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At least 19 recordsLinked to original sources
arXiv2026

Spectral and combinatorial methods for efficiently computing the rank of unambiguous finite automata

A zero-one matrix is a matrix with entries from $\{0, 1\}$. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties. Let $\mathcal{A}$ be a finite set of $n \times n$ zero-one matrices generating a monoid of zero-one matrices, and $m$ be the cardinality of $\mathcal{A}$. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by $\mathcal{A}$. By using linear-algebraic techniques, we show that this problem is in $\textsf{NC}$ and can be solved in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^2)$ space. We also provide a combinatorial algorithm finding a matrix of minimum rank in $\mathcal{O}(mn^4)$ time and $\mathcal{O}(n^3)$ space. As a byproduct, we show a very weak version of a generalisation of the Černý conjecture: there always exists a straight line program of size $\mathcal{O}(n^2)$ describing a product resulting in a matrix of minimum rank. For the special case corresponding to total DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of all states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time $\mathcal{O}(n^3 + mn^2)$ in this case.

cs.FL↗
arXiv2026

Confluence of conditional rewriting modulo

Sets of equations E play an important computational role in rewriting-based systems R. The equivalence relation =E induced by E introduces a partition of terms into E-equivalence classes on which rewriting computations, denoted ->R/E and called rewriting modulo E, are issued. This paper investigates confluence of ->R/E, usually called E-confluence, for conditional rewriting-based systems, where rewriting steps are determined by conditional rules. We rely on Jouannaud and Kirchner's framework to investigate confluence of an abstract relation R modulo an abstract equivalence relation E on a set A. We show how to particularize such a framework to be used with conditional systems. Then, we show how to define appropriate finite sets of conditional pairs to prove and disprove E-confluence. We introduce (i) Logic-based Conditional Critical Pairs, which do not require the use of (often infinitely many) E-unifiers to provide a finite representation of the local peaks considered in the abstract framework. We also introduce (ii) parametric Conditional Variable Pairs which are essential to deal with conditional rules in the analysis of E-confluence. Finally, we introduce (iii) Down Conditional Pairs which are often necessary to disprove E-confluence. Our results apply to well-known classes of rewriting-based systems, improving on previous results. As for unconditional systems, our results apply to Equational Term Rewriting Systems, first investigated by Huet and then by Jouannaud, and Jouannaud and Kirchner, among others. As for conditional systems, our results also apply to conditional rewrite theories and Maude.

cs.LO↗
arXiv2026

Physical Law Ecology: mapping multi-mechanism ecologies as the zeroth step of data-driven scientific discovery

Every data-driven equation discovery method assumes (implicitly and without verification) that the target system obeys a single governing law ($K{=}1$). Here we show that this assumption is the primary bottleneck limiting scientific discovery in multi-mechanism systems, and introduce Physical Law Ecology, a framework that makes $K^*$ (the number of coexisting independent mechanisms) itself the first quantity to be determined from data. The framework automatically mines a pool of topologically distinct candidate equations, constructs a continuous dominance weight field across parameter space, and discovers analytic evolution laws governing mechanism succession---with optional monotonicity constraints encoding irreversible physics. Across four unrelated systems (elastomer mechanics, pool boiling, galactic dynamics, and droplet evaporation), BIC consistently identifies $K^*{=}3$ independent governing topologies. Applied to 163 SPARC galaxies (3,269 spatially resolved measurements), the framework autonomously recovers three gravitational laws whose coexistence provides evidence against the single-universal-acceleration hypothesis of MOND ($p<10^{-34}$). In engineering applications, multi-law weighted prediction reduces error by 67-72\% over single-equation baselines while retaining full interpretability. By establishing the determination of $K^*$ as the zeroth step of scientific discovery-prior to and independent of equation search---this work opens a direction orthogonal to existing symbolic regression: not finding better equations, but mapping the ecology of mechanisms that govern complex systems.

cs.SC↗
arXiv2026

Faster modular composition using two relation matrices

Modular composition is the problem of computing the composition of two univariate polynomials modulo a third one. For a long time, the fastest algebraic algorithm for this problem was that of Brent and Kung (1978). Recently, we improved Brent and Kung's algorithm by computing and using a polynomial matrix that encodes a certain basis of algebraic relations between the polynomials. This is further improved here by making use of two polynomial matrices of smaller dimension. Under genericity assumptions on the input, this results in an algorithm using $\tilde{O}(n^{(ω+3)/4})$ arithmetic operations in the base field, where $ω$ is the exponent of matrix multiplication. With naive matrix multiplication, this is $\tilde{O}(n^{3/2})$, while with the best currently known exponent $ω$ this is $O(n^{1.343})$, improving upon the previously most efficient algorithms.

cs.SC↗
arXiv2026

Computing submatrices of the Hermite normal form of a structured polynomial matrix

Following several decades of successive algorithmic improvements, works from the 2010s have showed how to compute the Hermite normal form (HNF) of a univariate polynomial matrix within a complexity bound which is essentially that of polynomial matrix multiplication. Recently, several results on bivariate polynomials and Gröbner bases have highlighted the interest of computing determinants or HNFs of polynomial matrices that happen to be structured, with a small displacement rank. In such contexts, a small leading principal submatrix of the HNF often contains all the sought information. In this article, we show how the displacement structure can be exploited in order to accelerate the computation of such submatrices. To achieve this, we rely on structured linear algebra over the field thanks to evaluation-interpolation. This allows us to recover some rows of the inverse of the input matrix, from which we deduce the sought HNF submatrix via bases of relations.

cs.SC↗
arXiv2026

Quadratization of Autonomous Partial Differential Equations: Algorithmic Solutions

Quadratization for partial differential equations (PDEs) is a process that formally transforms a PDE with a nonquadratic right-hand side into a quadratic form by introducing auxiliary variables. Even though the existence and uniqueness of the solution of this quadratic form are, as of yet, unknown in the general case, this symbolic transformation has been used in diverse fields to simplify the analysis, simulation, and control of PDE models. This paper presents a rigorous definition of PDE quadratization, a sample case study on the solutions of quadratic representations, and theoretical contributions for the PDE quadratization problem of spatially one-dimensional PDEs, including results on existence and complexity. Its main focus, however, is introducing and analyzing QuPDE, an algorithm based on symbolic computation and discrete optimization that outputs a quadratization for any spatially one-dimensional polynomial or rational PDE. This algorithm is the first computational tool to find quadratizations for PDEs to date. We demonstrate QuPDE's performance by applying it to fourteen nonquadratic PDEs in diverse areas such as fluid mechanics, space physics, chemical engineering, and biological processes. QuPDE delivers a low-order quadratization in each case, uncovering quadratic transformations with fewer auxiliary variables than those previously discovered in the literature for some examples, and finding quadratizations for systems that had not been transformed to quadratic form before.

cs.SC↗
arXiv2026

Parallel Integration over Simple Radical Extensions

The parallel Risch (Risch--Norman) method is a fast heuristic for computing elementary integrals over towers of transcendental extensions. Its justification rests on two structural facts about the integral: a bound on its denominator and a description of the logarithms that can occur. Both are known for purely logarithmic towers (Davenport--Trager) and, in the form of a structure theorem, for arbitrary derivations on multivariate rational function fields (Bronstein). We extend both facts to a simple radical extension $L=K(y)$, $y^m=q$, of such a field. The key observations are that the integral closure of $F[t_1,\dots,t_n]$ in $L$ has an explicit basis, so that all factorisation can remain in a polynomial ring, and that the derivation has a well-defined pole order $δ_P\in\{0,1,e_P\}$ at every height-one prime $P$, so that pole orders of derivatives shift by $δ_P$. The denominator of the integral then has the same Hermite-type shape as in the transcendental case, while the admissible logands are precisely the $S$-units of the integral closure for an explicit finite set $S$ of primes; the latter can be larger than the set generated by irreducible polynomials, as the unit $x+\sqrt{x^2+1}$ shows. For $n=1$ we relate these $S$-units to torsion in the Jacobian and, for $m=2$, to the polynomial Pell equation, obtaining a complete description of the logarithmic part in genus~0. We describe the resulting algorithm and give examples.

cs.SC↗
arXiv2026

New bounds for the support of input-output equations in differential-algebraic systems

Given a polynomial dynamical system $\mathbf{x}'=\mathbf{f}(\mathbf{x},\mathbf{u})$ together with an observation function $y=g(\mathbf{x},\mathbf{u})$, where $\mathbf{x}=(x_1,\ldots,x_n)$, $\mathbf{u}=(u_1,\ldots,u_m)$ and $y$ are differential variables, and $\mathbf{f}=(f_1,\ldots,f_n)$, $g$ are polynomials with coefficients in a differential field, we study the problem of determining a minimal polynomial differential equation satisfied by the inputs $\mathbf{u}$ and the output $y$ which follows as a differential consequence of the system. We provide a characterization of a finite superset of the set of monomials appearing with non-zero coefficients in this input-output equation. Specifically, we establish an upper bound for the degree of the minimal polynomial and a family of inequalities that define a polytope containing its Newton polytope. These results extend recent work by Mukhina and Pogudin for systems with constant parameters, and enable the use of evaluation-interpolation techniques for the efficient computation of such eliminant polynomials.

math.AG↗
arXiv2026

Approximating CDTW Distance of Piecewise Algebraic Curves

Curves as input data naturally arise in a variety of fields including finance, seismology, medicine, spatio-temporal data mining, malicious activity detection, and more. A common way to analyze these data sets is to do similarity matching or clustering. The most common metrics used for measuring similarity of curves are Dynamic Time Warping (DTW) and Fréchet distance. These metrics are sensitive to sampling rate and outliers respectively, and do not yield robust outcomes. Continuous Dynamic Time Warping (CDTW) is a more robust distance metric that improves upon DTW and Fréchet distances. Existing algorithms for CDTW are either exact algorithms that focus on non-Euclidean norms and piecewise linear curves, or approximation algorithms limited to piecewise linear curves. We present an approximation algorithm for computing the CDTW distance under Euclidean norm between piecewise (higher degree) algebraic curves. That is, we present a fully polynomial-time approximation scheme (FPTAS) of multiplicative error $\varepsilon$, with $O \left( (m+n)^{\frac{19}{6}} (\frac{1}{\varepsilon})^{\frac{10}{3}} \log \left( \frac{ (m+n) }{\varepsilon^2} \right) \right)$ complexity, where $m$ and $n$ are the number of pieces of the two input curves.

cs.CG↗
arXiv2026

Towards a universal language of concepts: A survey

Humans can learn and generalize novel concepts from sparse data because they express knowledge in rich structural formats. In this paper, we propose that programs are a strong candidate for universal representation of concepts. We review computational models of concept learning that use programs as their concept representation and evaluate their contribution toward a universal representational language.

cs.AI↗
arXiv2026

Recurrences for permutations with long increasing subsequences

We prove two simple bivariate recurrences for the number of permutations with a long increasing subsequence. The two recurrences imply D-finiteness of the sequence in a certain range. As a consequence, we also obtain a proof of a conjecture posed by Kauers and Koutschan in 2023.

math.CO↗
arXiv2026

Degree bounds and synchronization in Gröbner basis computations for affine semi-regular systems

Determining the complexity of computing Gröbner bases is an important problem in both theory and practice, and solving degrees provide a central measure of this complexity. We study solving degrees and Gröbner basis computations for affine polynomial systems, with particular emphasis on semi-regular sequences. We first derive two upper bounds for the maximum Gröbner basis degree of the homogenized system. One is based on a regular initial subsequence of the highest-degree homogeneous parts. When these parts form a semi-regular sequence in nondecreasing degree order, the bound involves the $n$ smallest input degrees together with the largest one. The other bound is expressed in terms of the saturation exponent with respect to the homogenizing variable. Both are obtained by bounding the degree from which the Hilbert function of the quotient ring associated with the homogenized system is constant. We then compare the Buchberger-like Gröbner basis computations for an affine system, its homogenization, and its highest-degree homogeneous parts. The first degree fall is characterized by failure of injectivity of multiplication by the homogenizing variable. Before that point, choices of S-pairs and reducers in any computation can be matched in the others, and reduction sequences, remainders, intermediate bases, and leading monomials correspond under specialization. Cryptographic semi-regularity guarantees this correspondence until the step degree first reaches the degree of regularity. At that degree, affine reduction steps that preserve the sugar degree lift to homogeneous ones, yielding upper bounds on the algorithmic solving degree for a computation starting directly from the affine input.

math.AC↗
arXiv2026

Deciding superellipticity and computing the Weierstrass normal form

Let \( \mathcal{S}_{g,n} \subset \mathcal{M}_g \) be the locus of curves of genus \( g \geq 2 \) admitting a model \( y^n = h(x) \) with \( h \) separable; such curves $C$ have a cyclic group \( C_n \leq \operatorname{Aut}(C) \) of order \( n \) with \( C/C_n \cong \mathbb{P}^1 \). % We give an algorithm which, given an absolutely irreducible plane model \( F(x,y) = 0 \) of a curve \( C \) over a field \( k_0 \) of characteristic zero, decides for which \( n \) the curve lies in \( \mathcal{S}_{g,n} \) and returns a model \( y^n = h(x) \) together with the birational transformation to it.

math.AG↗
arXiv2026

The Indefinite Summation Problem for the Laurent Ring

This article solves the Indefinite Summation Problem (ISP) for the difference ring $(A, α)$, where $A$ is the Laurent ring of shift operators on the lattice $\Z^n$, and $α$ is any ring automorphism of $A$ of finite order. The solution translates to a finite procedure involving a matrix multiplication, where the size of the matrix can be estimated. It follows that the arithmetic complexity of the solution can also be determined. These results extend to a solution of the ISP for the ring of functions on $\Z^n$, on which $α$ acts by duality. The article points out that the solution to the ISP amounts to calculating the group cohomologies $H^i([α], A), i = 0, 1$, where $[α]$ is the cyclic group generated by $α$.

cs.SC↗
arXiv2026

Neural Symbollic Regression Using Deep Learning and Sparse Modelling

Symbolic Regression (SR) seeks to find succinct mathematical expressions that represent the fundamental relationships within data, providing interpretability and scientific understanding that exceeds that of black-box models. Nevertheless, traditional methods like Genetic Programming face challenges with scalability and are highly sensitive to noise, while sparse regression techniques such as SINDy rely significantly on predetermined feature libraries. In this work, we present a Neural Symbolic Regression (NSR) framework that treats neural networks as functional preconditioners for symbolic discovery. Our approach uses a decoupled pipeline: a neural network first learns a smooth, noise-robust approximation of the target function in an interaction- aware nonlinear feature space. LASSO is then applied to extract sparse, interpretable closed-form expressions. To improve predictive accuracy and symbolic fidelity by integrating distributed hyperparameter optimization with Ray Tune and ASHA scheduling. Experiments on the Nguyen benchmark suite show that our approach consistently outperforms SINDy and non-tuned neural baselines in RMSE, noise robustness, and out-of-distribution generalization. Ablation studies confirm the significance of feature interactions, neural depth, and tuning strategies. In general, this study presents a scalable and understandable neural-symbolic framework, creating a solid link between neural approximation and the discovery of sparse equations for scientific machine learning.

cs.LG↗
arXiv2026

Ontology-Guided Neuro-Symbolic Inference: Grounding Language Models with Mathematical Domain Knowledge

Language models exhibit fundamental limitations -- hallucination, brittleness, and lack of formal grounding -- that are particularly problematic in high-stakes specialist fields requiring verifiable reasoning. I investigate whether formal domain ontologies can enhance language model reliability through retrieval-augmented generation. Using mathematics as proof of concept, I implement a neuro-symbolic pipeline leveraging the OpenMath ontology with hybrid retrieval and cross-encoder reranking to inject relevant definitions into model prompts. Evaluation on the MATH benchmark with three open-source models reveals that ontology-guided context improves performance when retrieval quality is high, but irrelevant context actively degrades it -- highlighting both the promise and challenges of neuro-symbolic approaches.

cs.AI↗
arXiv2026

Standard bases for shift-stable groups and Subgroup Membership in wreath products

We develop a notion of standard bases for subgroups of the restricted direct product $G^{(\mathbb{N}^n)}$ that are stable under translation by $\mathbb{N}^n$, where $G$ is an arbitrary finite group. We construct an algorithm that computes standard bases for such subgroups and use them to solve several algorithmic problems, including membership, saturation, and variable elimination. Our approach is inspired by Buchberger's algorithm and the theory of Gröbner bases for ideals in polynomial rings. Building on the standard bases and our solutions to the algorithmic problems above, we prove that Subgroup Membership is decidable in wreath products $G \wr \mathbb{Z}^n$ for finite $G$ and $n \in \mathbb{N}$.

math.GR↗