Potential for significant improvements to the implicant selection heuristic in CAlC

Determine whether any implicant selection heuristic for the Cylindrical Algebraic Covering (CAlC) algorithm—specifically beyond the evaluated Size, Feature-Based, and Sum of Total Degrees (Sotd) criteria—can yield significant performance improvements in terms of solved instances on non-linear real arithmetic benchmarks. The aim is to ascertain if more effective selection strategies for choosing among candidate implicants (generated by evaluation, Boolean propagation, or exploration) exist that materially outperform the tested heuristics.

Background

In the experimental evaluation, the authors compared multiple implicant selection heuristics used to choose among candidate implicants produced during the CAlC search: Size (fewest constraints), Feature-Based (degree-based features), and Sotd (sum of total degrees), along with a reverse Sotd control. These heuristics guide which implicant is generalized into an implicit cell and thus impact the projection workload and overall runtime.

The reported results showed only modest differences among the meaningful variants (Size, Feature-Based, Sotd), with the virtual best solving only 18 more instances than Sotd on the QF_NRA set. Based on this small spread, the authors explicitly state uncertainty about whether substantial improvements over these heuristics are attainable, motivating a precise investigation into the existence of significantly better selection strategies.