MOTIVATION

P-SHIQ(D)

P-SHIQ(D)

Background to the research study

P-SHIQ is a probabilistic generalization of the DL SHIQ. It has a number of properties that make it an attractive formalism for managing uncertainty in OWL ontologies. First, it is very expressive as it supports arbitrary SHIQ concepts in probabilistic axioms. Second, any SHIQ ontology can be used as a basis for a P-SHIQ ontology which facilitates transition from classical to probabilistic ontological models. Thirdly, its reasoning tasks are decidable. (Pavel, 2008, 05-11)

Syntax And Semantics of SHIQ(d)

Syntax of P-SHIQ is an extension of the syntax of SHIQ with conditional constraints. Conditional constraints are expressions of the form (DjC)[l; u] where C and D are SHIQ concepts1. Conditional con straints can be used for representing uncertainty in both terminological (TBox) and assertional (ABox) knowledge. (Pavel, 2008, 05-11) A probabilistic TBox (PTBox) is a 2-tuple PT = (T;P) where T is a SHIQ TBox and P is a nite set of default2 condi tional constraints. Informally, a PTBox axiom (DjC)[l; u] means that \generally, if a randomly chosen individual belongs to C, its probability of belonging to D is between l and u".(Sirin, 2005, 15-22) A probabilistic ABox (PABox) is a nite set of strict condi tional constraints pertaining to a single probabilistic individual o. All constraints in a PABox are of the restricted form (Dj>)[l; u]. Informally, they mean that \o is a member of D with probability between l and u". (Gail, 1989, 1879-1886)

Syntactically, P-SHIQ+ differs from P-SHIQ in two main aspects. Firstly, the notion of a conditional constraint is extended. Secondly, instead of a collection of independent PABoxes in P-SHIQ, a P-SHIQ+ ontology contains only a single probabilistic component, namely, a collection of conditional constraints. One preliminary de nition is needed. Definition 3 (Entity). An entity in P-SHIQ+ is one of: role, concept, concept assertion, role assertion. (Sirin, 2005, 15-22) Intuitively, entities are the things that can appear in conditional constraints. Now we can de ne the extended syntax of conditional constraints. Definition 4 (Conditional Constraint). (Pavel, 2008, 05-11)

A conditional constraint in P-SHIQ+ is an expression of the form: ( j_)[l; u] where; _ are entities. Note that every conditional constraint in P-SHIQ is a well-formed conditional constraint in P-SHIQ+. No other probabilistic formulas are allowed. (Sirin, 2005, 15-22) A proba bilistic ontology in P-SHIQ+ has a simpler structure than in P-SHIQ: Definition 5 (Probabilistic Ontology). A probabilistic ontology in P-SHIQ+ is a tuple PO = (O;P) where O = (T;A) is a SHIQ ontology3 and P is a nite collection of conditional constraints. Any P-SHIQ ontology can be represented as a P-SHIQ+ ontology, however, the representation of PABox constraints is a bit di erent. Instead of implicitly attributing statements of the form (Cj>)[l; u] to a particular individual, it is represented as (C(a)j>)[l; u] and becomes an element of P. In other words, PABox constraints in P-SHIQ+ correspond to ground probabilistic formulas in FOPL2 while in P-SHIQ they do not. (Lukasiewicz, 2002, 35-88)

Semantics of P-SHIQ+

From an analysis of P-SHIQ alone, it may not ...