Operations on Knowledge Bases

From a purely syntactical point of view, a knowledge base is essentially a set of axioms and assertions. Therefore, one can apply standard set operations such as creating the union, intersection, or difference of two knowledge bases K1, K2. This allows then to determine – at a syntactical level – whether they are disjoint, have parts in common, or are actually the same. On the semantical level, however, the result and the practicabil- ity of these operations depends on the expressivity and the interpretations (models) of the single knowledge bases K_i. Obviously, two knowledge bases K₁ and K₂ that are satisfiable when viewed in isolation need not be satisfiable when building their union

K1∪ K2 because the may model contradicting knowledge. On the other hand, differ- enceK₁\ K₂ and intersection K₁∩ K₂ are uncritical regarding satisfiability for mono- tonic DLs because the result cannot be “larger” than the arguments (i.e., the result is a subset ofK1,K2anyway). Example 3.2and the next paragraph illustrates this.

Example 3.2

Consider the following KBs where the semicolon “;” delimits the TBox from the ABox and the comma “,” delimits axioms and assertions inside the TBox/ABox.

K₁ ={Person, Female, Woman≡PersonuFemale; Woman(CURIE)},

K₂ ={Person, Female, Woman≡PersonuFemale, Man≡Personu ¬Woman; Man(EINSTEIN)}.

The union, intersection, and difference are then

K₁∪ K₂={Person, Female, Woman≡PersonuFemale,

Man ≡Personu ¬Woman; Woman(CURIE), Man(EINSTEIN)},

K₁∩ K₂={Person, Female, Woman≡PersonuFemale},

K1\ K2={Woman(CURIE)}.

Example 3.2 is an innocuous one. None of the operations yields an inconsistent knowledge base, i.e., all concepts, axioms, and ABox assertions are still satisfiable in every resulting knowledge base; observe that they are also satisfiable in isolation. This is due to the fact that K1 and K2 do not model contradicting knowledge. Adding, for instance, Woman(CALLIOPE) to K₁ and Man(CALLIOPE) to K₂ would, however, re- sult in an inconsistency in the union K₁∪ K2 because the concepts Man and Woman were indirectly described as disjoint. That is, there can be no model in which the indi- vidual referred to by the name CALLIOPE14is both a man and a woman. Observe that

K₁and K2remain consistent when viewed in isolation. It is beyond the application of the union operator to knowledge bases to ensure that the resulting knowledge base is still satisfiable. Consequently, a consistency check should be done before committing an operation if an application requires a consistent knowledge base at any time.

Apart from dealing with knowledge bases by means of set operators, almost all ap- plications not only require the ability to read their content but to modify or edit them in order to accommodate new, revise existing, or retract obsolete knowledge [FMK+08]. In the AI research field this is well known as the belief revision and belief update prob- lem [Pep08]. In short, belief revision refers to the process of how to modify a knowl- edge base “in the light of new information that was previously inaccessible” and where the initial knowledge base needs to be modified because it is incomplete or parts have become obsolete or incorrect in light of the new information. In contrast, the belief up- date problem refers to the process of how to modify a knowledge base that needs to be brought up-to-date to changes in a dynamic domain (in the world) because it is out-of- date after changes have occurred in the domain. The update problem gets into focus in

Chapter 4when we discuss how to represent the effects of service and service operation invocations. In the context ofChapter 6, however, updating a KB is viewed from a data management point of view. More specifically, we will define the lower level storage layer of a knowledge base management system (KBMS) to provide update operations with direct semantics, meaning that operations directly add or delete axioms and asser- tions. To make this data-centric point of view more explicit, we classify belief revision and belief update as indirect update semantics and the latter as direct update semantics, see

Figure 3.1. In [HPSK06] the notion of edit semantics has been defined for ABoxes, which is equivalent to our notion of direct updates. We follow this definition but generalize it for the knowledge base as follows.

Definition 3.14 (Direct KB Update). Let L be a Description Logic and letK = T ∪ A be an L-knowledge base. Then, updating K by adding (deleting) a new (existing) L-syntactic

KB Storage Layer

KB Editing Layer

(e.g. Consistency Preservation, Belief Update)

U Uʹ reads Direct update semantics Indirect update semantics

Figure 3.1: Distinction between high level knowledge base updates and direct updates at the level of the storage layer.

instance ψ, written K +ψ (K −ψ), results in an updated KB K0 such that K0 = K ∪ {ψ}

(K0 = K \ {ψ}).

A direct KB update (or direct update for short) is a finite and non-empty set of these additions and deletes. Given K and a direct update U, K0 is the result of updating K with U, written

K =⇒U K0, obtained by applying all adds and deletes in U toK.

A direct update U is usually thought to be applied in an atomic way: either none or all additions and deletes are applied. Obviously, allowing direct updates onK may affect its entailments (i.e., the implicit knowledge). Moreover, applying U may result in an inconsistentK0 if U adds axioms or assertions that either contradict with existing knowledge or among each other (e.g., if an add of ψ and ¬ψis in U). Practical KBMSs

might, therefore, be equipped with a knowledge base editing15 layer on top of the stor- age layer as depicted inFigure 3.1. Its purpose is to transform a belief revision or belief update represented by U0 at the higher level (i.e., an indirect update, which may con- tradict withK), into a direct update U such that (i) consistency at DL level is preserved and (ii) the desired update semantics as defined by the actual Belief Revision or Belief Update approach is achieved. In order to achieve this, the editing layer may need to interrogate the knowledge base (i.e., read-access it), which is indicated by the dashed arrow inFigure 3.1. Finally, we make the following observation.

Observation 3.1. Given a monotonic DLLsuch asS HOI N (D)orS ROI Q(D), anL-KB

K, and a delete-only direct update U withK =⇒_U K0, ifKis consistent, so isK0.

The reason is that every delete can only reduce explicit and implicit knowledge. In other words, only addition of new axioms or assertions can lead to inconsistency. This is not the case in general for non-monotonic logics; thus, deletes would need to be considered as well by the consistency preservation mechanism.

15_{The term refers to the field of modifying a knowledge base either to resolve inconsistencies or in} response to a change request [FMK+08].