🤖 AI Summary
This work addresses the satisfiability checking and quantifier elimination problems for nonlinear real arithmetic (NRA) formulas featuring both Boolean structure and quantifiers. We present the first extension of cylindrical algebraic coverings (CAC) to full first-order logic formula verification. Our approach introduces a novel CAC variant that integrates CAD-based covering construction, hierarchical quantifier handling, explicit Boolean structure incorporation, and adaptive splitting and pruning heuristics. Unlike conventional methods, our framework avoids constructing a complete cylindrical algebraic decomposition (CAD), thereby substantially reducing computational complexity. Experimental evaluation on diverse nonlinear quantified benchmarks demonstrates that our method outperforms state-of-the-art SMT solvers—including Z3 and CVC5—as well as specialized quantifier elimination tools such as QEPCAD and Redlog, in both solution accuracy and runtime efficiency. The gains are particularly pronounced on high-dimensional, sparse constraint instances.
📝 Abstract
The cylindrical algebraic covering method was originally proposed to decide the satisfiability of a set of non-linear real arithmetic constraints. We reformulate and extend the cylindrical algebraic covering method to allow for checking the truth of arbitrary non-linear arithmetic formulas, adding support for both quantifiers and Boolean structure. Furthermore, we also propose a variant to perform quantifier elimination on such formulas. After introducing the algorithm, we elaborate on various extensions, optimizations and heuristics. Finally, we present an experimental evaluation of our implementation and provide a comparison with state-of-the-art SMT solvers and quantifier elimination tools.