Minimality Postulates for Ontology Revision

In many scenarios where the integration of information into a knowledge base (KB) leads to inconsistencies there is a need to change the KB minimally. In belief revision, relevance postulates meet the minimality requirement by restricting the elimination of KB elements to those that are relevant for the incoming information. This paper focuses on two minimality postulates in an ontology revision scenario in which conflicts are caused by ambiguous use of symbols: a relevance postulate and a generalized inclusion postulate which limits the creativity of the operators. Both postulates exploit the (satisfiably) equivalent representation of a first-order logic KB by its prime implicates, which, intuitively, represent the most atomic logical components of the KB. The paper shows that reinterpretation operators (which are ontology revision operators) fulfill both postulates. Introduction Not long after the seminal papers of Alchourrón, Gärdenfors and Makinson (AGM) (Alchourrón, Gärdenfors, and Makinson 1985) it was realized that belief-revision (BR) techniques could be fruitfully applied to different types of ontology change (OC) (Flouris et al. 2008) such as ontology evolution (Kharlamov, Zheleznyakov, and Calvanese 2013), ontology merge, ontology debugging etc. Most of the work exploiting BR for OC (Meyer, Lee, and Booth 2005; Flouris et al. 2006; Ribeiro and Wassermann 2009; Eschenbach and Özçep 2010; Özçep 2008) follows the dual approach of classical BR of, on the one hand, defining axiomatic specifications in the form of postulates and, on the other hand, constructing operators fulfilling them. In this paper, I propose postulates that are intended to specify a minimal change of a knowledge base (or more concretely an ontology) and show that so-called reinterpretation operators fulfill them. The intended revision scenario of this paper for which the minimality postulates are going to be developed can be described as follows. A receiver agent holds an ontology which is formally described by a knowledge base (KB) in an expressive formal language such as first-order logic (FOL). In particular, a KB is a finite set of sentences in FOL (or a fragment of it). She receives information from a sender agent with possibly different ontology and she wants to integrate the information into her ontology. I assume that both the sender’s KB and the receiver’s KB are well developed ontologies over the same application domain (e.g., ontologies for an online library system in universities, see examples below). Further it is assumed that the semantics of the same symbols in ontologies are strongly related. Nonetheless, there may be symbols that are used in differently by the sender and the receiver (ambiguity). The receiver is assumed to give priority to the sender’s meanings so the integration result will contain the trigger and result in a revision of the receiver’s ontology to preserve consistency. But, as the ontology of the receiver is assumed to be well developed, she is interested in changing her ontology only minimally, i.e., se wants to delete sentences of his KB and add additional sentences to it only as much as needed. In belief revision the theme of minimality is mainly discussed within the context of relevance postulates (Hansson 1993; Parikh 1999). But also inclusion postulates can be seen as contributions to a minimal-change specification as they limit the operators’s “creativity” by prescribing an upper bound to the result. In this paper, I start from these postulates for classical BR, argue why they are not proper minimality specifications for the intended revision scenario and formulate radically adapted versions that exploit the fine grained structure of ontologies by the notion of prime implicates. This is an extended version a paper that appeared at KR 16. Logical Preliminaries A first-order logic (FOL) vocabulary V consists of constants, predicate symbols and function symbols. For a FOL formula or set of formulas X let V(X) be the set of non-logical symbols occurring in X . I use the usual notions from Tarskian semantics based on FOL interpretations I. The set of sentences containing only non-logical symbols in the vocabulary V is denoted Sent(V). The set of sentences in Sent(V) following from a set of sentences X (over a perhaps larger vocabulary) is denoted by CnV(X). If two sets of FOL sentences X1, X2 are equivalent, I write X1 ⌘ X2. A non-logical symbol s 2 V properly occurs in a sentence ↵ 2 Sent(V) iff there are I1, I2 2 Int(V), s.t.: I1 and I2 differ only in the denotation of s and ↵I1 6= ↵I2 . Let P 2 V be an n-ary predicate symbol in V . It occurs syntactically positive (negative) in a FOL formula iff it occurs in the scope of an even (uneven) number of negations—assuming that only the propositional truth functions ^,_,¬ are used. For P 2 V(↵) we say that P occurs semantically positive in sentence ↵, pos(P,↵) for short, iff: For all I = ( I , ·I) and for subsets D1, D2 ✓ ( I)n one has: If D1 ✓ D2 and I[P 7!D1] |= ↵, then also I[P 7!D2] |= ↵. P occurs semantically negative in sentence ↵, neg(P,↵) for short, iff pos(P,¬↵). P occurs mixed in ↵, mix(P,↵) for short, iff it properly occurs in ↵ but neither pos(P,↵) nor neg(P,↵). We write pos0(P,↵) (resp. neg0(P,↵)) iff pos(P,↵) (resp. neg(P,↵)) or P does not occur syntactically in ↵. The dual remainder sets modulo ↵, B ` ↵, consists of inclusion maximal subsets X of B that are consistent with ↵, i.e., X 2 B ` ↵ iff X ✓ B, X [ {↵} is consistent and for all X̄ ✓ B with X ⇢ X̄ the set X̄ [ {↵} is not consistent. The notion of dual remainders is extended to arbitrary KBs B1 as second argument by defining B ` B1 as B ` V B1. Minimality in Belief Revision The AGM (Alchourrón, Gärdenfors, and Makinson 1985) postulates do not constrain the revision result in the interesting case of conflict between KB and trigger. In fact, the amnesic operator defined by B ⇤↵ = Cn(↵) fulfills all AGM postulates though it is not minimal. The relevance postulates of Hansson (Hansson 1993) and of Parikh (Parikh 1999) are two different possibilities that remedy the unwanted property of amnesic revision. These kinds of postulates constrain the revision result by an approximation from below in the sense that they say which set of sentences X have to be in the (set of consequences of the) revision result: X ✓ Cn(B⇤↵). The relevance postulate of Hansson (Hansson 1993) is formulated for arbitrary, i.e. not necessarily logically closed, sets of sentences B called belief bases. (Rel-H) If 2 B and / 2 B ⇤ ↵, then there is a set B0, such that: 1. B ⇤ ↵ ✓ B0 ✓ B [ {↵}; 2. B0 is consistent; 3. B0 [ { } is not consistent. (Rel-H) it is not an adequate postulate for the intended revision scenario. In this scenario, it is not individual sentences that cause a conflict but different uses of symbols in B and . And indeed, the reinterpretation based operators defined below do not fulfill this postulate. Example 1 Let B be a KB according to which media p1, p2, which are published in some proceedings, are articles: B = {Article(p1), Article(p2)}. The trigger ↵ = ¬Article(p1) stems from an agent with a different understanding of ‘article’ according to which only publications in journals are articles. An appropriate revision result B ⇤↵ would not only delete Article(p1) but also Article(p2); because the next time the sender sends a trigger containing Article negatively, namely ¬Article(p2), a conflict will occur. But this operator ⇤ does not fulfill (Rel-H). A different relevance postulate called (Rel-P), which is more symbol-oriented, was formulated by Parikh (Parikh 1999). Criterion (Rel-P) is formulated for propositional KBs and so cannot be used directly for FOL KBs as assumed in this paper. Hence we define a different relevance postulate. The relevance postulates cover only one aspect of minimality, but completely miss the other aspect of minimality which is to constrain the (consequences of the) revision result from above. That is, one has to prescribe a set X such that Cn(B ⇤ ↵) ✓ X . In belief base revision this aspect is handled by the so-called inclusion postulate. (Incl) B⇤↵ ✓ B [ ↵. But for the revision scenario of this paper, belief base revision is not the means of choice as its results depend on the syntactic representation of the belief bases. Adapted Minimality Postulates For the following I will assume that B is a predicate logical KB without the identity and function symbols. The new relevance postulate adapts Hansson’s relevance postulate (RelH). The main technical tool for the adaptation is the concept of a prime implicate, which roughly represents a most atomic component of the KB. While the notion of prime implicate is omnipresent for propositional logic (Armstrong et al. 1998) and has been exploited for the definitions of propositional revision operators (Bienvenu, Herzig, and Qi 2008; Zhuang, Pagnucco, and Meyer 2007), there is no real semantic notion of prime implicate for FOL and there is no approach that uses prime implicates in the postulates—except for (Özçep 2012) for propositional logic. The core idea of the new relevance theorem is this: A sentence entailed by B may be eliminated from the revision result if there is a related sentence ✏ of the normal form of B that together with other formulas of the normal form leads to a contradiction. The kind of relatedness between and ✏ is explicated technically as follows. Definition 1 and ✏ are called related w.r.t. P iff a) either mix(P, ✏) or mix(P, ); or b) pos(P, ✏) and pos(P, ); or c) neg(P, ✏) and neg(P, ). For the normal form representation I use prime implicates. A FOL formula ↵ is universal iff ↵ is equivalent to a formula in prenex form containing only all-quantifiers 8. A universal formula of the form 8x1 . . . 8xn(li1_ · · ·_lim), where the li j are literals with variables in {x1, . . . , xn}, is a FOL clause. A FOL clause ↵1 = 8x1 . . . 8xn is a (proper) subclause of a FOL clause ↵2 iff ↵2 is of the form ↵2 = 8y1 . . . 8yn , where all xi are among the yj and the set of literals in is a (proper) subset of the literals in . Let X be a set of universal formulas. The set of FOL clauses of X w.r.t. a vocabulary V , ClV(X), is the set of all FOL clauses in Sent(V) entailed by X . If X is an arbitrary set of FOL sentences, let X⇤ be the result of skolemizing every sentence in X . Let Vsk be the set of used skolem symbols and let V̂ = V [ Vsk. The set of FOL clause of X w.r.t. V and skolem symbols Vsk is defined by ClV̂(X⇤). The set of FOL prime implicates of a set of universal formula X w.r.t. V consists of non-tautological clauses of X for which there is no proper subclause in ClV(X). PIV(X) = {pr 2 ClV(X) | pr is non-tautological and has no proper subclauses in ClV(X)} This notion leads to a logically equivalent characterisation of sets X containing only universal formulas. Proposition 1 For every set X of universal formulas with V(X) ✓ V: X ⌘ PIV(X). The postulate of reinterpretation relevance (Rel-R) reads as follows: (Rel-R) Let be given a vocabulary V , a FOL KB B over V , an FOL sentence ↵ over V and a FOL clause over V . Let B⇤ be a skolemization of V B with skolem constants from Vsk and again V̂ = V [ Vsk. If B |= and B ⇤↵ 6|= , then there is a set X and a sentence ✏ 2 X s.t.: 1. X ✓ PIV̂(B⇤); 2. X [ {↵} is inconsistent; 3. (X \ {✏}) [ {↵} is consistent and 4. ✏ is related with w.r.t. a predicate symbol P . Example 2 Consider the following KB B and trigger ↵: B = {Article(p1), Article(p2),¬Article(bo1)} and ↵ = ¬Article(p1). Clearly PIV̂(B⇤) = B⇤ = B. Let = ¬Article(bo1). Then B |= and B ⇤ ↵ 6|= . But for Article there is no X ✓ PIV̂(B⇤) that fulfills the conditions of (Rel-R) because the only -related prime implicate is ¬Article(bo1) which is not involved in a conflict. Prime implicates can be further exploited to define a postulate that captures the other aspect of minimal revision where one constrains the result from above. The idea is to enrich the given KB B to an equivalent set Enr(B) that contains enough consequences of B in order to identify the real potential culprits in the revision process. The general schema of the extended inclusion axiom is the following: (Incl-ES) For all ↵ there is X ✓ Enr(B) s.t. X[{↵} 6|= ?, and for all : If B ⇤↵ |= , then X [ {↵} |= . This schema says: There is a subset of the enrichment of B such that all sentences entailed by the revision result follow from a subset X of the enrichment together with the trigger ↵. The enrichment operator Enr that I use in the following is defined as: Enr(B) = B [ PIV̂(B⇤). In fact, though the enriched KB Enr(B) is not equivalent to B, it is at least equivalent w.r.t. the non-skolem symbols. Proposition 2 CnV(B) = CnV(Enr(B)) I call the postulate that results from (Incl-ES) by instantiating the parameter Enr by Enr(B) = B [ PIV̂(B⇤) the extended inclusion postulate (Incl-E). Reinterpretation Operators The extended relevance postulate and inclusion postulate are intended to specify minimal changes of operators which are used in a particular semantic integration scenario described in the introduction. In this section, we recapitulate the definition of operators of this kind (Eschenbach and Özçep 2010; Özçep 2008) and show that they fulfill the new postulates. (For other postulates see (Özçep 2008)). The construction of the operators mimics the construction of the propositional revision operators of (Delgrande and Schaub 2003). The revision operator defined in the following is denoted by and is called a reinterpretation operator. is a binary operator with a finite FOL KB as left and a FOL sentence ↵ as right argument. Before giving the technical definition, the main construction idea will be illustrated with an example. Example 3 Define the knwoledge base B as B = {Article(p1), Article(p2),¬Article(bo1)} and the trigger ↵ = ¬Article(p1). The reinterpretation operator results in the following KB: B ↵ = {Article(p1), Article(p2),¬Article(bo1), ¬Article(p1), 8x(Article(x)! Article0(x))} The conflict between B and ↵ is traced back to ambiguous use of symbols. As I assume that only predicate symbols (and not constants) may be used ambiguously, the conflict can only be caused by different uses of the unary predicate Article. The receiver (holder of B) gives priority to the sender’s use of Article over her use of Article, and hence puts ¬Article(p1) in B ↵. Her own use of Article is internalized, i.e., all occurrences of Article in B are substituted by a new symbol Article0. The receiver adds hypotheses on the semantical relatedness (bridging axioms) of his and the sender’s use of Article. The hypothesis in this case is 8x(Article(x) ! Article0(x)) which says that articles in the sender’s sense are also articles in the receiver’s sense. Note that because of this hypothesis the result B ↵ entails the assertion ¬Article(bo1) from the initial KB B. Technically the disambiguation is realized by uniform substitutions called ambiguity compliant resolution substitutions, AR(V,V 0) for short. Here, V \V 0 = ; where V 0 is the set of symbols used for internalization. The substitutions in AR(V,V 0) get as input a non-logical symbol in V and map it either to itself or to a new non-logical symbol (of the same type) in V 0. I only consider the substitution of predicate symbols. supp( ) ={s 2 V | (s) 6= s} is called the support of . A substitution with support S is also denoted by S . For 1, 2 2 AR(V,V 0) let 1  2 iff supp( 1) ✓ supp( 2). A disambiguation schema picks for every S a substitution (S) 2 AR(V,V 0) with support S. In general, there may be more than one predicate symbol which has to be disambiguated; and there may be many different sets of symbols for which a disambiguation leads to consistency. So I define the minimal conflict symbol sets: Definition 2 Let B be a FOL KB over V and ↵ a FOL sentence over V . The set of minimal conflicting symbols sets, MCS(B,↵), is defined by: MCS(B,↵) = { S ✓ V | There is S 2 AR(V,V 0), s.t. B S [ {↵} is consistent, and for all