A Decision Tree Lifted Domain for Analyzing Program Families with Numerical Features (Extended Version)

Lifted (family-based) static analysis by abstract interpretation is capable of analyzing all variants of a program family simultaneously, in a single run without generating any of the variants explicitly. The elements of the underlying lifted analysis domain are tuples, which maintain one property per variant. Still, explicit property enumeration in tuples, one by one for all variants, immediately yields combinatorial explosion. This is particularly apparent in the case of program families that, apart from Boolean features, contain also numerical features with big domains, thus admitting astronomic configuration spaces. The key for an efficient lifted analysis is proper handling of variability-specific constructs of the language (e.g., feature-based runtime tests and #if directives). In this work, we introduce a new symbolic representation of the lifted abstract domain that can efficiently analyze program families with numerical features. This makes sharing between property elements corresponding to different variants explicitly possible. The elements of the new lifted domain are constraint-based decision trees, where decision nodes are labeled with linear constraints defined over numerical features and the leaf nodes belong to an existing single-program analysis domain. To illustrate the potential of this representation, we have implemented an experimental lifted static analyzer, called SPLNUM^2Analyzer, for inferring invariants of C programs. It uses existing numerical domains (e.g., intervals, octagons, polyhedra) from the APRON library as parameters. An empirical evaluation on benchmarks from SV-COMP and BusyBox yields promising preliminary results indicating that our decision trees-based approach is effective and outperforms the tuple-based approach, which is used as a baseline lifted analysis based on abstract interpretation.