A Survey on Product Operators in Abstract Interpretation

interpretation (6) has been widely applied as a general technique for the sound approximation of the semantics of computer programs. In particular, abstract domains (to represent data) and semantics (to represent data operations) approximate the concrete computation. When analyzing a program and trying to prove some property on it, the quality of the result is determined by the abstract domain choice. There is always a trade-off between accuracy and efficiency of the analysis. During the years, various abstract domains have been developed. An interesting feature of the abstract interpretation theory is the possibility to combine different domains in the same analysis. In fact, the abstract interpretation framework offers some standard ways to compose abstract domains, ensuring the preservation of the theoretical properties needed to guarantee the soundness of the analysis. These compositional methods are called domain refinements. A systematic treatment of abstract domain refinements has been given in (12, 14), where a generic refinement is defined to be a lower closure operator on the lattice of abstract interpretations of a given concrete domain. These kinds of operators on abstract domains provide high- level facilities to tune a program analysis in terms of accuracy and cost. Two of the most well-known domain refinements are the disjunctive completion (6, 9, 13, 15, 18) and the reduced product (6), but they are not the only ones. The reduced product can be seen as the most precise refinement of the simple Cartesian product. Moreover, the reduced cardinal power is introduced by (6). While the other domain refinements have been, since their introduction, widely used and explored, the reduced cardinal power has seen definitely less further developments since 1979, with the exception of (16). To verify our assertion, we looked for the number of scientific citations (in the abstract interpretation context) to some domain refinements in Google Scholar. We depicted the results of this search in Figure 1. In particular, we focused on the Cartesian product, the reduced product and the reduced cardinal power. Throughout the years, the number of citations increases in all three cases, but the absolute numbers are very different: just consider that the total citations of "Cartesian product" are 964, while the ones to "reduced cardinal power" are only 38.