On the complexities of consistency checking for restricted UML class diagrams

Abstract

a b s t r a c tAutomatic debugging of UML class diagrams helps in the visual specification of software systems because users cannot detect errors in logical consistency easily. This study focuses on the tractable consistency checking of UML class diagrams. We accurately identify inconsistencies in these diagrams by translating them into first-order predicate logic that is generalized by counting quantifiers and classify their expressivities by eliminating certain components. We introduce optimized algorithms that compute the respective consistencies of class diagrams of different expressive powers in P, NP, PSPACE, or EXPTIME with respect to the size of the class diagrams. In particular, owing to the restrictions imposed on attribute value types, the complexities of consistency checking of class diagrams decrease from EXPTIME to P and PSPACE in two cases: (i) when the class diagrams contain disjointness constraints and overwriting/multiple inheritances and (ii) when the class diagrams contain both these components along with completeness constraints. Additionally, we confirm the existence of a restriction of class diagrams that prevents any logical inconsistency.

Full Satisfiability of UML Class Diagrams

“…Three notions of UCD satisfiability have been proposed in the literature [19,5,6,20,9]. First, diagram satisfiability refers to the existence of a model, i.e., a FOL interpretation that satisfies all the FOL assertions associated to the diagram and where at least one class has a nonempty extension.…”

“…Third, we can check whether there is a model of an UML diagram that satisfies all classes and all relationships in a diagram. This last notion of satisfiability, referred here as full satisfiability and introduced in [8,9] is thus stronger than diagram satisfiability, since a model of a diagram that satisfies all classes is, by definition, also a model of that diagram. We now address the complexity of full satisfiability for UCDs with the standard set of constructs, i.e., UCD full .…”

“…Thus, one can in principle reason on UCDs. The reasoning tasks that one is interested in are, e.g., subsumption between two classes, i.e., the fact that each instance of one class is necessarily also an instance of another class, satisfiability of a specific class (or association) in the diagram, i.e., the fact that the information encoding that class (or association) in the diagram is not contradictory, diagram satisfiability, which requires that at least one class in the diagram is satisfiable, and full satisfiability of the diagram [8,9], i.e., the fact that there exists an instantiation of the diagram where all classes and associations of the diagram are non-empty.…”

“…The only works that addressed explicitly the complexity of full satisfiability of UCDs are [8,9], which include a classification of UCDs based on inconsistency triggers. Each inconsistency trigger is a pattern for recognizing possible inconsistencies of the diagram based on the interaction between different modelling constraints.…”

“…Each inconsistency trigger is a pattern for recognizing possible inconsistencies of the diagram based on the interaction between different modelling constraints. [8,9] introduce various algorithms for checking full satisfiability of UCDs with different expressive power, together with an analysis of their computational complexity (i.e., upper bounds are provided). In particular, checking full satisfiability in the following scenarios is showed to be in:…”