symbolic computation

Algebraic proof certificates (e.g., LPACs) suffer from severe size inflation due to redundant derivation steps across distinct variable sets, leading to high verification overhead and poor scalability. To address this, we propose an extension of LPAC featuring the first systematic mechanism for reusing and recycling proof fragments. Our approach introduces two novel inference rules—“fragment extraction” and “fragment application”—enabling efficient compression and reconstruction of linear combination structures. Built upon the practical algebraic calculus framework and incorporating linear-algebraic modeling, our system is fully integrated into the Pacheck 2.0 verifier. Experimental evaluation demonstrates that our method reduces proof size by 42% on average and accelerates verification time by 38%, significantly improving the compactness, verification efficiency, and practical applicability of algebraic proofs.

This work addresses the challenge of efficiently encoding and verifying mathematical proofs in zero-knowledge settings. We propose a structured representation based on $2 imes2$ matrix-valued polynomials: formal statements are mapped to multivariate polynomial matrices with integer coefficients, and compact numeric fingerprints are generated via random evaluation over a finite field; matrix homomorphism enables algebraic execution of logical operations—including hypothetical reasoning and variable substitution. Our key contribution is the first encoding of formula syntactic structure into computable polynomial-matrix “fingerprints”, supporting incremental, zero-knowledge proof derivation and verification. Experiments demonstrate that only the fingerprints of initial axioms require explicit verification; all subsequent formulas are derived efficiently from ancestral fingerprints, drastically reducing both communication and computational overhead in verification.

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 addresses the limitation in Lean 4 that its native multivariate polynomials employ a non-computable representation, hindering efficient execution of symbolic computations such as Gröbner basis algorithms and thus impeding large-scale algebraic reasoning. To overcome this, the authors propose a certificate-based hybrid verification framework that integrates a computable polynomial representation, leverages external computer algebra systems (e.g., SageMath, SymPy) to compute Gröbner bases, and formally verifies their correctness within Lean 4. This approach enables, for the first time, practical automated handling of large-scale polynomial problems in Lean 4, supporting essential reasoning tasks including remainder verification, ideal membership testing, and ideal equality. The method substantially enhances both the efficiency and practicality of formalized algebraic reasoning in proof assistants.

This work addresses the challenge of automatically generating geometric rules and cut-free proofs from classical modal formulas in modal logic, focusing on analytic inductive formulas and their first-order correspondents. We introduce the MASSA algorithm—the first method to jointly synthesize analytic geometric rules and canonical cut-free derivations in the G3K labelled sequent calculus directly from modal formulas. We rigorously prove MASSA’s termination and correctness for all analytic inductive formulas and establish the logical equivalence between the generated geometric rules and their first-order frame correspondents. Furthermore, we demonstrate the completeness of the SCAN algorithm for second-order quantifier elimination on this class of formulas, and extend our approach to general inductive formulas and modal logics with first-order quantifiers. The results significantly advance automated theorem proving and semantic correspondence analysis in modal logic.

This work proposes a composable program verification framework grounded in dependent type theory, designed to harmonize modular development with formal verification. The approach characterizes program interfaces using polynomial functors, models implementations via Kleisli morphisms of free monads, and encodes pre- and postconditions through dependent polynomials. Wiring diagrams are employed to support compositional reasoning about correctness. The key theoretical contribution lies in uncovering a lax monoidal structure mapping specifications to interfaces, together with compatible lax monoidal natural transformations between presheaves, thereby establishing a foundation for concurrent and relational verification scenarios. The entire framework has been formalized in Agda, demonstrating both its feasibility and strong compositional properties.

This work addresses the efficient computation of sum-of-squares multipliers (i.e., certificates) for non-negative univariate polynomials within Archimedean saturated quadratic modules, thereby verifying their membership. To this end, the authors propose a novel symbolic algorithm that leverages the natural generators introduced by Kuhlmann and Marshall, incorporates the Basic Lemma to decompose non-negative factors, and employs a systematic case analysis to achieve, for the first time, a constructive transformation from natural to primitive generators. This approach establishes a complete framework for certificate construction in univariate Archimedean saturated quadratic modules. Implementation in Maple demonstrates the algorithm’s effectiveness and superiority, successfully handling several instances where RealCertify fails.

Current large language models exhibit limited robustness in automatic formalization tasks when confronted with non-idealized, perturbed informal proofs, struggling to simultaneously maintain consistency and faithfulness. This work introduces the first robustness evaluation framework tailored to this task, proposing two types of perturbations—global style rewrites and local factual corruptions—and constructing a corresponding benchmark dataset. By integrating natural language perturbation generation, formal consistency verification, and automated faithfulness assessment, the study systematically evaluates seven state-of-the-art models. Results reveal that existing approaches are generally sensitive to global perturbations and fail to accurately capture localized modifications, exposing significant deficiencies in their robustness for proof formalization.

This work addresses the poor scalability of traditional symbolic methods in automatically proving high-dimensional polynomial inequalities by introducing NSPI, a novel neurosymbolic framework that deeply integrates large language models (LLMs), symbolic computation, and formal verification in Lean. The approach leverages an LLM to generate sum-of-squares (SOS) decomposition conjectures, which are then refined into exact representations through symbolic optimization and formally verified within Lean, thereby establishing an end-to-end pipeline from heuristic discovery to rigorous proof. By synergistically combining the exploratory power of neural models with the mathematical rigor of symbolic and formal methods, NSPI achieves significantly improved efficiency and scalability, demonstrating strong performance on benchmarks involving up to ten variables.

This work addresses a central challenge in system security: formally verifying that system designs and implementations satisfy intended safety properties and support security certification. The authors propose a systematic approach grounded in proof assistants, integrating interactive theorem proving and formal methods to precisely model and machine-check critical security properties across diverse domains—including system security, language-level security, secure compilation, and cryptography. By enabling rigorous, machine-verifiable proofs of correctness, this methodology significantly strengthens the formal assurance of security properties and provides a unified theoretical framework and toolchain for constructing verifiable and certifiable secure systems.