Forgetting is a non-standard reasoning problem that is concerned with creating restricted views of ontologies relative to a subset of the initial signature and preserving pertinent logical consequences up to the symbols in the restricted views. In this paper, we present an Ackermann-based approach for forgetting of concept and role symbols in ontologies expressible in the description logic ALCOIH +(O; u). The method is one of only few approaches that can eliminate role symbols, that can handle role inverse and ABox statements (via nominals), and the only approach so far providing support for forgetting in description logics with nominals. Despite the inherent di culty of forgetting for this level of expressivity, performance results with a prototypical implementation have shown very good success rates on real-world ontologies.
This paper presents a practical forgetting method for creating restricted views of
ontologies expressed in the language of the description logic ALCOIH +(O; u).
The work is motivated by the high demand for advanced techniques for
ontologybased knowledge processing. Research of forgetting, often referred to as uniform
interpolation (UI) in this area (or, variable elimination, projection or
secondorder quanti er elimination in knowledge representation and logic) has gained
signi cant momentum since the work of various groups developing forgetting
methods for the description logic ALC and logics weaker than ALC, e.g., [11,
18, 19, 21, 25{27], and the foundational studies of the properties of forgetting
for description logics by, e.g., [
Early work in the area primarily focused on forgetting concept symbols, as
role forgetting was realised to be signi cantly harder than forgetting of concept
symbols [
The work on forgetting for description logics is predated by work on
secondorder quanti er elimination [5{7], which can be traced to questions posed by
Boole and seminal work of [
The contribution of this paper is an approach for forgetting of concept and
role symbols in expressive description logics not considered so far. The method
accommodates ontologies expressible in the description logic ALCOIH and the
extension allowing positive occurrences of the least xpoint operator , the top
role O and role intersection u. This means the method is one of only few
approaches that can eliminate role symbols, while also handling role inverse and
ABox statements via nominals, and the only approach so far providing support
for forgetting in description logics with nominals. Being based on the Ackermann
approach to second-order quanti er elimination [
The Description Logic ALCOI H +(O; u) Let NC, NR and NO be mutually disjoint sets of concept symbols (names), role symbols (names) and individuals, respectively. Let N be a set of concept variables disjoint with NC, NR and NO. Concepts in ALCOIH(O; u) have one of the following forms: a j > j A j :C j C t D j 8R:C, where a 2 NO, A 2 NC, C and D are any ALCOIH(O; u)-concepts, and R is any role expression. Role expressions in ALCOIH(O; u) can be the top role O, a role symbol r 2 NR, the inverse r of a role symbol r, or a conjunction of these. The language of ALCOIH +(O; u) extends that of ALCOIH(O; u) with atomic least xpoint expressions of the form
X:C[X], where X 2 N , and C[X] is a concept expression in which X occurs
only positively (under an even number of negations). Moreover, -expressions
may occur only positively. Because of this restriction, -expressions can be
simulated in ALCOIH(O; u) with auxiliary concept symbols as in [
The semantics of the ALCOIH(O; u)-fragment is as expected. Intuitively,
X:C[X] denotes the least general concept C w.r.t. a concept expression for
which C C[C ] holds, where C[C ] is a concept expression obtained from
replacing every occurrence of X in C[X] by C . A formal de nition of the
semantics of xpoint expressions can be found in [
We assume without loss of generality that a TBox T is a set of concept axioms of the form C v D, where C and D are closed concepts, i.e., contain no concept variable not in the scope of a . An RBox R is a set of role axioms of the form R v S, where R and S is a role symbol or an inverted role symbol. Nominals in a description logic make ABox assertions super uous, since these can be captured by TBox inclusions via nominals. An ALCOIH +(O; u)-ontology is therefore assumed to be the union of the TBox and the RBox in this paper. Theorem 1. Reasoning in ALCOIH
+(O; u) is decidable.
Theorem 1 follows from the decidability results of guarded xpoint logic [
In the sequel we describe the normal form on which our forgetting method works. A TBox clause is a disjunction of ALCOIH +(O; u)-concepts, in which role expressions can be a role symbol, an inverted role symbol, or a conjunction of these. A role literal is either a role symbol or an inverted role symbol, or their negations. An RBox clause is a disjunction of two role literals of complementary polarity. RBox clauses are transformed from the clausi cation of role axioms, where role negation is introduced.
A clause that contains occurrences of a designated concept symbol and role symbol S is called an S-clause. Given an S-clause C, an occurrence of S is positive in C if it is under an even number of negations. Otherwise it is negative. A role symbol that occurs immediately under a (negated) universal role restriction is assumed to be negative (positive). C is positive (negative) w.r.t. S if every occurrence of S in C is positive (negative). A set N of clauses is positive (negative) w.r.t. S if every occurrence of S in N is positive (negative).
By sig(E) we denote the concept and role symbols occurring in E, where E ranges over concepts, clauses, and ontologies. is assumed to be the concept and role symbols to be forgotten in this paper. Let S be any concept or role symbol and let I and I0 be interpretations. We say I and I0 are equivalent up to S, or Sequivalent, if I and I0 coincide but di er possibly in the interpretations of S. More generally, I and I0 are equivalent up to , or -equivalent, if I and I0 are the same but di er possibly in the interpretations of the symbols in . De nition 1 (Forgetting in ALCOIH +(O; u)-Ontologies). Let O and O0 be ALCOIH +(O; u)-ontologies and let the signature be sig(O). O0 is the solution of forgetting -symbols in O, if the following conditions hold: (i) sig(O0) sig(O) and sig(O0) \ = ;, and (ii) for any interpretation I: I j= O0 i I0 j= O, for some interpretation I0 -equivalent to I.
This states that the given ontology O and the forgetting solution O0 are equivalent up to the interpretations of the symbols in . It follows that if both O0 and O00 are solutions of forgetting symbols in from O, then they are equivalent.
The forgetting process in our method consists mainly of three phases: the reduction to a set of ALCOIH +(O; u)-clauses, the forgetting phase and the de ner elimination phase (see Figure 1).
Ontology O
Process to set of clauses N Σ = {S1, ..., S2} into
Analyser Forgetting Solution O0
Elimination of Definer Symbols
R Σ Σ
C
Transform to r-reduced form
Ackermann to forget r Transform to A-reduced form
Ackermann to forget A
Initially given, as the input to the method, are an ontology O of TBox and RBox axioms expressible in ALCOIH +(O; u), and a set that contains the concept and role symbols to be forgotten. The -symbols are entirely determined by the users and their application demands; can thus be any subset of the signature of the initial ontology as the user wishes. Once received by the system, O is transformed into a set N of clauses, i.e., the normal form of our method.
The forgetting phase is an iteration of several rounds in which individual concept and role symbols are eliminated. An important feature of the method is that concept symbols and role symbols are forgotten in a focused way, that is, the rules for concept forgetting and the rules for role forgetting are mutually independent; concept and role symbols can thus be forgotten in any desired order. In the forgetting phase, in order for the forgetting rules to be applied to eliminate S 2 , the S-clauses must be transformed into S-reduced form. Thus, whether the -symbols are eliminable depends entirely on whether the current set of clauses can be transformed into reduced form. This reduction is performed using two Ackermann-based calculi working independently, which also lead to goal-oriented elimination of the concept and role symbols. The calculi are described in the next sections. Equivalence-preserving simpli cation rules are applied throughout the forgetting process, ensuring that the current clauses are always in simpler representations for e ciency. What the method returns, if the forgetting is successful, is an ontology O0 that does not contain the symbols in , i.e., the returned ontology is formulated using only symbols in sig(O)n .
Previous research has shown that the success rates of forgetting depend very
much on the order in which the -symbols are forgotten [
We refer to the maximal symbol of w.r.t. the forgetting order as the pivot in our method. Following (and starting with the pivot), the method forgets the -symbols one by one. If the pivot is successfully eliminated from N , we attempt to forget the next symbol in , which has become the new pivot. Otherwise the pivot remains in , agged as a currently non-forgettable symbol, and the next symbol in becomes the pivot. When the end of the ordering has been reached, the calculus is applied to the agged symbols, attempting again to eliminate them. Success will always be pursued until a forgetting solution is found. Though the ordering provides useful guidance during the forgetting process, it does not generally guarantee the success of forgetting. On the symbols not eliminated, a di erent ordering, not attempted before, will be used. When all possible orderings have been attempted and there are still -symbols remaining, this means that the method fails to nd a forgetting solution.
Theorem 2. For any ALCOIH +(O; u)-ontology O and any sig(O), the method always terminates and returns a set O0 of clauses. If O0 does not contain any -symbols, the method was successful. O0 is then a solution of forgetting the symbols in from O. If neither O nor O0 uses , O0 is -equivalent to O in ALCOIH(O; u). Otherwise, it is -equivalent to O in ALCOIH +(O; u). 4
We rst present the calculus for forgetting of concept symbols in ontologies
expressible in ALCOIH +(O; u). The calculus extends the calculus of [
N ; C1 t A; : : : ; Cn t A
A
N:C1t:::t:Cn provided: (i) A does not occur in the Ci, and (ii) N is negative w.r.t. A. PurifyC;p NA N is positive w.r.t. A. N>
N ; C1[A] t A; : : : ; Cn[A] t A
A
N X:(:C1t:::t:Cn)[X] provided: (i) the Ci are negative w.r.t.
A, and (ii) N is negative w.r.t. A.
PurifyC;n
NA N is negative w.r.t. A.
N:> Fig. 2. Forgetting concept symbol A in a set N of ALCOIH De nition 2 (A-Reduced Form). Suppose A is a concept symbol. A clause is in A-reduced form if it is negative w.r.t. A, or it has the form A t C, where C is an ALCOIH +(O; u)-concept that (Case I) does not contain any occurrence of A, or (Case II) is negative w.r.t. A. A set N of clauses is in A-reduced form if every A-clause in N is in A-reduced form.
The Ackermann and Purify rules, given in Figure 2, are the forgetting rules that lead to the elimination of concept symbols in ALCOIH +(O; u)-clauses. Suppose A 2 NC is the pivot and C is a concept expression, then NCA denotes the set obtained from N by replacing every occurrence of A by C. We refer to the clauses of the form Ci t A (1 i n) as positive premises of the rule. The Ackermann rule is applicable (to forget A in N ) only if N is in A-reduced form. Theorem 3 (Ackermann Lemma for Concept Forgetting). Let I be an arbitrary ALCOIH +(O; u)-interpretation. For A the pivot, when the NonCyclic AckermannC rule is applicable, the conclusion of the rule is true in I i for some interpretation I0 A-equivalent to I, the premises are true in I0. The same is true for the Cyclic AckermannC rule and the PurifyC;p(n) rules.
This states that eliminating the pivot symbol in NC with the AckermannC and PurifyC rules preserves equivalence up to the pivot. Given the pivot A and a set N of clauses in A-reduced form, the Non-Cyclic Ackermann rule is applied when A does not occur in the Ci (1 i n) of the positive premises. The Cyclic Ackermann rule is applied when A occurs in the Ci but only negatively, e.g., A occurs in a cyclic clause :8r:A t A. Fixpoints are introduced in this case in order to facilitate nite representation of the result, where every occurrence of A in :C1 t : : : t :Cn is replaced by X, a fresh concept variable, and every occurrence of A in N is replaced by X:(:C1 t : : : t :Cn)[X]. The Purify rule is applied whenever A is pure in N , i.e., N is positive (or negative) w.r.t. A.
N ; C t :(D1 t : : : t Dn) N ; C t :D1; : : : ; C t :Dn provided: A has positive occurrences in :(D1 t : : : t Dn).
SkolemizationO
N ; C t :8O:D N ; :b t :D t 8O:C provided: (i) b is a fresh nominal, and (ii) A occurs positively in :8O:D. Concept Surfacing
N ; C t 8R:D N ; (8R :C) t D provided: (i) A does not occur positively in C, and (ii) A occurs positively in 8R:D.
N ; :a t C1 t : : : t Cn N ; :a t C1 j : : : j :a t Cn provided: A occurs positively in C1 t : : : t Cn.
SkolemizationR
N ; :a t :8R:C N ; :a t :8R::b; :b t :C provided: (i) b is a fresh nominal, and (ii) A occurs positively in :8R:C.
Sign Switching
N (N:AA):A:A provided: (i) Sign Switching has not been performed on A, and (ii) N is closed to other rewrite rules.
Clauses are usually not given in reduced form initially. Figure 3 lists the
set of rewrite rules for nding the A-reduced form of a set of clauses for A the
pivot. The Case Splitting rule splits the derivation into several branches. The
intermediate forgetting solution returned at the end of a symbol elimination
round is the disjunction of the solutions of each branch in the derivation. New
compared to [
The main contribution of this paper is a calculus for forgetting of role symbols in ontologies expressible in ALCOIH +(O; u). Since role symbols occur in the TBox and the RBox, both need to be processed when role symbols are forgotten. De nition 3 (r-Reduced Form). Suppose r is a role symbol. A clause is in r-reduced form if it has the form C t 8r:D or C t :8(r u Q):D, where C and D are ALCOIH +(O; u)-concepts that do not contain any occurrence of r and Q is an ALCOIH +(O; u) role expression that does not contain any occurrence of r; or it has the form :S t r or S t :r, where S 2 fs; s g for s (6= r) a role symbol. A set N of clauses is in r-reduced form if every r-clause in N is in r-reduced form.
As in concept forgetting, the pivot-clauses are rst transformed into
pivotreduced form, so that the Ackermann rule for role forgetting becomes applicable.
Finding pivot-reduced form of a clause is not always possible unless de ner
symbols are introduced. De ner symbols are specialised concept symbols that do not
occur in the present ontology [
N ; C t 8r :D N ; D t 8r:C provided: r is a role symbol that does not occur in C and D.
N ; :S t r N ; S t :r N ; :S t r N ; S t :r provided: S is either a role symbol or an inverted role symbol.
Since the underlying language accommodates role inverse, the calculus includes two Role Surfacing rules, shown in Figure 4, in order to reformulate expressions without occurrences of inverses of the pivot in the respective TBox and RBox clauses, so that the other rules in the calculus do not need to cater for role inverse. The Role Surfacing rules are applied after de ner symbols have been introduced, and before the application of the forgetting rules.
PT+ PR+ z }| { z }| { N ; C1 t :8r u Q1:D1; : : : ; Ck t :8r u Qk:Dk ; :T 1 t r; : : : ; :T u t r ;
PT PR C1 t 8r:D1; : :}:|; Cm t 8r:Dm{; z:r t S1; :}:|: ; :r t Sn z { N ; T -BlockH(1; m); : : : ; T -BlockH(k; m) ;
R-BlockC(1; m); : : : ; R-BlockC(u; m) ; R-BlockR(1; n); : : : ; R-BlockR(u; n) ? T -BlockH(j; m) is the set (1 fCj t CY t :8Hj:(Dj t DY )jY j
k): [m]g, where CY = 8<:F> Ci iofthYer=wi;se, :i2Y
DY = <8:F> :Di iofthYer=wi;se, and :i2Y Hj = S1 u : : : u Sn u Qj if PR 6= ;, otherwise (PR = ;) Hj = O u Qj. ? R-BlockC(l; m) is the set fC1 lt 8T l:D1; : : : ; Cm t 8T l:Dmg (1 l u). ? R-BlockR(l; n) is the set f:T t S1; : : : ; :T l t Sng (1 l u).
provided: r is the pivot that does not occur in N , and N is in r-reduced form. PurifyR
Nr provided: N is positive (negative) w.r.t. r, where r is the pivot. N(:)O
Fig. 5. Forgetting role symbol r in a set N of ALCOIH Theorem 4 (Ackermann Lemma for Role Forgetting). Let I be an arbitrary ALCOIH +(O; u)-interpretation. For r the pivot, when the AckermannR or PurifyR rule in Figure 5 is applicable then the conclusion of the rule is true in I i for some interpretation I0 r-equivalent to I, the premises are true in I0.
Given a set N of clauses with r 2 NR being the pivot, once N has been transformed into r-reduced form, we apply the AckermannR rule and the PurifyR rule given in Figure 5 to eliminate r. By de nition, clauses in r-reduced form have four distinct forms. We refer to the clauses of the form Cj t :8r u Qj :Dj (1 j k) as positive TBox premises (denoted by PT+) and clauses of the form Ci t 8r:Di (1 i m) as negative TBox premises (denoted by PT ) of the rule. We refer to the clauses of the form :T l t r (1 l n) as positive RBox premises (denoted by PR+) and clauses of the form :r t Si (1 i u) as negative RBox premises (denoted by PR) of the rule. Since it is possible for r to occur in any of these premises (and the premises do not necessarily wholly exist), there are several situations where the AckermannR rule is applicable. For di erent PT and PR the AckermannR rule is performed as follows. are Ccoamsebi(nIe)d: IwfiPthT e6=ve;ryancdlaPusRe 6 =in;P,tT+he(nifPPTT+an6=d ;P)Ran(tdheevneergyatcivlaeupsereimnisPesR+) (if PR+ 6= ;), which leads to the elimination of r, and a forgetting solution O0 where r 62 sig(O0). Speci cally, combining PT and PR with one of the positive TBox premises (PT+) leads to a set of TBox clauses (denoted by T -BlockH(j; m)), where Hj is the conjunction of Qj and the Si (1 i n) occurring in PR (Hj := Qj u S1 u : : : u Sn), m denotes the number of negative TBox premises (jPT j), and j refers to the positive TBox premise with which PT and PR combine. [m] denotes the set f1; : : : ; mg, and fCj t CY t :8Hj :(Dj t DY )jY [m]g is the set that contains all clauses corresponding to every assignment of Y .
The following example illustrates this case. Given = frg and a set N of
clauses in r-reduced form with +P,Tco=mbfiAn1intg 8PrT:B1; A2 t 8r:B2g, PR = f:r t
s1; :r t s2 g, and A t :8r:B 2 PT and PR with A t :8r:B leads
to T -BlockH(j; m) = fA t CY t :8(s1 u s2 ):(B t DY )jY [
|C{zY} |D{zY} 2. A t A1 t:8(s1 u s2 ):(B t :B1)
|C{zY} |D{zY} 3. A t A2 t:8(s1 u s2 ):(B t :B2) |C{zY} |D{zY} when Y = ; when Y = f1g when Y = f2g 4. A t A1 t A2 t:8(s1 u s2 ):(B t :B1 t :B2) | C{zY } | D{zY } when Y = f1; 2g
Combining PT with one of the positive RBox premises leads to a set of TBox clauses (denoted by R-BlockC(l; m)), where m are the negative TBox premises and l refers to the positive RBox premise with which PT combines; combining PR with one of the positive RBox premises leads to a set of RBox clauses (denoted by R-BlockR(l; n)), where n are the negative RBox premises and l refers to the positive RBox premise with which PR combines.
Case (II): If PT 6= ; and PR = ;, then combining PT with one of the positive TBox premises leads to the same result as in Case (I) (T -BlockH(j; m)), only that a top role (O) replaces S1 u : : : u Sn, i.e., H = O. Combining PT with one of the positive RBox premises leads to the same result as in Case (I), i.e., R-BlockC(l; m) (1 l u). Case (III): If PT = ; and PR 6= ;, then combining pPrRemwisitehs loenaedsotfotthheepsoasmiteivreesTulBtsoxaspirnemCaissees(Ia)n,di.ew.,itTh-BonloecokfHt(hje; mpo)s(i1tive jRBokx) and R-BlockR(l; n) (1 l u) respectively. Case (IV): The case PT = ; and PR = ; can be seen as an instance of the case where r is pure. The pivot is eliminated using the PurifyR rule in this case. +
The pivot is forgotten in the ontology once every positive premise (in PT and PR+) has been combined with PT (if PT 6= ;) and PR (if PR 6= ;). Given a set of clauses in pivot-reduced form with m negative TBox premises (jPT j = m), + n negative RBox premises (jPRj = n), k positive TBox premises (jPT j = k), and u positive RBox premises (jPR+j = u), combining PT and PR with all positive TBox premises yields a set of k 2m clauses (exponential growth); combining PT and PR with all positive RBox premises yields a set of um + un clauses (polynomial growth). The size of the forgetting solution therefore depends largely on the number of the negative TBox premises (m).
We implemented a prototype of the forgetting method in Java using the OWLAPI1, and conducted a number of experiments on real-world ontologies that are taken from the NCBO BioPortal2 and Oxford Ontology3 repositories to evaluate the practicality of the method. The experiments were run on a desktop with an Intelr Coretm i7-4790 processor, and four cores running at up to 3.60 GHz and 8 GB of DDR3-1600 MHz RAM. The ontologies used for our evaluation were restricted to ALCOIH(O; u)-fragments, and any sub-concepts beyond the scope of ALCOIH(O; u) were replaced by >. Consequently, 180 and 200 ontologies of various sizes were selected from the NCBO BioPortal and Oxford Ontology Library, respectively. We repeated the experiments 100 times on each ontology and averaged the results to verify the accuracy of our ndings.
To t in with real-world applications such as computing logical di erence between ontologies and predicate hiding, where forgetting a small number of symbols is in demand, we set up a series of experiments where we forgot 10% and 30% of concept and role symbols that were randomly selected in each ontology. There are also situations where it would be of interest to forget a large number of symbols; ontology reuse is such an example [12{15]. We therefore set up a series of experiments with a large number of concept and role symbols to be forgotten (80% of the concept symbols and 50% of the role symbols in the initial signature). The heuristic for determining the order of eliminating concept and roles symbols ( C and R) was also tested. The -symbols were eliminated in the order as returned by the OWL-API function that gets all concept symbols in the ontology, when the heuristic was not applied. We started with the evaluation of concept symbol forgetting, where a timeout of 15 minutes was imposed.
Input Setting Results Setting Results (B3Oo5in4onPtAAoolxvoriggot.ym)als 16286|000Σ0((|A(318(00%v0%%%g))).) 33337777C Ti31m72194313098e........38239436(39614727s600602e69c.) T126i54100m3.......43.64100e8%%%%%%%o%ut Su11999899c00261348c00......e..677343s00%%%%%%s%%R. 1186F770037......i38..2800x32%%%%%%p%%. 214|02ΣA(((|1v530(00g%%%%.)))) 33337777R 2D1311111....3e0630091fi////oooonoooonnnnennnntttrtttttoooooooio...n..... 11T2200882200i......m..63115395510262e1684104736(ssssssssseeeeeeeeeccccccc.....cc..).. S11u11111100c000000c00000000e........s00000000s%%%%%%%%R. C112211l4455a6644..u......11005555s%%%%%%%%e↑ (8o7On5xAAfoxvrigod.m)s 11230846884(((1A38000v%%%g))). 33337777 21419248191617406326........473418274969057852126203 113191398747.....5...5052055%%%%%%%% 7899877698878981........35505555%%%%%%%% 11112100031469........55385550%%%%%%%% 12555A(((1v35000g%%%.))) 33337777 33235223....09589477////oooooooonnnnnnnnttttttttoooooooo........ 11113389338746........013359127893391927697872sssssssseeeeeeeecccccc..cc...... 111111110000000000000000........00000000%%%%%%%% 1177226644......99..229966%%%%%%%%
The results (of forgetting only concept symbols) obtained from forgetting 10%, 30% and 80% of the concept symbols from the respective BioPortal and Oxford ontologies are shown in Figure 6, which is quite revealing in several ways. The most notable observation to emerge from the results is that, with the 1 http://owlapi.sourceforge.net/ 2 http://bioportal.bioontology.org/ 3 http://www.cs.ox.ac.uk/isg/ontologies/ heuristically determined elimination order (indicated by 3), our implementation was successful (forgot all -symbols) in 96.7% of the BioPortal-cases within a limited period of time, with 8.3% of them using xpoints in the result. In the case of 10% of the concept symbols speci ed to be forgotten, the success rate rose to 100%. Even without the heuristic (7), the forgetting solution could be found by our implementation in 92.6% of the cases. Since the Oxford ontologies were more than twice as large as the BioPortal ontologies and a larger set of concept symbols was speci ed to be forgotten, a reduction in the performance was expected. The implementation was unable to compute the solution in 11.5% of the Oxford-cases, and in 13.8% of the solved cases xpoints occurred in the result. The use of the heuristic boosted the overall success rate by 4% and 9.2%, and improved the time e ciency by 124.1% and 136.9% in the BioPortal and Oxford ontologies, respectively.
We also evaluated the performance of forgetting di erent number of role symbols with the same ontologies used for the evaluation of concept forgetting (using a timeout of 5 minutes). The results (of forgetting only role symbols) are shown in Figure 7, from which it can be seen that our method successfully eliminated the role symbols in in all cases. The time used for forgetting role symbols, as expected, was signi cantly longer than forgetting the same number of concept symbols, despite the 100% success rate. Because of the nature of the AckermannR rule, role symbol forgetting leads to growth of clauses in the forgetting solution, which was however modest (Clause ") compared to the theoretical worst case. De ner symbols were introduced only in a small proportion of the ontologies to help conversion to reduced form. This indicates that most clauses in the ontologies were at. By m=onto, we mean that there were m de ner symbols introduced in each ontology, and by n onto we mean that n ontologies out of the total introduced the de ner symbol. 7
In this paper we have developed a method of concept and role forgetting for expressive description logics with nominals, non-empty TBoxes and ABoxes, and a rich language for expressing properties of roles. The method can handle role inclusion statements, role conjunctions and role inverses. This is extremely useful from the perspective of ontology engineering as it increases the arsenal of tools available to create restricted views of ontologies. The results of the evaluation on real-world ontologies has shown that often xpoints and role conjunction are not needed to express forgetting solutions, and overall the performance results are very positive.
Currently beyond the scope of our method are forgetting transitive roles. We
can extend the method to eliminate concept symbols in the presence of transitive
roles, but the interaction between transitive roles and role hierarchy inclusion
statements can lead to results where it is not clear how to represent them nitely.
The problem of extending the method to handle number restrictions remains
completely open; possibly the techniques of [