X-KIF New Knowledge Modeling Language

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1 Proceedings of I-MEDIA 07 and I-SEMANTICS 07 Graz, Austria, September 5-7, 2007 X-KIF New Knowledge Modeling Language Michal Ševčenko (Czech Technical University in Prague Abstract: This paper proposes X-KIF a new modeling language intended for implementing large, formal knowledge bases. The language originated from the SUO-KIF language, which has been used for implementing a large family of ontologies based on the Suggested Upper Merged Ontology (SUMO). We address some of the problems with the SUO-KIF language, and try to formally define the semantics of some higher-order constructs frequently used in SUMO. Key Words: KIF, knowledge representation, knowledge modeling language, ontology Category: I Introduction The field of knowledge representation and modeling has long history that went through several stages with different knowledge modeling paradigms. In the 80s and 90s, the primary paradigm was some sort of higher order logic, attempting to maximize expressive power of the language, giving up the completeness of inference. These efforts resulted in languages like KIF [Genesereth 1991] or CycL [CycL]. The last decade brought motion called Semantic Web, which focuses on less expressive, but highly scalable structures, with expressive power not exceeding first-order logic. The primary languages of the Semantic Web are RDF and OWL. The Semantic Web paradigm, however, is not intended for AI-intensive knowledge representation. Semantic Web also promotes strongly decentralistic approach to ontology development. In the meanwhile, another effort within the IEEE Standard Upper Ontology Working Group resulted in the Suggested Upper Merged Ontology (SUMO) [Niles and Pease 2001]. SUMO defines about a thousand common-sense concepts, accompanied with axioms that partially define their meaning. SUMO uses simplified dialect of the KIF language called SUO-KIF [SUO-KIF]. SUMO concepts have been mapped to the synsets of the WordNet lexical database, making it a valuable computational linguistic tool. In our recent project Knowledge support for modeling and simulation [Ševčenko 2003], we decided to use the alternative SUMO project rather than the Semantic Web languages. We didn t want to completely give up the AI-intensive approach to knowledge representation, and also the more centralistic approach, based on standardized upper ontologies, seemed more appropriate. However, the

2 306 M. Sevcenko: X-KIF - New Knowledge Modeling... original SUO-KIF language has couple of problems as well, so we decided to develop a new knowledge modeling language X-KIF, having modern features known from the Semantic Web languages, while retaining the expressivity and philosophy of the SUO-KIF language. The key enhancements with respect to the SUO-KIF language are the following: 1. X-KIF has namespace support. This allows different parties to develop ontologies concurrently without concern of name clashes with different author s concepts. 2. All concepts in X-KIF must be explicitly defined. This prevents errors like defining new concept by misspelling its name in an axiom. 3. X-KIF provides support for organizing ontologies into modular sections, making large knowledge bases more manageable. 4. Abstract data model of X-KIF permits for different concrete syntaxes, including XML-based syntax for information exchange, binary data format for scalable, high-performance randomly-accessible representation, and pseudotext-based format for entering axioms by knowledge engineers. 5. Knowledge base axioms are first-class objects, which can be associated with metadata such as documentation or inference engine hints. 6. Data model is appropriate for wide range of resources, from large, weakly formalized resources like the Wordnet, to frame-based ontologies or highlyformalized ontologies like SUMO. In Section 2, we describe the overall structure of X-KIF, and particularly its meta-level layer. Section 3 describes the abstract syntax of X-KIF that is used to state formal semantics of concepts. This syntax is similar to the SUO-KIF language syntax. In Section 4 and 5, we address some problems with higherorder semantics of X-KIF stated in [Pease 2003], and propose possible solutions for these problems. 2 X-KIF structure X-KIF serves for description of knowledge bases, which are collections of concepts and axioms. Concepts are fixed points to which formal semantic is attached using axioms. Axioms may be simple facts stating e.g. that certain concept is a subclass of another concept, or complex logical formulas stated in first- or higher-order logic. Concepts denote objects in the universe of discourse, i.e. in some interesting part of the world to be modeled. Concepts may denote classes of objects,

3 M. Sevcenko: X-KIF - New Knowledge Modeling such as motors, instances of those classes, such as motor in my car, or relations and functions among objects. Note that unlike first-order logic, functions and relations are considered part of the universe of discourse. The X-KIF data model is split into two layers, see Figure 1. The lower, axiomatic layer, is used to describe the actual semantics of concepts in the knowledge base. The semantics is represented by logical sentences, specified in Section 3. The axiomatic layer is very similar to the SUO-KIF language. The second, meta-level layer, is used to organize large ontologies into modular parts, and to provide means for naming concepts and axioms, so that they can be referenced from logical sentences. meta-level (ontologies, sections, namespaces) axiomatic level (sentences) Figure 1: Two layers of X-KIF The meta-level layer has been added to X-KIF to facilitate management of large knowledge bases. The layer allows to organize concepts and axioms hierarchically into a system of sections. Each section may contain concepts and axioms, and may be further divided into subsections to an arbitrary depth. Ontology is a special kind of section, containing implementation of some compact, well defined part of the knowledge base, and may be therefore thought of as a separate package that is authored and distributed independently. Ontology, however, may refer to concepts and axioms defined in different ontologies. For a conceptual view of knowledge base structure, see Figure 2. A knowledge base section may be associated with a Uniform Resource Identifier (URI), which defines a namespace for all objects defined within that section, including itself. For ontologies, the namespace URI is mandatory. Every knowledge base object, that is, section, concept or axiom, has a local identifier, which must be unique within the object s namespace. The local identifier of an object may be combined with the object s namespace URI to form object s globally unique URI. For instance, if concept Human is contained in section Objects of ontology SUMO, and SUMO has namespace URI org.sumo.1.0, the URI of con-

4 308 M. Sevcenko: X-KIF - New Knowledge Modeling... Knowledge Base Ontology A Ontology B contains 0..1 Knowledge Base Object id * refers Section A 1 Section A 11 Section A 2 Ontology C has 0..2 URI * Section Concept Axiom contains 1..2 has Ontology Sentence * Concepts Axioms (a) (b) Figure 2: Meta-level layer of X-KIF: (a) conceptual view, (b) UML diagram cept Human is org.sumo.1.0#human. Note that the ontology SUMO has also its URI, which is different from the URI of its namespace. URI of SUMO may be for example org.sumo.1.0#sumo. The namespace URI of an ontology is intended to refer to specific version of the ontology. If a different version of some ontology is published, it should be given different URI. It is therefore convenient to incorporate some kind of version information or release date directly into the URI of an ontology. A section may also have associated alternative URI called last version URI,which is intended to refer to the last known version of that section. Different versions of the same section have therefore identical last version URIs. When knowledge engineer references a concept in external section, he may decide whether it is more appropriate to refer to definition in particular version of that section, and uses its primary URI, or whether the most recent version of the section should be used, and uses the last version URI instead. All relations depicted in Figure 2b are meta-level relations that are not specified using axioms, and therefore are not visible from the axiomatic level of X-KIF. Also note that axioms may refer not only to concepts, but also to sections or other axioms. This allows, for example, to attach metadata such as documentation to concepts, sections and axioms uniformly. 3 Abstract syntax of sentences In this section, we specify the abstract grammar describing logical sentences comprising axioms. The grammar contains the following terminal symbols: logical connectives,,,,, quantifiers and, logicalequalitysign, parentheses ( and ), variables, which we denote with letters x, y, z, possibly with indexes,

5 M. Sevcenko: X-KIF - New Knowledge Modeling row variables, typeset in bold, such as a, b, literal constants, and finally, the knowledge base objects, usually typeset in sans-serif, such as Human. There are two basic non-terminal symbols. Terms, which we denote with letter t, possibly with indexes, and sentences, which we denote with Greek letters ϕ and ψ. Terms are defined inductively as follows. 1. All knowledge base objects are terms. 2. Literal constants are terms. Literal constants refer to certain well-known objects in the universe of discourse, such as numbers, characters, or strings. 3. Variables are terms. 4. Expression f (t 1,...,t n ), where f is function symbol, and t 1,...,t n are terms or row variables, is term. Sentences are defined inductively as follows. 5. Expression r (t 1,...,t n ), where r is either a relation symbol, or a variable, and t 1,...,t n are terms or row variables, is sentence. This particular case of sentence is called atomic sentence or atom. 6. Expression r (...,ϕ,...), where r is either a relation symbol, or a variable, and ϕ is a sentence, is a sentence. The predicate may contain other terms or row variables as arguments, but may not contain more than one sentence argument.thistypeofsentenceiscalledhigher-order atom. 7. A variable is a sentence. 8. Expression t 1 t 2,wheret 1 and t 2 are terms, is a sentence called equation. 9. If ϕ, ψ are sentences, then ϕ, ϕ ψ, ϕ ψ, ϕ ψ, ϕ ψ are also sentences. 10. If ϕ is sentence and x is a variable, then xϕ, xϕ are also sentences. 11. If ϕ is sentence, then (ϕ) is also a sentence. Parentheses may be used to clarify priority of logical connectives and quantifiers. Note the difference between plain variables and row variables. While plain variables are intended to refer to objects in the universe of discourse, row variables denote sequences of objects, which may be used inside expressions 4, 5 and 6. Note that row variables are not terms; in particular, they cannot be used in expression 8.

6 310 M. Sevcenko: X-KIF - New Knowledge Modeling... 4 Semantics of X-KIF In this section, we describe the model-based semantics of X-KIF, by extending commonly-used semantics of first-order predicate logic. Formal definition of semantics of first-order predicate calculus uses a notion of interpretation. Interpretation M is a total function from sentences to truth values: M : Λ {true, f alse}, where Λ is set of all well-formed sentences of the language. In first-order logic, every interpretation must satisfy the following properties: M (ϕ) M ( ϕ), M (ϕ ψ) istrueiffm (ϕ) istrueandm (ψ) is true, M (ϕ ψ) istrueiffm (ϕ) istrueorm (ψ) istrue,m (ϕ ψ) is true iff M ( ϕ ψ) is true, M (ϕ ψ) is true iff M (ϕ ψ) is true and M (ψ ϕ) is true. M ( x : ϕ) is true iff M (ϕ ) is true for every sentence obtained by replacing all occurrences of x in ϕ with some ground term, and M ( x : ϕ) =M ( x : ϕ). M is said to be a model of sentence ϕ if M (ϕ) =true, denoted as M = ϕ. Asentenceissaidtobesatisfiable if it has a model. A set of sentences T is satisfiable if there exists an interpretation that is a model for all sentences in T. Asentenceψ is said to be entailed by a set of sentences T, written as T = ψ, if each interpretation that is a model for all sentences in T,isalsoamodelforψ. We can extend the notion of interpretation to higher-order sentences. We shall call a higher-order interpretation H, which must satisfy at least those properties of M. Additionally, we require that if H = P (...,ϕ,...), then H = P (..., ϕ,...). That is, if certain sentence holds in some context, represented by predicate symbol and its arguments other than the sentence, the negation of the same sentence may not hold in the same context. This requirement holds only for positive atoms. It is therefore possible that both P (ϕ) and P ( ϕ) have the same model. We also require that all contexts are closed under entailment, thatis,ifh = P (...,ϕ,...), and ϕ = ψ, thenh = P (...,ψ,...). For instance, P (ϕ) = P (ϕ ψ). The notion of entailment is then extended to higher-order interpretation: a higher-order sentence ψ is entailed by higher-order set of sentences T, if all higher-order interpretations that model all sentences in T also model ψ. To actually test the entailment, we usually use some inference system. Inference system is a set of syntactic rules that allows to generate new sentences from given set of sentences. If the system allows to generate sentence ψ from a set of sentences T, we denote T ψ. An inference system is said to be sound if all sentences generated from T are entailed by T,thatisT ϕ T = ϕ. An inference system is said to be complete if it guarantees to generate every sentence that is entailed, that is, T = ϕ T ϕ. In first-order logic, there exists an inference system called resolution, which is known to be both sound and complete. We extend the resolution to accommodate higher-order sentences of X-KIF.

7 M. Sevcenko: X-KIF - New Knowledge Modeling First-order resolution contains the following rule: {ϕ A, ψ B} (ϕ ψ) θ, wherea and B are atoms, and θ is the most general unifier of A and B. We extend the inference system with the following extra rules: 1. {ϕ P (...,α,...),ψ P (...,β,...)} (ϕ ψ) θ, ifαθ βθ. The sentences αθ and βθ must be closed sentences (not containing free variables), although the sentences α and β need not be closed. It is therefore possible to resolve literals believes (x,teacher (x)) and believes (John, teacher (John)), even though the first literal contains a sentence that is not closed (is bound to the outer context). 2. ϕ P (...,ψ,...) ϕ P (..., ψ,...). This rule specifies that if a clause contains a positive higher-order literal, the literal may be replaced with negative literal, that has the sentence argument negated as well. This is, in fact, an application of resolution rule for general axiom P (...,ψ,...) P (..., ψ,...), which must hold for all predicate symbols, according to our definition of higher-order semantics. Other operations are done the same way as in first-order resolution. For instance, if a variable is to be unified with a sentence argument, the sentence is treated as term expression, and is simply substituted for the variable. 5 Semantics of other higher-order X-KIF extensions In the previous section, we described the semantics of one particular extension of X-KIF, namely the usage of nested sentences. In this section, we describe how we treat other higher-order extensions, namely row variables and usage of variable in the predicate position. Generally speaking, we consider the axioms using these two extra features as axiom schemata, that is, structures that describe sets of axioms. Each such axiom schema can be therefore instantiated to generate a set of axioms that no longer contain higher-order features. Although the instantiation process may be thought of a kind of preprocessing step that transforms a particular theory before it is sent to some inference system to construct a proof, we propose to directly modify the inference system itself to accommodate the extensions. Semantically, the inference system extension corresponds to generating axioms from their corresponding schemata on the fly as they are needed during the proof. The proposed extensions are implemented as follows: 1. Variables in the predicate position. For sentences using variables for predicate symbol, we simply extend the unification procedure of the resolution. In first order logic, two literals t 1 (...)and t 2 (...)canberesolved only if t 1 and t 2 are the same predicate symbols. The extension is that t 1

8 312 M. Sevcenko: X-KIF - New Knowledge Modeling... and t 2 can be arbitrary terms that are unifiable. For instance, the axioms instance(issibling,symmetricrelation) and instance (p,symmetricrelation) p (x, y) p (y, x), allow to deduce fact issibling (x, y) issibling (y, x). This simple extension does not introduce full higher-order semantics of quantification over subsets, but allows to specify axiom schemata for predicates in a natural way. In fact, the latter axiom is an axiom schema that can be instantiated, using the extended resolution rule, for every symmetric relation, into plain first order axiom. Similar technique can be used for functions. 2. Row variables. We use row variables mostly for specifying axiom schemata for concrete relations, which usually have arity bound with some reasonable number. It is therefore feasible to simply replace axiom schema that uses row variable with a small number of axioms, replacing the row variable with a sequence of plain variables. Sometimes, row variables are used in axioms where the length of sequence denoted by row variable cannot be bound. For instance, in SUMO row variables are used for specifying semantics of lists, which may be of arbitrary length. We therefore propose a simple extension of the unification procedure, which should allow to unify terms containing expressions with row variables. For instance, term f (1, a) can be unified with term f (x, y, 3) with substitution x 1, a y, 3. 6 Conclusion In this paper, we introduced new knowledge modeling language X-KIF, which combines expressive power of the SUO-KIF language with modern features of contemporary XML-based languages like OWL. The language can represent wide variety of resources like OWL ontologies, SUMO, or the WordNet. We also defined the semantics of some higher-order extensions of the language. References [CycL] The syntax of CycL ; [Genesereth 1991] Genesereth, M.: Knowledge interchange format ; Proc. Principles of Knowledge Representation and Reasoning: Proceedings of the Second International Conference, pages Morgan Kaufmann, (1991). [Niles and Pease 2001] Niles, I., Pease, A.: Towards a standard upper ontology ; Proc. of the 2nd International Conference on Formal Ontology in Information Systems (FOIS-2001), Ogunquit, Maine, (Oct 2001). [Pease 2003] Pease, A.: The sigma ontology development environment ; Working Notes of the IJCAI-2003 Workshop on Ontology and Distributed Systems. Volume 71 of CEUR Workshop Proceeding series, (2003). [SUO-KIF] Standard upper ontology knowledge interchange format ; [Ševčenko 2003] Ševčenko, M.: Intelligent user support for modeling and simulation of dynamic systems ; Technical Report DC-PSR , Czech Technical University in Prague, (Jan 2003).

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