second-order quantifier elimination

Cylindrical Algebraic Decomposition (CAD) faces efficiency and applicability bottlenecks in real quantifier elimination (QE), particularly when handling multiple equality constraints and rational function inputs. Method: We propose two core improvements: (1) a multivariate resultant projection strategy replacing iterative resultant computation, significantly accelerating projection under multiple equalities; and (2) a rigorous reformulation and formal correctness proof of the McCallum–Brown double-equation reconstruction method, extended for the first time to arbitrary quantified structures, alongside the first CAD framework supporting full quantification over rational functions—overcoming prior limitations restricted to satisfiability checking. Contribution/Results: Experiments demonstrate substantial speedups on QE problems involving multiple equalities. Moreover, our approach provides the first reliable, general-purpose symbolic algorithm for nonlinear real arithmetic within SMT solvers.

This work presents the first quantifier elimination procedure for the complex field within an ordered ring language extended with symbols for the imaginary unit, real and imaginary parts, and complex conjugation. By reducing the complex quantifier elimination problem to its real counterpart and heuristically reconstructing the results according to complex semantics, the authors establish a dedicated quantifier elimination framework for complex numbers. This approach overcomes the traditional limitation of such methods to real-closed fields. A prototype implementation has been integrated into the open-source system Logic1, and its efficacy and practicality are demonstrated through multiple illustrative examples.

This work addresses the double-exponential complexity inherent in quantifier elimination via Fourier–Motzkin elimination (FME) and cylindrical algebraic decomposition (CAD) by introducing treewidth—a graph-theoretic measure of sparsity—into the quantifier elimination framework. The authors propose a dynamic programming approach based on tree decompositions of variable dependency graphs, which leverages the sparse structure of input formulas. By integrating parameterized algorithms with FME and CAD, the method handles both linear real arithmetic (LRA) and nonlinear real arithmetic (NRA) formulas. Theoretically, the approach achieves single-exponential complexity when the treewidth is bounded by a constant. Experimental evaluation demonstrates substantial performance improvements over state-of-the-art heuristics on benchmarks with low treewidth.

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.

This paper addresses the classical problem of computing Boolean solutions to Boolean equations—an issue rooted in 19th-century algebraic logic and closely related to modern Boolean unification. We establish, for the first time, a higher-order logical semantics for this problem, formalizing it as a second-order quantified predicate logic problem, and systematically uncover its deep connections with first-order reasoning, second-order quantifier elimination, and Craig interpolation. Our main contributions are: (1) a proof that the set of Boolean solutions is recursively enumerable for first-order inputs; (2) several constructive solution algorithms based on Craig interpolation; and (3) a modeling and solving framework for Boolean solutions under vocabulary constraints. Collectively, these results bridge Boolean unification with higher-order logic and automated reasoning, advancing both theoretical foundations and algorithmic methodology.

This study addresses the inefficiency and poor interpretability of traditional cylindrical algebraic decomposition (CAD) methods in quantifier elimination problems involving multiple equality constraints, particularly in characterizing the relationship between parameters and unknowns. The authors propose a refined partitioning strategy for the parameter space that explicitly distinguishes between cases yielding finitely many versus infinitely many solutions. Under specific conditions, this approach substantially simplifies the equality projection steps in CAD, thereby overcoming limitations inherent in existing theoretical frameworks. The method not only enables an explicit description of parameter-dependent solution structures but also significantly enhances computational efficiency and result interpretability in applications such as approximation theory, classification of robotic singularities, and modeling of biochemical systems.

This work addresses the high degree of manual effort and tediousness inherent in existing automated reasoning algorithms—such as those for hyper-exponential quantifier elimination—for complexity analysis. The paper proposes a higher-order abstract interpretation framework grounded in operator semantics, which automatically abstracts symbolic programs into numerical recurrence relations. By integrating termination analysis, fixed-point theory, and SMT solving techniques, the method enables fully automated derivation and verification of asymptotic upper bounds on computational complexity. This approach substantially reduces human intervention while significantly enhancing the automation, efficiency, and scalability of complexity analysis for intricate algorithms.

This work addresses the problem of second-order quantifier elimination by proposing a method that explicitly constructs a logically equivalent first-order formula together with its corresponding second-order witness function. Building upon the clause-set-based SCAN saturation algorithm, we extend it for the first time to support the automatic computation of witness functions, thereby enabling effective witness generation for a broad class of applicable formulas while preserving the original soundness guarantees of logical equivalence. This advancement not only broadens the applicability of the SCAN algorithm within computational logic but also demonstrates the feasibility and practical utility of the proposed approach through an implemented prototype system.

This work addresses key limitations in automated theorem proving for plane geometry—namely, reliance on manual intervention and susceptibility to division-by-zero errors. We propose a fully automated method integrating complex-number identity modeling with elimination ideals. Geometric statements are first algebraized; real-domain constraints are handled via slack variables, and linear polynomial criteria are introduced to rigorously ensure conclusion validity—thereby eliminating the division-by-zero pitfalls inherent in classical complex-number methods. The approach leverages symbolic computation to systematically clear denominators, rewrite variables, and perform ideal-theoretic elimination. The algorithm has been implemented in Mathematica, Maple, Giac, and GeoGebra; notably, a prototype tool is embedded in the experimental version of GeoGebra. To our knowledge, this is the first fully automated realization of the complex-number method for geometric theorem proving, achieving substantial improvements in both proof efficiency and usability.

This paper addresses the quantifier normalization problem for Boolean combinations of polynomial equalities and inequalities over ℂ and ℝ. We introduce the notion of “Boolean elimination” and construct a unified normal form containing exactly one existential and one universal quantifier (∃∀ or ∀∃), which equivalently reduces any such Boolean combination to a single polynomial equation while preserving linear degree bounds. We prove that purely existential or purely universal normal forms are impossible over ℂ, thereby establishing the theoretical optimality of this two-quantifier structure. Our approach integrates tools from algebraic number theory, model theory, and polynomial system analysis, and we extend the results to ℚ. Compared to classical quantifier elimination, this paradigm drastically simplifies the Boolean structure while achieving an optimal trade-off between quantifier complexity and expressive power.