM(O) which characterizes the semantics of an ontology O Corollary 1 Let O and O be two consistent SHIQ ontologies It holds that 1 Let α be a concept axiom or assertion written in S(O) I(F) = α for each forest-like model F FM(O, sub(α)) iff I = α for each model I Mod(O) 2 FM(O O ) = FM(O, sub(o )) FM(O, sub(o)) 9 The Corollary 1 affirms the semantic equivalence between Mod(O) and FM(O) in the sense that each axiom/assertion which is satisfied by Mod(O) is satisfied by FM(O), and conversely This result allows us to replace a possibly infinite set Mod(O) with a finite set FM(O) in constructions presented in the next sections At first glance a proof of the only-if direction from Item 1 in Corollary 1 is challenging since Mod(O) is much larger than FM(O, sub(α)) However, this can be overcome by using the argument by absurdity since inconsistency of O { α} wrt FM(O, sub(α)) implies inconsistency of O { α} wrt Mod(O) We refer the readers to Appendix for a complete proof 4 Revision Operation The main goal of the present section is to define a revision operation which allows for revising a consistent ontology O by axioms from another consistent ontology O Since no change is needed if O O is consistent, we always assume that O O is inconsistent Such a revision operation returns a set of completion forests of which a revised ontology should admit in order to take into account new knowledge from O and to be semantically as close as possible to O These properties on revision operation are captured by the AGM postulates rephrased for DL ontologies [23] To reach this goal, we need to introduce a distance between two completion forests which yields a total pre-order over them and allows one to talk about similarity between two ontologies This distance is an extension of that defined over completion trees [10] At the end of the section, we show that our revision operation satisfies actually all AGM postulates We begin this section by extending isomorphisms defined over completion trees to those defined over completion forests Definition 4 (Isomorphism) Let F=(G, T x 1,, T x n ) and F = (G, T x 1,, T x n ) be two forest-like models with G = (V, E, L), G = (V, E, L ), T x i = V i, L i, E i and T x j = V j, L j, E j (1 i, j n) Let V = V V 1 V n and V = V V 1 V n Let E = E E 1 E n and E = E E 1 E n We use succ(x) to denote the set of successors of a node x in a tree T x i or T x j with 1 i, j n T x i and T x j are isomorphic for 1 i, j n if there is a bijection π from V i to V j such that (i) π( x i ) = x j ; and (ii) for each node x V i, we have π(x ) succ(π(x)) for each x succ(x) F and F are isomorphic if there is a bijection π from V to V such that (i) π( x i ) = x j for each x i V; and (ii) for each T x i F, two trees T x i and T π( x i ) are isomorphic In this case, we say that π is an isomorphism between F and F

10 F and F are equivalent if there is an isomorphism π between F and F such that L(x) = L (π(x)) for each x V, and L( x, y ) = L ( π(x), π(y) ) for each x, y E where L(x) = L i (x) (resp L(x) = L j (x)) if x V i (resp x V j ), and L( x i) = L( x i ) (resp L( x i ) = L ( x j )) if x i V (resp x j V ) Note that if there exists a bijection π between two trees T x i and T x j as described in Definition 4 then the restriction of π to succ(x), denoted π succ(x), is a bijection from succ(x) to succ(π(x)) In fact, π succ(x) is an injective from succ(x) to succ(π(x)) due to the first item in Definition 4 It holds that π succ(x) is also a subjective from succ(x) to succ(π(x)) since if there is some y succ(π(x)) such that y = π(y ) for some y succ(x ) with x x then π(y ) succ(π(x )) due to the first item in Definition 4 This implies that π(y ) succ(π(x)) succ(π(x )) =, which is not possible Remark 1 Let F FM(O, sub(o )) and F FM(O, sub(o)) two forest-like models with the sets of root nodes V and V It is needed to import all individuals from O (included in sub(o)) to O and reversely when revising O by new axioms from O Therefore, for each individual a there are a unique tree T x i of F and a unique tree T x j of F such that a L( x i ) L( x j ) This implies that there is a bijection ϕ from V to V such that ϕ( x i ) = x j iff a L( x i ) L( x j ) for some individual a Since the notion of isomorphism refers only to the structure of completion forests, we can always obtain such an isomorphism between any two completion forests by adding empty nodes and edges to these completion forests This is similar to what we have made between two completion trees [10] In the following, we introduce a distance between two isomorphic completion forests This distance refers to the label of nodes and edges of forest-like models, and relies on the operator of symmetric difference, denoted, which is defined as follows: S S = (S S ) \ (S S ) for any two sets S and S Definition 5 (Distance) Let F=(G, T x 1,, T x n ) and F = (G, T x 1,, T x n ) two forest-like models with G = (V, E, L), G = (V, E, L ), T x i = V i, L i, E i and T x j = V j, L j, E j for 1 i, j n Let ϕ be a bijection from V to V such that ϕ( x i ) = x j iff there is some individual a satisfying a L( x i ) L ( x j ) The distance between F and F, denoted d(f, F ), is defined as follows: d(f, F ) = n d(t x i, T ϕ( x i ) ) + i=1 where d(t, T ) = max ( L( x, y ) x,y E L ( ϕ(x), ϕ(y) ) )} min { max ( L(x) π Π(T,T ) x,y E L (π(x)) + L( x, y ) L ( π(x), π(y) ) + L(y) L (π(y)) )} 10, and Π(T, T ) is the set of all isomorphisms between two trees T and T We have defined a distance over forest-like models whose sets of root nodes should be associated by a bijection ϕ according to Remark 1 This restriction is justified by the fact that (i) it would compare the labels of two nodes which contain the same individual rather than two different individuals, and (ii) contrary to a concept which may represent a set of individuals and occurs in several times in a forest-like model, a named individual occurs once in a forest-like model As any distance, d(f, F ) should allow one to measure the difference between two forests F and F that is it must satisfy identity, symmetry and triangle inequality properties For this purpose, we use an operator, namely max, which represents the greatest difference between two triples (composed of an edge and two nodes) associated by an isomorphism π between F and F This operation max allows us to ensure the identity property but it is not sufficient for guaranteeing the triangle inequality property d(f, F ) d(f, F ) + d(f, F ) For this reason, we must use a further operator, namely min, which allows for choosing an isomorphism π from all isomorphisms between F and F (one of which gets involved to determine d(f, F )+d(f, F )) such that the greatest difference between triples associated by π is smallest Lemma 3 The function d(f, F ) in Definition 5 satisfies identity, symmetry and triangle inequality properties Proof sketch The symmetry property holds due to commutativity of the operator of symmetric difference over two sets The identity property is a consequence of the property S S = iff S = S and the operator max over each term L( x, y ) L( ϕ(x), ϕ(y) ) The operator min which refers to the set of all isomorphisms between two comparable forests and transitivity of isomorphisms crossing forests-like models help to establish the triangle inequality In fact, to show d(f, F ) d(f, F )+d(f, F ), we pick up an isomorphism π 1 from Π(F, F ) such that π 1 yields d(f, F ), and an isomorphism π 2 from Π(F, F ) such that π 2 yields d(f, F ) From π 1 and π 2, we can define an isomorphism π 2 π 1 from F to F which must yield a value greater than d(f, F ) A complete proof of this lemma can be found in Appendix In addition, computing the distance by using directly Definition 5 can lead to an exponential blow-up since there may be an exponential number of different isomorphisms between two completion forests This issue will be discussed and solved in Section 6 We now show that the distance in Definition 5 yields a total pre-order over a set of isomorphic completion forests To do this, we define a relation F F over isomorphic completion forests including a forest F 0 containing only empty labels as follows: F F iff d(f 0, F) d(f 0, F )

11 Professor supervises( Student) Someone who is not a professor cannot supervise any student and ( 1 teachescourse) does not teach more than a course Student Course, Student Professor, Student Researcher, Student Expert A student is not a course nor a professor nor a researcher nor an expert Course Professor, Course Researcher, Course Expert A course is not a professor nor a researcher nor an expert Table 2: Axioms added into the ontology UNI {{Alex}, Professor, Researcher, Expert, Student, Course, supervisesstudent, 2 teachescourse} x {{Alex}, Professor, Researcher, Expert, Student, Course, supervisesstudent, 2 teachescourse} x F 1 : supervises y teaches z teaches w F 2 : supervises y teaches z teaches w {Student} X {Course} X {Course} X {Student} X {Course} X {Course} X {{Alex}, Professor, Researcher, Expert, Student, Course, supervisesstudent, 2 teachescourse} x {{Alex}, Professor, Researcher, Expert, Student, Course, supervisesstudent, 2 teachescourse} x F 3 : supervises teaches teaches F supervises teaches teaches 1: y z w y z w {Student} X {Course} X {Course} X {Student} X {Course} X {Course} X where X = { Professor, Expert, Researcher, supervises( Student), ( 1 teachescourse)} Figure 5: Completion forests yielding models of UNI (F 1, F 2, F 3 ) and of {δ 1, δ 2 } (F 1 ) Lemma 4 The relation is a total pre-order over isomorphic completion forests All of the above notions provide sufficient elements to define a revision operation for a SHIQ ontology O by another ontology O Definition 6 (Revision Operation) Let O and O be two consistent ontologies in SHIQ A set of forest-like models of the revision of O by O, denoted FM(O O ), is defined as follows: FM(O O ) = {F FM(O, sub(o)) F 0 FM(O, sub(o )), F FM(O, sub(o)), F FM(O, sub(o )) : d(f, F 0) d(f, F )} Intuitively, among the forest-like models in FM(O, sub(o)), FM(O O ) retains only those which are closest to forest-like models from FM(O, sub(o )) thanks to the operation d(f 1, F 2 ) that characterizes the difference between F 1 and F 2 11 Example 4 Consider again the ontology UNI of Example 1 As for simplification, assume that O is an ontology obtained by adding into UNI the axioms from Table 2, and O consists of δ 1, δ 2 By applying the new tableau algorithm over O, the set FM(O, sub(o )) contains 3 forest-like models F 1, F 2, and F 3 in Figure 5 By running the new tableau algorithm over O, we obtain a forest-like model F 1 FM(O, sub(o)) illustrated in Figure 5 among other forest-like models By applying the distance formula introduced in Definition 5, we obtain d(f 1, F 3 ) = 4 and d(f 1, F 1 ) = d(f 1, F 2 ) = 2 According to Definition 6, FM(O O ) contains a unique forest-like model F 1 Note that d(f x, F y ) > 2 for all F y FM(O, sub(o )) and F x FM(O, sub(o)) with F x F 1 As mentioned in Section 1, our goal is to propose a revision operation that ensures the principle of minimal change introduced by Alchourrón, Gärdenfors and Makinson [1] as postulates in belief revision framework Katsuno and Mendelzon [20] have rephrased these postulates for propositional knowledge bases, namely (R1)-(R6), and shown that the existence of a total pre-order over models of a propositional knowledge base is equivalent to (R1)-(R6) The argument of the proof has not required any specific structure of models of a propositional knowledge base The revision operation according to Definition 6 provides directly satisfaction of the principle of minimal change Indeed, FM(O O ) retains only forest-like models from FM(O, sub(o)) which are closest to forest-like models from FM(O, sub(o )) according to the distance between completion forests This distance infers the total pre-order over forest-like models This observation allows us to get straightforwardly the result saying that revision postulates imply a total pre-order over forest-like models since we consider only models such as forest-like models over which a total pre-order exists already What remains to be proved is that the revision operation in Definition 6 sat-

12 (P1) I(F) = α for each forest-like model F FM(O O ) and each axiom α O (P2) If FM(O, sub(o )) FM(O, sub(o)) then FM(O O ) = FM(O, sub(o )) FM(O, sub(o)) (P3) If O is consistent then FM(O O ) (P4) If FM(O 1, sub(o 1)) = FM(O 2, sub(o 2)) and FM(O 1, sub(o 1 )) = FM(O 2, sub(o 2 )) then FM(O 1 O 1) = FM(O 2 O 2) (P5) FM(O O ) FM(O, sub(o) sub(o )) FM(O (O O )) (P6) If FM(O O ) FM(O, sub(o) sub(o )) then FM(O (O O )) FM(O O ) FM(O, sub(o) sub(o )) Table 3: Revision postulates (P1)-(P6) isfies revision postulates To do this, we use the revision postulates (G1)-(G6) that were formulated by Qi, Liu and Bell [23] for DL ontologies To provide a precise idea about how the postulates (G1)-(G6) are translated into those formulated in our setting, we consider the following postulate (G2) taken from the Qi, Liu and Bell s paper [23]: (G2) If Mod(O) Mod(O ), then Mod(O O ) = Mod(O) Mod(O ) where the revised ontology of O by O is denoted by O O According to Corollary 1 and Definition 6, we can replace Mod(O) and Mod(O O ) with FM(O, sub(o )) and FM(O O ), respectively, to obtain revision postulates that refer only to computable structures We now rephrase all postulates by Qi, Liu and Bell in our setting as indicated in Table 3 Intuitively, (P1) guarantees that all axioms from O can be inferred from the revised ontology This postulate relies on the fact that new knowledge should be more privileged than the existing ones Such new knowledge would reflect a better understanding of the application domain in question (P2) says that the initial ontology O is not changed if O O is consistent (P3) ensures that the resulting ontology is consistent if the revising ontology O is consistent (P4) says that the revision operation should be independent of the syntax of ontologies In fact, if we replace O 1 and O 1 with O 2 and O 2 such that they admit the same forest-like models then revision ontologies admit the same forest-like models as well Regarding (P5) and (P6), we use Figure 6 to illustrate how they can ensure the principle of minimal change For the sake of brevity, we take ontologies O, O and O such that sub(o) = sub(o ) = sub(o ) In this case, we can write FM(X ) for FM(X, sub(x )) where X, X {O, O, O } The stripe portion of FM(O ) in Figure 6 represents FM(O O ) since FM(O O ) FM(O ) according to Definition 6 If the minimal principle is not guaranteed then there exists some F 0 FM(O )\FM(O O ) such that F 0 is closer to FM(O) than any F FM(O O ) Since (P5) and (P6) hold for any ontology O, we can take O such that F 0 FM(O ) This implies that F 0 must belong to FM(O (O O )) since F 0 FM(O O ) and it is closest to FM(O) among those belonging to FM(O O ) However, (P5) and (P6) entail FM(O (O O )) = FM(O O ) FM(O ) This forces F 0 to become some F 0 FM(O O ), which is a contradiction 12 FM(O) d(f 0, FM(O)) d(f 0, FM(O)) FM(O ) FM(O ) Figure 6: (P5) & (P6) ensure the principle of minimal change We now obtain the AGM postulates in our setting Theorem 1 The revision operation FM(O O ) described in Definition 6 satisfies the postulates (P1)-(P6) We refer the readers to Appendix for a complete proof of the theorem 5 Computing The Revised Ontology In this section, we present a procedure for constructing a SHIQ ontology O that admits at least forest-like models in FM(O O ) It has turned out [9] that there may not exist a DL-lite ontology which admits exactly a given set of models By the following example, we show that it is also the case for SHIQ ontologies Example 5 Reconsider Example 4 with FM(O O ) = {F 1} Assume that there exists an ontology Ô with sub(ô) = { supervisesstudent, supervises( Student), ( 2 teachescourse), ( 1 teachescourse), Professor, Professor, Researcher, Researcher, Expert, Expert, Student, Student, Course, Course} which admits the unique F 1 as forest-like model By running the new tableau algorithm over Ô, FM(Ô) must contain F 1 and another F having one node {x} with L(x) = {{Alex}, Professor, Researcher, Expert, Student, Course, supervises( Student), ( 1 teachescourse)}, which is a contradiction Therefore, there may not exist a SHIQ ontology which admits exactly a given set of forest-like models To address this issue, we are borrowing the notion of maximal approximation from De Giacomo and colleagues work F 0 F 0