symbolic manipulation

We introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity $k$ within a specified operator basis and determines which admit closed-form antiderivatives within the same class. This allows us to compute the "integrability fraction" $ρ(k)$ (the fraction of functions whose derivatives lie within the same class), which we do for five operator bases including combinations of rational functions, powers, exponentials, logarithms and trigonometric functions. We find that $ρ(k)$ declines at high complexity and that the operator basis has a dramatic effect -- in particular, adding the logarithm boosts $ρ(k)$ by a factor of $\sim$3 and produces or exacerbates a clear peak at $k=6$. We also deploy ESI as a novel integration algorithm, identifying three integrals that resist SymPy, Mathematica, RUBI, FriCAS, Maxima and Giac under all tested strategies. When an antiderivative can be found by multiple methods, ESI often returns the simplest form. These results reveal that the landscape of symbolic integrability is shaped primarily by the choice of operators, and that exhaustive enumeration can systematically discover integrable forms -- including novel ones -- that elude computer albegra systems.

Quantum computing lacks symbolic modeling and reasoning theories supporting automated verification, hindering the practical deployment of formal verification tools. To address this, we propose Symbolic Operator Logic (SOL), the first framework embedding classical first-order logic into a quantum operator language—enabling recursive symbolic definitions of quantum data and operations, as well as deductive reasoning about their properties. SOL modularly incorporates classical theories—including modal Boolean algebras and group theory—thereby allowing off-the-shelf automated verification techniques to be directly applied to quantum systems. The framework provides a scalable, composable infrastructure for quantum theorem proving in proof assistants such as Lean and Coq. By bridging the critical gap between classical formal methods and quantum verification, SOL significantly enhances the capability to formally verify quantum algorithms and protocols.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses the challenges of ensuring termination, semantic reliability, and completeness for user-defined modules in algebraic simplification libraries. We propose a generic algebraic modeling paradigm based on dependent types, formalizing algebraic structures via free algebras (fral) and variable extensions (frex), and enforcing simplification rules at the type level using dependently typed languages (Idris2/Agda). This guarantees strong normalization, semantic soundness, and completeness under a given equational theory for both built-in and user-defined modules. We introduce a novel “interface–implementation” separation, enabling dual modularity: reuse of foundational infrastructure (term representation, reflection, certification) and compositional nesting of existing simplification modules. We experimentally implement verified simplifiers for monoids and their variants (commutative, involutive), demonstrating feasibility, scalability, and high reusability within real-world theorem-proving environments.

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 proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

Many equations in science lack closed-form solutions, and existing symbolic regression methods rely on input-output data, making it difficult to solve equations using only their mathematical form. This work proposes a Symbolic Equation Solver (SES), which, for the first time, frames equation solving as a data-free optimization problem over a differentiable symbolic model. SES constructs an objective function solely from the equation’s structure and its initial or boundary conditions, without requiring any observed data. By leveraging symbolic expression generation techniques, the method directly recovers compact and accurate analytical solutions from the equation itself. It demonstrates success across diverse equation types—including algebraic equations, those involving transcendental terms, ordinary differential equations, and partial differential equations—thereby overcoming the traditional dependence of symbolic regression on training data.

This work proposes ViSA-R2, a novel framework designed to recover analytical solutions from 2D steady-state physical field visualizations and their derivatives, thereby enabling AI-driven scientific reasoning. The method establishes an end-to-end pipeline that emulates the physicist’s reasoning process—structured as structure identification, analytical hypothesis formulation, parameter derivation, and consistency verification—and introduces a self-validating, solution-centered chain-of-thought mechanism. Built upon Qwen3-VL (8B) and SymPy, the study also releases ViSA-Bench, the first benchmark supporting vision-language models for symbolic reasoning in physics, evaluated through multi-dimensional metrics including numerical accuracy, structural similarity, and character-level precision. Experiments demonstrate that ViSA-R2 significantly outperforms both open- and closed-source vision-language models across 30 linear steady-state scenarios, achieving high-fidelity symbolic expression inference.