Full Satisfiability of UML Class Diagrams

Abstract

Abstract. UML class diagrams (UCDs) are the de-facto standard formalism for the analysis and design of information systems. By adopting formal language techniques to capture constraints expressed by UCDs one can exploit automated reasoning tools to detect relevant properties, such as schema and class satisfiability and subsumption between classes. Among the reasoning tasks of interest, the basic one is detecting full satisfiability of a diagram, i.e., whether there exists an instantiation of the diagram where all classes and associations of the diagram are non-empty and all the constraints of the diagram are respected. In this paper we establish tight complexity results for full satisfiability for various fragments of UML class diagrams. This investigation shows that the full satisfiability problem is ExpTime-complete in the full scenario, NP-complete if we drop isa between relationships, and NLogSpace-complete if we further drop covering over classes. 1

Querying UML Class Diagrams

“…The above results have been refined and extended to full satisfiability of UCDs by Artale et al [2] and Kaneiwa et al [30]; upper (resp., lower) bounds are obtained by a reduction to (resp., from) satisfiability of UCDs. In [2], classes of UCDs were identified for which statisfiability is np-complete and nlogspacecomplete by restricting the constructs allowed in the diagrams. In [30], it has also been shown that there exists a fragment of UCDs for which full satisfiability is in ptime.…”

“…The exptime membership is obtained by providing a polynomial translation of UCD constructs to DLR if d KBs [16]. The above results have been refined and extended to full satisfiability of UCDs by Artale et al [2] and Kaneiwa et al [30]; upper (resp., lower) bounds are obtained by a reduction to (resp., from) satisfiability of UCDs. In [2], classes of UCDs were identified for which statisfiability is np-complete and nlogspacecomplete by restricting the constructs allowed in the diagrams.…”

“…Despite its expressiveness, the language of UCDs is often insufficient to express all constraints the designer would like to enforce. For this reason, the OMG consortium devised the Object Constraint Language (OCL) 2 , which allows for expressing arbitrary constraints on UCDs. OCL is a powerful language, but is not widely adopted due to its complex syntax and ambiguous semantics.…”

Finite satisfiability of UML class diagrams with constrained class hierarchy

“…Consistency of class diagrams is studied in Kaneiwa and Satoh [2010] and Artale et al [2010]. Kaneiwa and Satoh [2010] provide upper bounds by introducing optimized algorithms for deciding class diagram consistency for various combinations of constraints.…”

“…Correctness of a class diagram involves consistency [Andre et al 2000;Berardi et al 2005;Queralt and Teniente 2008;Artale et al 2010;Kaneiwa and Satoh 2010;Jarrar and Heymans 2008] and finite satisfiability [Lenzerini and Nobili 1990;Thalheim 2000;Calvanese and Lenzerini 1994;Boufares and Bennaceur 2004;Maraee and Balaban 2007;Cabot et al 2008;Berrabah and Boufarès 2008]. …”