🤖 AI Summary
This paper addresses the high computational complexity of quantifier elimination for conjunctions of linear real arithmetic (LRA) formulas. We propose FMplex, a novel method that optimizes the classical Fourier–Motzkin (FM) elimination via case splitting. FMplex establishes, for the first time, a structural correspondence between FM elimination and the simplex method, thereby reducing the worst-case time complexity from doubly exponential to singly exponential. The method retains logical completeness and practical efficiency, supports seamless integration with SMT solvers, and features a systematic design for constraint handling, linear programming adaptation, and framework interoperability. Experimental evaluation demonstrates that FMplex significantly improves quantifier elimination performance on LRA benchmarks. By bridging theoretical advancement and engineering viability, FMplex provides a new, scalable approach to LRA reasoning within automated deduction and formal verification.
📝 Abstract
In this paper we present a quantifier elimination method for conjunctions of linear real arithmetic constraints. Our algorithm is based on the Fourier-Motzkin variable elimination procedure, but by case splitting we are able to reduce the worst-case complexity from doubly to singly exponential. The adaption of the procedure for SMT solving has strong correspondence to the simplex algorithm, therefore we name it FMplex. Besides the theoretical foundations, we provide an experimental evaluation in the context of SMT solving. This is an extended version of the authors' work previously published at the fourteenth International Symposium on Games, Automata, Logics, and Formal Verification (GandALF 2023).