On the interaction between sharing and linearity

In the analysis of logic programs, abstract domains for detecting sharing and linearity information are widely used. Devising abstract unification algorithms for such domains has proved to be rather hard. As a consequence, only correct algorithms are available at the moment, which partially exploit the information conveyed by the abstract domains. In this paper, we define a new (infinite) domain ShLin which can be thought of as a general framework from which other domains can be easily derived by abstraction. ShLin makes the interaction between sharing and linearity apparent. Moreover, we provide a simple and elegant optimal abstract unification algorithm on ShLin and we lift it to two well-known abstractions of ShLin. Namely, to the classical Sharing× Lin abstract domain and to the more precise ShLin abstract domain by Andy King. As a consequence, we obtain optimal abstract unification algorithms for such domains.