algebraic manipulation

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.

This paper addresses the open problem of synthesizing guarded loops with polynomial invariants for formal verification. Prior approaches are limited to affine, unguarded loops; ours is the first algebraic-geometry-driven method supporting nondeterministic branching loops with inequality guards, polynomial update maps, and arbitrary polynomial invariants. Our core contributions are threefold: (1) introducing a novel class of synthesizable invariants; (2) reducing loop synthesis to solving systems of multivariate polynomial equations over the rationals; and (3) integrating algebraic-geometric techniques—such as Gröbner bases and elimination theory—with SMT solvers to efficiently compute solution spaces. We implement a prototype system and evaluate it on multiple benchmarks. Experimental results demonstrate its ability to synthesize finite-loop programs satisfying complex polynomial invariants, confirming correctness, effectiveness, and scalability.

To address the challenges of high formula complexity and poor recovery of true solutions in high-dimensional symbolic regression, this paper proposes an iterative variable-combination dimensionality reduction method grounded in functional dependency analysis. The method automatically identifies fixed algebraic combinations—such as sums, products, or ratios—among input variables and replaces them with new synthetic variables, thereby substantially reducing the search-space dimensionality while preserving semantic equivalence. Its core innovation lies in rigorously formalizing the validity of variable substitutions as a functional dependency verification problem and tightly coupling this verification with symbolic regression solvers to achieve end-to-end automated dimensionality reduction. Experiments demonstrate that the method robustly identifies effective variable combinations and consistently improves formula recovery rates and convergence speed for leading symbolic regression algorithms—including SR-Tree, AI-Feynman, and DeepSymbolic—across multiple benchmark datasets. On average, recovered expressions exhibit over 35% lower complexity.

This work addresses the challenge of reliably translating formalized mathematics into natural language that is both precise and readable. It proposes a symbol-to-text framework based on an intermediate language architecture, extending conventional syntactic sugar mechanisms to support general mathematical expressions. By unifying formal systems such as Agda, Lean, and Rocq through Dedukti and integrating Grammatical Framework (GF), the approach ensures grammatical correctness and expressive diversity across multiple languages. The resulting system, Informath, generates fluent, accurate, and multilingual mathematical narratives at relatively low development cost, effectively rendering AI-generated or automatically formalized proofs into comprehensible expository text.

This paper uncovers the algebraic essence underlying convex analysis, Gaussian probability, and quadratic structure. To this end, we introduce Graphical Quadratic Algebra (GQA)—a novel algebraic framework based on chordal graphs—that uniformly models quadratic relations, Gaussian stochastic processes, and nondeterministic Gaussian processes via rotation-invariant quadratic generators. We provide the first sound and complete axiomatic characterization of three fundamental models: least-squares estimation, Gaussian randomness, and nondeterminism—revealing their shared conditional algebraic structure. Our method integrates string diagram theory, categorical semantics, and formal semantics of probabilistic programming. Theoretical contributions include soundness and completeness proofs for all three models within GQA. Applications demonstrate efficacy in linear regression, probabilistic programming, and noisy circuit modeling.

This work investigates the creative space of mathematical proofs under constraints, with a particular focus on the impact of non-constructive reasoning. We introduce a strategy ablation methodology that integrates our custom-built Meno automated formalization tool with Goedel Prover embeddings to systematically explore both formal and informal proof spaces for foundational theorems from *Analysis I* within the Lean theorem prover. Our experiments successfully generate a novel class of machine-produced proofs, revealing that these proofs cluster along low-dimensional submanifolds in a high-dimensional representation space and significantly diverge from human-constructed proof trajectories. This study provides the first quantitative characterization of the structural differences between machine-generated and human proofs.

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 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 exponential blowup arising from structural mismatches in conversions between conjunctive normal form (CNF) and algebraic normal form (ANF) by introducing power-term polynomial algebra as a novel intermediate representation. The proposed framework seamlessly integrates CNF clauses with structured monomials, enabling symbolic manipulation without introducing auxiliary variables or expanding into conventional ANF. By defining power terms, local rewriting rules, and a systematic product reconstruction mechanism, it establishes a unified formalism that preserves the compactness of CNF while supporting the algebraic operability of ANF. The resulting canonical representation compactly encodes disjunctive clauses and facilitates efficient local transformations, offering a new paradigm for hybrid reasoning and structure-aware bidirectional conversion between CNF and ANF.

This work addresses the scalability bottlenecks in formal verification of gate-level highly optimized arithmetic circuits, which stem from state-space explosion and intermediate polynomial blowup. To overcome these challenges, the paper proposes a parameterized symbolic algebraic verification framework that uniformly models general arithmetic circuits—supporting arbitrary combinations of addition and multiplication—as multivariate polynomials. The approach integrates existing techniques through a parameterized algebraic reduction theory, with key innovations including a unified parameter space encompassing multiple state-of-the-art methods, along with novel algorithms for HA-preserving extraction, density-aware elimination detection, and conservative polynomial size estimation. Experimental results on standard multipliers and practical arithmetic units such as multiply-adders and dot-product engines demonstrate that the proposed method achieves state-of-the-art verification performance.