Bisimulation-Based Comparisons for Interpretations in Description Logics

Bisimulation-Based Comparisons for Interpretations in Description Logics

Ali Rezaei Divroodi and Linh Anh Nguyen Institute of Informatics, University of Warsaw

Banacha 2, 02-097 Warsaw, Poland {rezaei,nguyen}@mimuw.edu.pl

Abstract. We study comparisons between interpretations in descrip- tion logics with respect to “logical consequences” of the form of semi- positive concepts (like semi-positive concept assertions). Such compar- isons are characterized by conditions similar to the ones of bisimulations.

The simplest among the considered logics is a variant of PDL (propo- sitional dynamic logic). The others extend that logic with inverse roles, nominals, quantified number restrictions, the universal role, and/or the concept constructor for expressing the local reflexivity of a role. The stud- ied problems are: preservation of semi-positive concepts with respect to comparisons, the Hennessy-Milner property for comparisons, and mini- mization of interpretations that preserves semi-positive concepts.

1 Introduction

Bisimulation is a natural notion of equivalence arose in modal logic [23–25] and state transition systems [20, 11]. It can be viewed as a binary relation associating state transition systems which behave in the same way in the sense that one system simulates the other and vice versa. Kripke models in modal logic are a special case of labeled state transition systems.

Bisimulations have widely been studied for various variants of modal logic like dynamic logic, temporal logic, hybrid logic and, in particular, also for description logics (DLs) [13, 6, 14, 21]. They have been used for analyzing the expressivity of a wide range of modal logics (see, e.g., [2] for details), for minimizing state transition systems, as well as for concept learning in DLs (e.g., [19, 22, 10, 5]).

Bisimilarity between two states is usually defined by three conditions (the states have the same label, each transition from one of the states can be simu- lated by a similar transition from the other, and vice versa). For bisimulation between two pointed-models, the initial states of the models are also required to be bisimilar. When converse is allowed, two additional conditions are required for bisimulation [2]. Bisimulation conditions for dealing with graded modalities were studied in [4, 3, 12]. In the field of hybrid logic, the bisimulation condition for dealing with nominals is well known (see, e.g., [1]). In DLs, such condi- tions are used for dealing with inverse roles, (quantified) number restrictions and nominals, respectively. There are also bisimulation conditions for dealing with individuals, the universal role and theSelf constructor in DLs [7, 22].

In modal logic, bisimulation invariance has the form: if two states are bisimi- lar then they satisfy the same set of formulas (i.e., all modal formulas are invari- ant w.r.t. bisimulation). For the converse, the Hennessy-Milner property states that, in finitely branching Kripke models, two states are bisimilar iff they sat- isfy the same set of formulas. This property can be generalized for non-finitely branching Kripke models (see, e.g., [14]).

Simulation is a notion with weaker conditions than bisimulation. It is only

“one way”, while bisimulation is “two way”. In the most common understanding, the “ways” are related with the “transitions” but not w.r.t. comparison between the sets of atomic formulas satisfied at the considered states. Such simulation preserves positive existential formulas (see, e.g., [2]).

What variant of bisimulation can be used to talk about preservation of pos- itive formulas, which may use both existential and universal modal operators?

Defining positive formulas to be the ones without⊥(falsity),¬(negation) and

→(implication), in [15] Nguyen gave a bisimulation-based comparison between Kripke models that preserves positive formulas in basic serial monomodal logics.

In [17] he extended the preservation result also for serial regular grammar log- ics and proved the corresponding Hennessy-Milner property. Such bisimulation- based comparison uses the conditions of bisimulation for “transitions” and com- pares the sets of atomic formulas satisfied at the considered states. Bisimulation- based comparison between Kripke models is worth studying, because it can be used for minimizing a Kripke model w.r.t. the set of logical consequences be- ing positive formulas. For example, after constructing a least Kripke model of a positive modal logic program in a serial modal logic [15, 17, 8], one can minimize it w.r.t. positive formulas to obtain a minimal Kripke model that characterizes the program w.r.t. positive consequences. Such minimization is also applicable to (non-serial) DLs [16, 18].

In this paper, we study bisimulation-based comparisons between interpreta- tions in DLs. The simplest among the considered logics is ALCreg, a variant of PDL (propositional dynamic logic). The others extend that logic with in- verse roles, nominals, quantified number restrictions, the universal role, and/or the concept constructor for expressing the local reflexivity of a role. The studied problems are: preservation of semi-positive concepts with respect to comparisons, the Hennessy-Milner property for comparisons, and minimization of interpreta- tions that preserves semi-positive concepts. The class of semi-positive concepts differs from the class of positive concepts in that, in the recursive definition, it allows also ⊥. This is involved with non-seriality.

Apart from [15, 17, 8], bisimulation-based comparisons for modal logics were studied also in [9] (and possibly other works). In [9] the notion is studied at an abstract level for coalgebraic modal logics under the name Λ-simulation, and the term “positive formula” is used instead of “semi-positive formula”. As mentioned before, the term “simulation” traditionally has another meaning, and in our opinion⊥should not be referred to as “positive”. At an abstract level, the work [9] does not have a result like a Hennessy-Milner property. In the current work, to guarantee a Hennessy-Milner property, roles in semi-positive concepts

have a specific syntax due to the presence of the test operator. The definition of semi-positive concepts itself in the current work is not trivial (e.g., we have that ifC is a semi-positive concept then≤n r.¬C is also a semi-positive concept).

Our results on preservation of semi-positive concepts and the Hennessy- Milner property w.r.t. comparisons may overlap to a certain degree with the known ones. However, our results on “characterizing bisimulation by semi- positive concepts” and “minimization preserving semi-positive concepts” are completely novel.

2 Notation and Semantics of Description Logics

Our languages use a finite set Σ_C ofconcept names(atomic concepts), a finite setΣRofrole names(atomic roles), and a finite setΣI ofindividual names. Let Σ=ΣC∪ΣR∪ΣI. We denote concept names by letters likeA andB, denote role names by letters likerands, and denote individual names by letters likea andb.

We consider some (additional)DL-featuresdenoted byI (inverse),O(nom- inal),Q(quantified number restriction),U (universal role), Self. Aset of DL- featuresis a set consisting of some or zero of these names.

LetΦbe any set of DL-features and letLstand forALCreg. The DL language LΦ allowsrolesandconceptsdefined inductively as follows:

– ifr∈Σ_R thenris a role ofL_Φ – ifA∈ΣC thenA is a concept ofLΦ

– ifR andS are roles ofLΦ andC is a concept ofLΦ then

• ε,R◦S ,RtS,R^∗ andC? are roles ofLΦ

• >,⊥,¬C,CtD, CuD,∃R.C and∀R.C are concepts ofLΦ

• ifI∈ΦthenR⁻ is a role ofLΦ

• ifO∈Φanda∈ΣI then{a}is a concept of LΦ

• ifQ∈Φ,r∈Σ_R andnis a natural number then≥n r.Cand≤n r.C are concepts ofLΦ

• if{Q, I} ⊆Φ,r∈Σ_R andnis a natural number then≥n r⁻.C and≤n r⁻.C are concepts ofL_Φ

• ifU ∈ΦthenU is a role ofLΦ

• ifSelf∈Φandr∈ΣR then∃r.Selfis a concept of LΦ.

We use letters likeRand S to denote arbitrary roles, and use letters likeC andD to denote arbitrary concepts. We refer to elements ofΣRalso asatomic roles. Let Σ_R^± = Σ_R∪ {r⁻ | r ∈ Σ_R}. From now on, by basic roles we refer to elements of Σ_R^± if the considered language allows inverse roles, and refer to elements ofΣ_Rotherwise. In general, the language decides whether inverse roles are allowed in the considered context.

Aninterpretation I = h∆^I,·^Ii consists of a non-empty set ∆^I, called the domain of I, and a function·^I, called the interpretation function of I, which maps every concept nameAto a subsetA^Iof∆^I, maps every role namerto a binary relation r^I on ∆^I, and maps every individual nameato an element a^I

(R◦S)^I =R^I◦S^I (RtS)^I =R^I∪S^I (R^∗)^I = (R^I)^∗

(C?)^I ={hx, xi |C^I(x)}

ε^I ={hx, xi |x∈∆^I} U^I =∆^I×∆^I (R⁻)^I = (R^I)⁻¹

>^I =∆^I

⊥^I =∅ (¬C)^I =∆^I\C^I (CtD)^I =C^I∪D^I (CuD)^I =C^I∩D^I

{a}^I ={a^I}

(∃r.Self)^I ={x∈∆^I |r^I(x, x)}

(∃R.C)^I={x∈∆^I | ∃y[R^I(x, y) andC^I(y)]

(∀R.C)^I={x∈∆^I | ∀y[R^I(x, y) impliesC^I(y)]}

(≥n R.C)^I={x∈∆^I |#{y|R^I(x, y) andC^I(y)} ≥n}

(≤n R.C)^I={x∈∆^I |#{y|R^I(x, y) andC^I(y)} ≤n}

Fig. 1.Interpretation of complex roles and complex concepts.

of∆^I. The interpretation function·^I is extended to complex roles and complex concepts as shown in Figure 1, where #Γ stands for the cardinality of the set Γ. We writeC^I(x) to denotex∈C^I, and writeR^I(x, y) to denotehx, yi ∈R^I. An interpretationI is said to beserialinL_Φ if, for every basic roleRofL_Φ and everyx∈∆^I, there existsy∈∆^I such thathx, yi ∈R^I.

We say that a role R is in the converse normal form(CNF) if the inverse constructor is applied in R only to role names and the role U is not under the scope of any other role constructor. Since every role can be translated to an equivalent role in CNF,¹ in this paper we assume that roles are presented in the CNF.

3 Positive and Semi-Positive Concepts

LetL^pos_Φ be the smallest set of concepts andL^pos_Φ,∃,L^pos_Φ,∀ be the smallest sets of roles defined recursively as follows:

– ifr∈ΣR thenris a role ofL^pos_Φ,∃andL^pos_Φ,∀,

– ifI∈Φandr∈Σ_R thenr⁻ is a role ofL^pos_Φ,∃andL^pos_Φ,∀, – ifR andS are roles ofL^pos_Φ,∃andC is a concept ofL^pos_Φ

thenε,R◦S ,RtS, R^∗ andC? are roles ofL^pos_Φ,∃, – ifR andS are roles ofL^pos_Φ,∀andC is a concept ofL^pos_Φ

thenε,R◦S ,RtS, R^∗ and (¬C)? are roles ofL^pos_Φ,∀, – ifA∈ΣC thenA is a concept ofL^pos_Φ ,

– ifO∈Φanda∈ΣI then{a} is a concept ofL^pos_Φ ,

1 For example, ((rts⁻)◦r^∗)⁻= (r⁻)^∗◦(r⁻ts).

– ifSelf∈Φandr∈Σ_R then∃r.Selfis a concept ofL^pos_Φ ,

– ifC is a concept ofL^pos_Φ ,R is a role ofL^pos_Φ,∃ andS is a role ofL^pos_Φ,∀ then

• >,CtD,CuD,∃R.C and∀S.C are concepts ofL^pos_Φ ,

• ifQ∈Φ,r∈Σ_R andnis a natural number then≥n r.Cand≤n r.(¬C) are concepts ofL^pos_Φ ,

• if{Q, I} ⊆Φ,r∈Σ_R andnis a natural number then≥n r⁻.C and≤n r⁻.(¬C) are concepts ofL^pos_Φ ,

• ifU ∈Φthen∀U.C and∃U.C are concepts ofL^pos_Φ .

A concept of L^pos_Φ is called a positive concept of LΦ. We introduce both L^pos_Φ,∀ and L^pos_Φ,∃ due to the test constructor of roles. The concepts ∃(A?).B and

∀((¬A)?).B are positive concepts; they are equivalent to AuB and AtB, respectively. That the concept≤n R.(¬A) is positive should not be a surprise, as∀R.Ais equivalent to≤0R.(¬A).

LetL^sp_Φ be the smallest set of concepts and L^sp_Φ,∃, L^sp_Φ,∀ be the smallest sets of roles defined analogously to the case ofL^pos_Φ ,L^pos_Φ,∃,L^pos_Φ,∀except that⊥is also allowed as a concept ofL^sp_Φ. We call concepts ofL^sp_Φ semi-positive conceptsofLΦ.

4 Bisimulation-Based Comparisons for Interpretations

LetIandI⁰be interpretations. A binary relationZ⊆∆^I×∆^I0 is called anLΦ- comparisonbetweenI andI⁰ if the following conditions hold for everya∈ΣI, A∈ΣC,r∈ΣR,x, y∈∆^I, x⁰, y⁰∈∆^I0:

Z(a^I, a^I0) (1)

Z(x, x⁰)⇒[A^I(x)⇒A^I0(x⁰)] (2) [Z(x, x⁰)∧r^I(x, y)]⇒ ∃y⁰∈∆^I0[Z(y, y⁰)∧r^I0(x⁰, y⁰)] (3) [Z(x, x⁰)∧r^I0(x⁰, y⁰)]⇒ ∃y∈∆^I[Z(y, y⁰)∧r^I(x, y)], (4) ifI∈Φthen

[Z(x, x⁰)∧r^I(y, x)]⇒ ∃y⁰∈∆^I0[Z(y, y⁰)∧r^I0(y⁰, x⁰)] (5) [Z(x, x⁰)∧r^I0(y⁰, x⁰)]⇒ ∃y∈∆^I[Z(y, y⁰)∧r^I(y, x)], (6) ifO∈Φthen

Z(x, x⁰)⇒[x=a^I ⇒x⁰ =a^I0], (7) ifQ∈Φthen

ifZ(x, x⁰) holds then, for every role namer, there exists a bijection h:{y|r^I(x, y)} → {y⁰|r^I0(x⁰, y⁰)}such thath⊆Z, (8) if{Q, I} ⊆Φthen (additionally)

ifZ(x, x⁰) holds then, for every role namer, there exists a bijection h:{y|r^I(y, x)} → {y⁰|r^I0(y⁰, x⁰)}such thath⊆Z, (9)

ifU ∈Φthen

∀x∈∆^I∃x⁰∈∆^I0Z(x, x⁰) (10)

∀x⁰∈∆^I0∃x∈∆^IZ(x, x⁰), (11)

ifSelf∈Φthen

Z(x, x⁰)⇒[r^I(x, x)⇒r^I0(x⁰, x⁰)]. (12) For example, ifΦ={Q, I}then only the conditions (1)-(6), (8) and (9) (and all of them) are essential.

By (2’), (7’), (12’) we denote the conditions obtained respectively from (2), (7), (12) by replacing the second implication (⇒) by equivalence (⇔). If the conditions (2), (7), (12) are replaced by (2’), (7’), (12’) then the relation Z is called anLΦ-bisimulationbetweenI andI⁰ [7].

Proposition 4.1.

1. The relation{hx, xi |x∈∆^I}is an LΦ-comparison between I andI. 2. IfZ1 is an LΦ-comparison between I0 andI1, andZ2 is anLΦ-comparison

betweenI1 andI2, thenZ₁◦Z₂ is anLΦ-comparison between I0 andI2. 3. If Z is a set of LΦ-comparison between I and I⁰ then S

Z is also an LΦ- comparison betweenI andI⁰.

The proof of this proposition is straightforward.

Lemma 4.2. Let I and I⁰ be interpretations and Z be an LΦ-comparison be- tween I andI⁰. Then the following properties hold for every conceptC of L^sp_Φ, every roleR of L^sp_Φ,∃, every roleS of L^sp_Φ,∀, every x, y∈∆^I, every x⁰, y⁰ ∈∆^I0, and every a∈ I:

Z(x, x⁰)⇒[C^I(x)⇒C^I0(x⁰)] (13) [Z(x, x⁰)∧R^I(x, y)]⇒ ∃y⁰ ∈∆^I0[Z(y, y⁰)∧R^I0(x⁰, y⁰)] (14) [Z(x, x⁰)∧S^I0(x⁰, y⁰)]⇒ ∃y∈∆^I[Z(y, y⁰)∧S^I(x, y)]. (15) See the appendix for a proof of this lemma.

A concept C of LΦ is said to be preserved by LΦ-comparisons if, for any interpretations I, I⁰ and any LΦ-comparison Z between I and I⁰, if Z(x, x⁰) holds and x ∈ C^I then x⁰ ∈ C^I0. The following theorem follows immediately from the assertion (13) of Lemma 4.2.

Theorem 4.3. All concepts of L^sp_Φ are preserved byLΦ-comparisons.

Corollary 4.4. All concepts ofL^pos_Φ are preserved by LΦ-comparisons.

LetI andI⁰ be interpretations,x∈∆^I andx⁰∈∆^I0. Define that:

– xisequivalent tox⁰w.r.t. (concepts of)LΦ, denoted byx≡Φx⁰, if, for every conceptC ofLΦ,x∈C^I iffx⁰ ∈C^I0;

– xisless than or equal tox⁰ w.r.t. concepts ofL^sp_Φ (resp.L^pos_Φ ), denoted by x≤^sp_Φ x⁰ (resp.x≤^pos_Φ x⁰), if, for every conceptCofL^sp_Φ (resp.L^pos_Φ ),x∈C^I impliesx⁰∈C^I0;

– xisequivalent to x⁰ w.r.t. concepts ofL^sp_Φ, denoted byx≡^sp_Φ x⁰, if x≤^sp_Φ x⁰ andx⁰≤^sp_Φ x.

We say that an interpretationI is finitely branching(orimage-finite) w.r.t.

LΦif, for everyx∈∆^Iand every basic roleRofLΦ, the set{y∈∆^I |R^I(x, y)}

is finite. We say thatI isunreachable-objects-free(w.r.t.LΦ) if every element of

∆^Iis reachable from somea^I(witha∈ΣI) via a path consisting of edges being instances of basic roles (ofLΦ). The following theorem comes from our work [7].

Theorem 4.5 (The Hennessy-Milner Property). Let I and I⁰ be finitely branching interpretations (w.r.t. LΦ) such that, for every a∈ ΣI, a^I ≡Φ a^I0. Suppose that if U ∈Φ then either ΣI 6=∅ and both I,I⁰ are finite, or both I, I⁰ are unreachable-objects-free. Then, for every x∈∆^I andx⁰ ∈∆^I0,x≡Φ x⁰ iff there exists an LΦ-bisimulationZ between I andI⁰ such thatZ(x, x⁰)holds.

In particular, the relation{hx, x⁰i ∈∆^I×∆^I0 |x≡Φx⁰}is an LΦ-bisimulation between I andI⁰.

In the rest of this section we present theorems similar to the Hennessy-Milner property that are related toLΦ-comparisons and/or semi-positive concepts.

Theorem 4.6. Let I and I⁰ be finitely branching interpretations (w.r.t. LΦ) such that, for every a ∈ Σ_I, a^I ≤^sp_Φ a^I0. Suppose that if U ∈ Φ then either Σ_I 6=∅and bothI,I⁰are finite, or bothI,I⁰are unreachable-objects-free. Then, for every x∈∆^I and x⁰ ∈∆^I0, x≤^sp_Φ x⁰ iff there exists an L_Φ-comparison Z between I andI⁰ such thatZ(x, x⁰)holds. In particular, the relation {hx, x⁰i ∈

∆^I×∆^I0 |x≤^sp_Φ x⁰} is anLΦ-comparison betweenI andI⁰. See the appendix for a proof of this theorem.

Analyzing the proof of Theorem 4.6, it can be seen that, in the caseQ /∈Φ,⊥ is only used for showing that there existsy∈∆^I such thatr^I(x, y) holds when proving the condition (4). If I is a serial interpretation then that property is guaranteed. Therefore, we also have the following theorem, whose proof is very similar to the proof of Theorem 4.6.

Theorem 4.7. Let I and I⁰ be finitely branching interpretations (w.r.t. LΦ) such that I is serial and, for every a∈Σ_I,a^I ≤^pos_Φ a^I0. SupposeQ /∈Φand if U ∈Φthen eitherΣ_I 6=∅and bothI,I⁰ are finite, or bothI,I⁰ are unreachable- objects-free. Then, for everyx∈∆^I andx⁰ ∈∆^I0,x≤^pos_Φ x⁰ iff there exists an L_Φ-comparison Z between I and I⁰ such that Z(x, x⁰) holds. In particular, the relation{hx, x⁰i ∈∆^I×∆^I0 |x≤^pos_Φ x⁰}is anL_Φ-comparison betweenI andI⁰.

5 Characterizing Bisimulation by Semi-Positive Concepts

In the case Q∈Φ, there is a closer relationship between semi-positive concepts andLΦ-bisimulation from the semantic point of view.

Theorem 5.1. Let I and I⁰ be finitely branching interpretations (w.r.t. L_Φ) such that, for every a∈Σ_I,a^I≡^sp_Φ a^I0. SupposeQ∈Φ and ifU ∈Φthen both I and I⁰ are unreachable-objects-free. Then, for every x ∈ ∆^I and x⁰ ∈ ∆^I0, x≡^sp_Φ x⁰iff there exists anL_Φ-bisimulationZ betweenIandI⁰such thatZ(x, x⁰) holds. In particular, the relation {hx, x⁰i ∈ ∆^I ×∆^I0 | x ≡^sp_Φ x⁰} is an LΦ- bisimulation between I andI⁰.

See the appendix for a proof of this theorem.

Corollary 5.2. Let I and I⁰ be finitely branching interpretations (w.r.t. LΦ) such that, for every a∈ΣI,a^I≡^sp_Φ a^I0. SupposeQ∈Φ and ifU ∈Φthen both I and I⁰ are unreachable-objects-free. Then, for every x ∈ ∆^I and x⁰ ∈ ∆^I0, x≡^sp_Φ x⁰ iff x≡Φx⁰.

This corollary follows from Theorems 5.1 and 4.5.

Example 5.3. We show that the assumption Q ∈ Φ of Theorem 5.1 is neces- sary. Let Φ = ∅, Σ_I = {a}, Σ_C = {A, B}, Σ_R = {r} and let I, I⁰ be the interpretations specified as follows.

– ∆^I = {u, v₀, v₁, v₂}, a^I =u, r^I = {hu, v₀i,hu, v₁i,hu, v₂i}, A^I ={v₁, v₂}, B^I={v2},

– ∆^I0 ={u, v0, v2},a^I0 =u,r^I0 ={hu, v0i,hu, v2i}andA^I0 =B^I0 ={v2}.

Notice that I⁰ is obtained from I by deleting v1. Observe that there are LΦ- comparisons between I and I⁰ as well as between I⁰ and I, but there is no LΦ-bisimulations betweenI andI⁰. In particular,a^I≡^sp_Φ a^I0, buta^I6≡Φa^I0.C

The point of the above example is that, whenQ /∈Φ, ifv0,v1,v2are pairwise differentr-successors ofu,v0≤^sp_Φ v1andv1≤^sp_Φ v2 then the edgehu, v1i ∈r^I is not essential for the semantics of semi-positive concepts. Also note that, when Q /∈ Φ, if v and v⁰ are different r-successors of u such that v ≡^sp_Φ v⁰ then the edgehu, v⁰i ∈r^I is not essential for the semantics of semi-positive concepts.

SupposeQ /∈Φand letI be a finitely branching interpretation. We say that I isL^sp_Φ-tidyif it is unreachable-objects-free and, for everyx, y, y⁰, y⁰⁰∈∆^I and every basic roleR ofLΦ,

– if{hx, yi,hx, y⁰i} ⊆R^I andy≡^sp_Φ y⁰ theny=y⁰,

– if{hx, yi,hx, y⁰i,hx, y⁰⁰i} ⊆R^I,y≤^sp_Φ y⁰andy⁰≤^sp_Φ y⁰⁰theny=y⁰ory⁰ =y⁰⁰ or (Self∈Φandy⁰ =x).

Theorem 5.4. SupposeQ /∈Φ. Let I andI⁰ be finitely branching and L^sp_Φ-tidy interpretations such that, for everya∈ΣI,a^I≡^sp_Φ a^I0. Then, for everyx∈∆^I and x⁰ ∈∆^I0,x≡^sp_Φ x⁰ iff there exists anLΦ-bisimulation Z between I andI⁰ such that Z(x, x⁰) holds. In particular, the relation{hx, x⁰i ∈∆^I×∆^I0 |x≡^sp_Φ x⁰} is anLΦ-bisimulation between I andI⁰.

See the appendix for a proof of this theorem.

6 Auto-Bisimulation and Minimization

In this section, we recall some results of our manuscript [7], not published in [6].

AnLΦ-bisimulation betweenI and itself is called anLΦ-auto-bisimulation ofI. AnLΦ-auto-bisimulation ofIis said to be thelargestif it is larger than or equal to (⊇) any otherLΦ-auto-bisimulation ofI.

Proposition 6.1. For every interpretation I, the largestLΦ-auto-bisimulation

of I exists and is an equivalence relation. C

Given an interpretation I, by ∼_Φ,I we denote the largest L_Φ-auto- bisimulation of I, and by≡_Φ,I we denote the binary relation on ∆^I with the property that x≡_Φ,I x⁰ iffxisLΦ-equivalent tox⁰.

Theorem 6.2. For every finitely branching interpretationI,≡_Φ,Iis the largest L_Φ-auto-bisimulation of I (i.e. the relations ≡_Φ,I and∼_Φ,I coincide).

An interpretationI is said to be minimal among a class of interpretations if I belongs to that class and, for every other interpretation I⁰ of that class,

#∆^I ≤ #∆^I0 (the cardinality of ∆^I is less than or equal to the cardinality of∆^I0).

A concept assertion of L_Φ (resp. L^sp_Φ) is an expression of the form C(a), whereCis a concept ofLΦ(resp.L^sp_Φ). We say that an interpretationI satisfies a concept assertion C(a) ifa ∈ C^I. We say that I satisfies the same concept assertions ofLΦ(resp.L^sp_Φ) as an interpretationI⁰if, for every concept assertion C(a) ofLΦ (resp.L^sp_Φ),I satisfiesC(a) iffI⁰ satisfiesC(a).

6.1 The Case without Qand Self

Thequotient interpretationI/∼Φ,I ofI w.r.t.∼Φ,I is defined as usual:

– ∆^I/∼Φ,I = {[x]_∼Φ,I | x ∈ ∆^I}, where [x]_∼Φ,I is the abstract class of x w.r.t.∼Φ,I

– a^I/∼Φ,I = [a^I]_∼Φ,I, fora∈Σ_I

– A^I/∼Φ,I ={[x]_∼Φ,I |x∈A^I}, forA∈ΣC

– r^I/∼Φ,I ={h[x]∼Φ,I,[y]∼Φ,Ii | hx, yi ∈r^I}, forr∈ΣR.

Theorem 6.3. Suppose Φ⊆ {I, O, U} and letI be an unreachable-objects-free interpretation. IfI/_∼Φ,I is finitely branching then it is a minimal interpretation that satisfies the same concept assertions of LΦ asI.

6.2 The Case with Qand/or Self

For the case when Q ∈ Φ or Self ∈ Φ, in order to obtain a result similar to Theorem 6.3, we introduce QS-interpretations as follows.

AQS-interpretationis a tupleI=h∆^I,·^I,Q^I,S^Ii, where

– h∆^I,·^Iiis an interpretation,

– Q^I is a function that maps every basic role to a function∆^I×∆^I →Nsuch thatQ^I(R)(x, y)>0 iffhx, yi ∈R^I, whereNis the set of natural numbers, – S^I is a function that maps every role name to a subset of∆^I.

IfI is a QS-interpretation then we redefine (∃r.Self)^I={x∈∆^I|x∈S^I(r)}

(≥n R.C)^I={x∈∆^I|Σ{Q^I(R)(x, y)|C^I(y)} ≥n}

(≤n R.C)^I={x∈∆^I|Σ{Q^I(R)(x, y)|C^I(y)} ≤n}.

Other notions for interpretations remain unchanged for QS-interpretations.

ForIbeing an interpretation, thequotient QS-interpretationofIw.r.t.∼Φ,I, denoted byI/^QS_∼Φ,I, is the QS-interpretationI⁰=h∆^I0,·^I0,Q^I0,S^I0isuch that:

– h∆^I0,·^I0iis the quotient interpretation ofI w.r.t.∼_Φ,I – for every basic roleR and everyx, y∈∆^I,

Q^I0(R)([x]∼Φ,I,[y]∼Φ,I) = max

x⁰∈[x]∼Φ,I

#{y⁰∈[y]∼Φ,I | hx⁰, y⁰i ∈R^I} – for every role namer,

S^I

0(r) ={[x]∼Φ,I | hx, xi ∈r^I}.

Note that, in the case whenQ∈Φ, we have

Q^I0(R)([x]∼Φ,I,[y]∼Φ,I) = #{y⁰∈[y]∼Φ,I | hx, y⁰i ∈R^I}.

Here is a counterpart of Theorem 6.3, with no restrictions onΦ:

Theorem 6.4. Let I be an unreachable-objects-free interpretation. IfI/^QS_∼Φ,I is finitely branching then it is a minimal QS-interpretation that satisfies the same concept assertions of LΦ asI.

7 Minimization Preserving Semi-Positive Concepts

SupposeΦ⊆ {O, U,Self}and letI be a finitely branching interpretation such that it is also unreachable-objects-free whenU ∈Φ. ByTidy^sp_Φ(I) we denote the maximalL^sp_Φ-tidy sub-interpretation ofI obtained by modifying I as follows:

– For eachr∈ΣR, if {hx, yi,hx, y⁰i} ⊆r^I,y ≡^sp_Φ y⁰,y 6=y⁰ and y⁰ 6=xthen delete the pairhx, y⁰ifromr^I.

– For eachr∈ΣR, if{hx, yi,hx, y⁰i,hx, y⁰⁰i} ⊆r^I,y ≤^sp_Φ y⁰,y⁰ ≤^sp_Φ y⁰⁰, y 6≡^sp_Φ y⁰,y⁰6≡^sp_Φ y⁰⁰ and (Self∈/Φor y⁰6=x) then delete the pairhx, y⁰ifromr^I. – Delete from the domain ofI all elements not reachable from anya^I (with

a∈Σ_I) via a path consisting of edges being instances of basic roles of L_Φ. Lemma 7.1. Suppose Φ ⊆ {O, U,Self} and let I be a finitely branching in- terpretation such that it is also unreachable-objects-free when U ∈ Φ. Then Tidy^sp_Φ(I)satisfies the same concept assertions ofL^sp_Φ asI.

Proof. LetI⁰ =Tidy^sp_Φ(I) and letZ,Z⁰be the smallest binary relations such that the following conditions hold for everya∈Σ_I,r∈Σ_R,x, y∈∆^I,x⁰, y⁰ ∈∆^I0:

– Z(a^I, a^I) andZ⁰(a^I, a^I),

– Z(x, x⁰)∧r^I(x, y)∧r^I0(x⁰, y⁰)∧y≤^sp_Φ y⁰⇒Z(y, y⁰), – Z⁰(x⁰, x)∧r^I(x, y)∧r^I0(x⁰, y⁰)∧y⁰≤^sp_Φ y⇒Z⁰(y⁰, y).

It is easy to see thatZ is anLΦ-comparison betweenI andI⁰, andZ⁰ is an LΦ-comparison between I⁰ andI. Therefore, by Theorem 4.3,I⁰ and I satisfy

the same concept assertions of L^sp_Φ. C

Theorem 7.2. Suppose Φ⊆ {O, U,Self}. Let I0 and I₀⁰ be finitely branching interpretations such that they are also unreachable-objects-free when U ∈Φand they satisfy the same concept assertions ofL^sp_Φ. LetI=Tidy^sp_Φ(I0),I2=I/∼Φ,I

if Self∈/ Φ, and I2 =I/^QS_∼Φ,I if Self ∈Φ. Then I2 satisfies the same concept assertions of L^sp_Φ asI₀⁰ and #∆^I2 ≤#∆^I00.

Proof. LetI⁰ =Tidy^sp_Φ(I₀⁰). By Lemma 7.1, I andI⁰ satisfy the same concept assertions ofL^sp_Φ. Consequently, by Theorem 5.4, there exists anLΦ-bisimulation between I and I⁰. By Theorem 4.5, it follows that I and I⁰ satisfy the same concept assertions of LΦ. If Self ∈/ Φ then let I₂⁰ = I⁰/_∼Φ,I0, else let I₂⁰ = I⁰/^QS_∼

Φ,I0. By Theorems 6.3 and 6.4, #∆^I2 = #∆^I20. Since #∆^I20 ≤ #∆^I00, it

follows that #∆^I2 ≤#∆^I00. C

Theorem 7.3. Suppose Q ∈Φ. Let I and I⁰ be finitely branching interpreta- tions such that they are also unreachable-objects-free whenU ∈Φand they satisfy the same concept assertions of L^sp_Φ. Then I2 = I/^QS_∼Φ,I is a QS-interpretation that satisfies the same concept assertions of L^sp_Φ asI⁰ and#∆^I2≤#∆^I0. Proof. Let I₂⁰ = I⁰/^QS_∼

Φ,I0. By Theorem 5.1, there exists an LΦ-bisimulation between I and I⁰. By Theorem 4.5, it follows that I and I⁰ satisfy the same concept assertions ofLΦ. Hence, by Theorem 6.4, #∆^I2= #∆^I20. Since #∆^I02≤

#∆^I0, it follows that #∆^I2 ≤#∆^I0. C

Notice that minimization of interpretations that preserves semi-positive con- cepts for the case whenQ /∈ΦandI∈Φis not investigated in this section.

8 Conclusions

We have studied bisimulation-based comparisons between interpretations in a reasonably systematic way for a large class of useful description logics and ob- tained novel results on “characterizing bisimulation by semi-positive concepts”

and “minimization preserving semi-positive concepts”.

Acknowledgments.This work was supported by the Polish National Science Centre (NCN) under Grant No. 2011/01/B/ST6/02759.

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A Proofs

Proof of Lemma 4.2

We prove this lemma by induction on the structures ofC,Rand S.

Consider the assertion (14). SupposeZ(x, x⁰) andR^I(x, y) hold. By induction on the structure ofRwe prove that there existsy⁰ ∈∆^I0 such thatZ(y, y⁰) and R^I0(x⁰, y⁰) hold. The base case occurs whenR is a role name and the assertion for it follows from (3). The induction steps are given below.

– CaseR=εis trivial.

– CaseR=R₁◦R₂, whereR₁ andR₂ are roles ofL^sp_Φ,∃: We have that (R₁◦ R2)^I(x, y) holds. Hence, there existsz∈∆^Isuch thatR₁^I(x, z) andR^I₂(z, y) hold. By the inductive assumption of (14), there existsz⁰ ∈ ∆^I0 such that Z(z, z⁰) and R₁^I0(x⁰, z⁰) hold, and there exists y⁰ ∈ ∆^I0 such that Z(y, y⁰) and R₂^I0(z⁰, y⁰) hold. Since R^I₁⁰(x⁰, z⁰) and R₂^I0(z⁰, y⁰) hold, we have that (R1◦R2)^I0(x⁰, y⁰) holds, i.e.R^I0(x⁰, y⁰) holds.

– CaseR=R1tR2, where R1 andR2 are roles ofL^sp_Φ,∃, is trivial.

– CaseR=R^∗₁, whereR₁ is a role of L^sp_Φ,∃: Since R^I(x, y) holds, there exists x0, . . . , xk ∈∆^I such thatx0 =x, xk =y and, for 1 ≤i≤k, R^I₁(x_i−1, xi) holds. Letx⁰₀=x⁰. For each 1≤i≤k, sinceZ(x_i−1, x⁰_i−1) andR^I₁(x_i−1, xi) hold, by the inductive assumption of (14), there exists x⁰_i ∈ ∆^I such that Z(x_i, x⁰_i) and R^I₁⁰(x⁰_i−1, x⁰_i) hold. Hence,Z(x_k, x⁰_k) and (R^∗₁)^I0(x⁰₀, x⁰_k) hold.

Lety⁰ =x⁰_k. Thus,Z(y, y⁰) andR^I0(x⁰, y⁰) hold.

– CaseR = (D?), where D is a concept of L^sp_Φ: By the definition of (D?)^I, we have thatD^I(x) holds andx=y. By the inductive assumption of (13), D^I0(x⁰) holds, and thereforeR^I0(x⁰, x⁰) holds. By choosingy⁰=x⁰, we have thatZ(y, y⁰) andR^I0(x⁰, y⁰) hold.

– CaseI∈ΦandR=r⁻: The assertion for this case follows from (5).

The assertion (15) can be proved analogously as for (14) except for the case S = (¬C)?, where C is a concept of L^sp_Φ. The proof for this case is as follows.

Suppose Z(x, x⁰) andS^I0(x⁰, y⁰) hold. Thus, (¬C)^I0(x⁰) holds andx⁰ =y⁰. By the contrapositive of the inductive assumption of (13), it follows that (¬C)^I(x) holds. By choosing y=x,Z(y, y⁰) andS^I(x, y) hold.

Consider the assertion (13). SupposeZ(x, x⁰) andC^I(x) hold, whereC is a concept ofL^sp_Φ. We show that C^I0(x⁰) holds. The cases when C is of the form

>,⊥,A,DtD⁰ or DuD⁰ are trivial.

– CaseC=∃R.D, whereR is a role ofL^sp_Φ,∃ andDis a concept of L^sp_Φ: Since (∃R.D)^I(x) holds, there existsy ∈∆^I such that R^I(x, y) and D^I(y) hold.

By the inductive assumption of (14) (proved earlier), there existsy⁰ ∈∆^I0 such thatZ(y, y⁰) andR^I0(x⁰, y⁰) hold. By the inductive assumption of (13), D^I0(y⁰) holds. Therefore,C^I0(x⁰) holds.

– CaseC = ∀S.D, where S is a role of L^sp_Φ,∀ and D is a concept of L^sp_Φ: Let y⁰ be an arbitrary element of∆^I0 such thatS^I0(x⁰, y⁰) holds. We show that

D^I0(y⁰) holds. By the inductive assumption of (15) (proved earlier), there existsy∈∆^I such thatZ(y, y⁰) andS^I(x, y) hold. Since (∀S.D)^I(y) holds, it follows thatD^I(y) holds. Therefore, by the inductive assumption of (13), it follows thatD^I0(y⁰) holds.

– CaseO∈ΦandC={a}: Since{a}^I(x) holds, we have thatx=a^I. By the condition (7), it follows thatx⁰ =a^I0. HenceC^I0(x⁰) holds.

– CaseSelf ∈Φ and C =∃r.Self: Since (∃r.Self)^I(x) holds, we have that r^I(x, x) holds. By the condition (12), it follows thatr^I0(x⁰, x⁰) holds. Hence C^I0(x⁰) holds.

– CaseQ∈Φ and C = (≥n r.D), where D is a concept of L^sp_Φ: By the con- dition (8), there exists a bijectionh:{y |r^I(x, y)} → {y⁰|r^I0(x⁰, y⁰)}such thath⊆Z. Since (≥n r.D)^I(x) holds, there exist pairwise differenty1, . . . , yn ∈∆^I such thatr^I(x, yi) and D^I(yi) hold for every 1≤i≤n. For each 1≤i≤n, lety⁰_i=h(yi). Thus,Z(yi, y⁰_i) holds. By the inductive assumption of (13), it follows thatD^I0(y⁰_i) holds. Sincer^I0(x⁰, y⁰) and D^I0(y⁰_i) hold for 1 ≤ i ≤n, and yi 6=yj for 1 ≤ i 6=j ≤ n, it follows that (≥n r.D)^I0(x⁰) holds, which meansC^I0(x⁰) holds.

– Case{Q, I} ⊆ΦandC= (≥n r⁻¹.D), whereD is a concept ofL^sp_Φ, can be proved analogously to the above case.

– Case Q ∈ Φ and C = (≤ n r.(¬D)), where D is a concept of L^sp_Φ: For the sake of contradiction, supposeC^I0(x⁰) does not hold. Thus, (¬C)^I0(x⁰) holds, which means (≥(n+ 1)r.(¬D))^I0(x⁰) holds. By the condition (8), there exists a bijection h : {y | r^I(x, y)} → {y⁰ | r^I0(x⁰, y⁰)} such that h⊆ Z. Since (≥(n+ 1)r.(¬D))^I0(x⁰) holds, there exist pairwise different y⁰₁, . . . , y_n+1⁰ ∈ ∆^I0 such that r^I0(x⁰, y⁰_i) and (¬D)^I0(y_i⁰) hold for all 1 ≤ i≤n+ 1. For each 1≤i≤n+ 1, let yi =h⁻¹(y⁰_i). Sincehis a bijection, y₁, . . . , y_n+1are pairwise different, and by the definition ofh,r^I(x, y_i) holds for every 1≤i ≤n+ 1. For 1 ≤i ≤n+ 1, since (¬D)^I0(y⁰_i) holds, by the contrapositive of the inductive assumption of (13), it follows that (¬D)^I(y_i) holds. Thus, (¬C)^I(x) holds, which contradicts the assumption thatC^I(x) holds. Therefore,C^I0(x⁰) holds.

– Case{Q, I} ⊆ΦandC= (≤n r⁻¹.(¬D)), whereD is a concept ofL^sp_Φ, can be proved analogously to the above case.

– CaseU ∈Φ andC=∀U.D, whereD is a concept ofL^sp_Φ: Lety⁰ ∈∆^I0. By the condition (11), there existsy∈∆^I such thatZ(y, y⁰) holds. SinceC^I(x) holds, it follows thatD^I(y) holds. By the inductive assumption of (13), it follows thatD^I0(y⁰) holds. HenceC^I0(x⁰) holds.

– Case U ∈ Φ and C = ∃U.D, where D is a concept of L^sp_Φ: Since C^I(x) holds, there exists y ∈ ∆^I such that D^I(y) holds. By the condition (10), there existsy⁰ ∈∆^I0 such thatZ(y, y⁰) holds. By the inductive assumption of (13), it follows thatD^I0(y⁰) holds. HenceC^I0(x⁰) holds.

Proof of Theorem 4.6

First, supposeZis anLΦ-comparison betweenIandI⁰ such thatZ(x, x⁰) holds.

We show thatx≤^sp_Φ x⁰. LetC be an arbitrary concept of L^sp_Φ such that C^I(x)