$$\hbox {OCL}_\textsf {FO}$$OCLFO: first-order expressive OCL constraints for efficient integrity checking

OCL is the standard language for defining constraints in UML class diagrams. Unfortunately, as we show in this paper, full OCL is so expressive that it is not possible to check general OCL constraints efficiently. In particular, we show that checking general OCL constraints is not only not polynomial, but not even semidecidable. To overcome this situation, we identify $$\hbox {OCL}_\textsf {FO}$$OCLFO, a fragment of OCL which is expressively equivalent to relational algebra (RA). By equivalent we mean that any $$\hbox {OCL}_\textsf {FO}$$OCLFO constraint can be checked through a RA query (which guarantees that $$\hbox {OCL}_\textsf {FO}$$OCLFO checking is efficient, i.e., polynomial), and any RA query encoding some constraint can be written as an $$\hbox {OCL}_\textsf {FO}$$OCLFO constraint (which guarantees expressiveness of $$\hbox {OCL}_\textsf {FO}$$OCLFO). In this paper we define the syntax of $$\hbox {OCL}_\textsf {FO}$$OCLFO, we concisely determine its semantics through set theory, and we prove its equivalence to RA. Additionally, we identify the core of this language, i.e., a minimal subset of $$\hbox {OCL}_\textsf {FO}$$OCLFO equivalent to RA.