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  • Guessing Gr{ö}bner Bases of Structured Ideals of Relations of Sequences
    arXiv.cs.SC Pub Date : 2020-09-11
    Jérémy BerthomieuPolSys; Mohab Safey El DinPolSys

    Assuming sufficiently many terms of a n-dimensional table defined over a field are given, we aim at guessing the linear recurrence relations with either constant or polynomial coefficients they satisfy. In many applications, the table terms come along with a structure: for instance, they may be zero outside of a cone, they may be built from a Gr{\"o}bner basis of an ideal invariant under the action

  • PolyAdd: Polynomial Formal Verification of Adder Circuits
    arXiv.cs.SC Pub Date : 2020-09-07
    Rolf Drechsler

    Only by formal verification approaches functional correctness can be ensured. While for many circuits fast verification is possible, in other cases the approaches fail. In general no efficient algorithms can be given, since the underlying verification problem is NP-complete. In this paper we prove that for different types of adder circuits polynomial verification can be ensured based on BDDs. While

  • On FGLM Algorithms with Tropical Gröbner bases
    arXiv.cs.SC Pub Date : 2020-09-04
    Yuki IshiharaXLIM; Tristan VacconXLIM; Kazuhiro Yokoyama

    Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article

  • Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations
    arXiv.cs.SC Pub Date : 2020-09-03
    Vladimir P. Gerdt; Daniel Robertz; Yuri A. Blinkov

    For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations. Our approach is applicable to regular solution grids. For the grids of this type we give a new definition of s-consistency for finite difference approximations which

  • On a non-archimedean broyden method
    arXiv.cs.SC Pub Date : 2020-09-03
    Xavier DahanXLIM; Tristan VacconXLIM

    Newton's method is an ubiquitous tool to solve equations, both in the archimedean and non-archimedean settings -- for which it does not really differ. Broyden was the instigator of what is called "quasi-Newton methods". These methods use an iteration step where one does not need to compute a complete Jacobian matrix nor its inverse. We provide an adaptation of Broyden's method in a general non-archimedean

  • Computing critical points for invariant algebraic systems
    arXiv.cs.SC Pub Date : 2020-09-02
    Jean-Charles FaugèrePolSys; George LabahnSCG; Mohab Safey El DinPolSys; Éric SchostSCG; Thi Xuan VuPolSys, SCG

    Let $\mathbf{K}$ be a field and $\phi$, $\mathbf{f} = (f_1, \ldots, f_s)$ in $\mathbf{K}[x_1, \dots, x_n]$ be multivariate polynomials (with $s < n$) invariant under the action of $\mathcal{S}_n$, the group of permutations of $\{1, \dots, n\}$. We consider the problem of computing the points at which $\mathbf{f}$ vanish and the Jacobian matrix associated to $\mathbf{f}, \phi$ is rank deficient provided

  • Homotopy techniques for solving sparse column support determinantal polynomial systems
    arXiv.cs.SC Pub Date : 2020-09-02
    George LabahnSCG; Mohab Safey El DinPolSys; Éric SchostSCG; Thi Xuan VuPolSys, SCG

    Let $\mathbf{K}$ be a field of characteristic zero with $\overline{\mathbf{K}}$ its algebraic closure. Given a sequence of polynomials $\mathbf{g} = (g_1, \ldots, g_s) \in \mathbf{K}[x_1, \ldots , x_n]^s$ and a polynomial matrix $\mathbf{F} = [f_{i,j}] \in \mathbf{K}[x_1, \ldots, x_n]^{p \times q}$, with $p \leq q$, we are interested in determining the isolated points of $V_p(\mathbf{F},\mathbf{g})$

  • Robots, computer algebra and eight connected components
    arXiv.cs.SC Pub Date : 2020-08-31
    Jose CapcoJKU; Mohab Safey El DinPolSys; Josef SchichoRISC

    Answering connectivity queries in semi-algebraic sets is a long-standing and challenging computational issue with applications in robotics, in particular for the analysis of kinematic singularities. One task there is to compute the number of connected components of the complementary of the singularities of the kinematic map. Another task is to design a continuous path joining two given points lying

  • Towards a noncommutative Picard-Vessiot theory
    arXiv.cs.SC Pub Date : 2020-08-25
    G. DuchampLIPN; Viincel Hoang Ngoc Minh; Vu Nguyen Dinh

    A Chen generating series, along a path and with respect to $m$ differential forms, is a noncommutative series on $m$ letters and with coefficients which are holomorphic functions over a simply connected manifold in other words a series with variable (holomorphic) coefficients. Such a series satisfies a first order noncommutative differential equation which is considered, by some authors, as the universal

  • Exact $p$-adic computation in Magma
    arXiv.cs.SC Pub Date : 2020-08-24
    Christopher Doris

    We describe a new arithmetic system for the Magma computer algebra system for working with $p$-adic numbers exactly, in the sense that numbers are represented lazily to infinite $p$-adic precision. This is the first highly featured such implementation. This has the benefits of increasing user-friendliness and speeding up some computations, as well as forcibly producing provable results. We give theoretical

  • Computing singular elements modulo squares
    arXiv.cs.SC Pub Date : 2020-08-24
    Przemysław Koprowski

    The group of singular elements was first introduced by Helmut Hasse and later it has been studied by numerous authors including such well known mathematicians as: Cassels, Furtw\"{a}ngler, Hecke, Knebusch, Takagi and of course Hasse himself; to name just a few. The aim of the present paper is to present algorithms that explicitly construct groups of singular and $S$-singular elements (modulo squares)

  • Computing the Real Isolated Points of an Algebraic Hypersurface
    arXiv.cs.SC Pub Date : 2020-08-24
    Huu Phuoc Le; Mohab Safey El Din; Timo de Wolff

    Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations

  • A Simple and Fast Algorithm for Computing the $N$-th Term of a Linearly Recurrent Sequence
    arXiv.cs.SC Pub Date : 2020-08-20
    Alin Bostan; Ryuhei Mori

    We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and $\mathsf{M}(d)$ denotes the number of arithmetic operations for computing the product of two polynomials of degree $d$. The state-of-the-art algorithm, due to Charles Fiduccia

  • Competition Report: CHC-COMP-20
    arXiv.cs.SC Pub Date : 2020-08-07
    Philipp RümmerUppsala University, Sweden

    CHC-COMP-20 is the third competition of solvers for Constrained Horn Clauses. In this year, 9 solvers participated at the competition, and were evaluated in four separate tracks on problems in linear integer arithmetic, linear real arithmetic, and arrays. The competition was run in the first week of May 2020 using the StarExec computing cluster. This report gives an overview of the competition design

  • An Experiment Combining Specialization with Abstract Interpretation
    arXiv.cs.SC Pub Date : 2020-08-07
    John P. GallagherRoskilde University, Denmark and IMDEA Software Institute, Spain; Robert GlückCopenhagen University, Denmark

    It was previously shown that control-flow refinement can be achieved by a program specializer incorporating property-based abstraction, to improve termination and complexity analysis tools. We now show that this purpose-built specializer can be reconstructed in a more modular way, and that the previous results can be achieved using an off-the-shelf partial evaluation tool, applied to an abstract interpreter

  • From Big-Step to Small-Step Semantics and Back with Interpreter Specialisation
    arXiv.cs.SC Pub Date : 2020-08-07
    John P. GallagherRoskilde University, Denmark and IMDEA Software Institute, Spain; Manuel HermenegildoIMDEA Software Institute, Spain; Bishoksan KafleIMDEA Software Institute, Spain; Maximiliano KlemenIMDEA Software Institute, Spain; Pedro López GarcíaIMDEA Software Institute, Spain; José MoralesIMDEA Software Institute, Spain

    We investigate representations of imperative programs as constrained Horn clauses. Starting from operational semantics transition rules, we proceed by writing interpreters as constrained Horn clause programs directly encoding the rules. We then specialise an interpreter with respect to a given source program to achieve a compilation of the source language to Horn clauses (an instance of the first Futamura

  • Proceedings 8th International Workshop on Verification and Program Transformation and 7th Workshop on Horn Clauses for Verification and Synthesis
    arXiv.cs.SC Pub Date : 2020-08-06
    Laurent FribourgCNRS & ENS Paris-Saclay, France; Matthias HeizmannUniversity of Freiburg, Germany

    The proceedings consist of a keynote paper by Alberto followed by 6 invited papers written by Lorenzo Clemente (U. Warsaw), Alain Finkel (U. Paris-Saclay), John Gallagher (Roskilde U. and IMDEA Software Institute) et al., Neil Jones (U. Copenhagen) et al., Michael Leuschel (Heinrich-Heine U.) and Maurizio Proietti (IASI-CNR) et al.. These invited papers are followed by 4 regular papers accepted at

  • Trace Logic for Inductive Loop Reasoning
    arXiv.cs.SC Pub Date : 2020-08-04
    Pamina Georgiou; Bernhard Gleiss; Laura Kovács

    We propose trace logic, an instance of many-sorted first-order logic, to automate the partial correctness verification of programs containing loops. Trace logic generalizes semantics of program locations and captures loop semantics by encoding properties at arbitrary timepoints and loop iterations. We guide and automate inductive loop reasoning in trace logic by using generic trace lemmas capturing

  • Graphical Conditions for Rate Independence in Chemical Reaction Networks
    arXiv.cs.SC Pub Date : 2020-07-17
    Elisabeth DegrandLifeware; François FagesLifeware; Sylvain SolimanLifeware

    Chemical Reaction Networks (CRNs) provide a useful abstraction of molecular interaction networks in which molecular structures as well as mass conservation principles are abstracted away to focus on the main dynamical properties of the network structure. In their interpretation by ordinary differential equations, we say that a CRN with distinguished input and output species computes a positive real

  • Formal Power Series on Algebraic Cryptanalysis
    arXiv.cs.SC Pub Date : 2020-07-29
    Shuhei Nakamura

    In cryptography, attacks that utilize a Gr\"{o}bner basis have broken several cryptosystems. The complexity of computing a Gr\"{o}bner basis dominates the overall computing and its estimation is important for such cryptanalysis. The complexity is given by using the solving degree, but it is hard to decide this value of a large scale system arisen from cryptography. Thus the degree of regularity and

  • Parameter identifiability and input-output equations
    arXiv.cs.SC Pub Date : 2020-07-28
    Alexey Ovchinnikov; Gleb Pogudin; Peter Thompson

    Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input-output equations and, in particular, characteristic sets of differential ideals. The precise relation between identifiability and input-output

  • On Bergman's Diamond Lemma for Ring Theory
    arXiv.cs.SC Pub Date : 2020-07-27
    Takao Inoué

    This expository paper deals with the Diamond Lemma for ring theory, which is proved in the first section of G.M. Bergman, The Diamond Lemma for Ring Theory, Advances in Mathematics, 29 (1978), pp. 178--218. No originality of the present note is claimed on the part of the author, except for some suggestions and figures. Throughout this paper, I shall mostly use Bergman's expressions in his paper.

  • Cyclotomic Identity Testing and Applications
    arXiv.cs.SC Pub Date : 2020-07-26
    Nikhil Balaji; Sylvain Perifel; Mahsa Shirmohammadi; James Worrell

    We consider the cyclotomic identity testing problem: given a polynomial $f(x_1,\ldots,x_k)$, decide whether $f(\zeta_n^{e_1},\ldots,\zeta_n^{e_k})$ is zero, for $\zeta_n = e^{2\pi i/n}$ a primitive complex $n$-th root of unity and integers $e_1,\ldots,e_k$. We assume that $n$ and $e_1,\ldots,e_k$ are represented in binary and consider several versions of the problem, according to the representation

  • Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator
    arXiv.cs.SC Pub Date : 2020-07-24
    Pavel TrutmanCIIRC; Safey El Din MohabSU; Didier HenrionLAAS; Tomas PajdlaCIIRC

    The Inverse Kinematics (IK) problem is to nd robot control parameters to bring it into the desired position under the kinematics and collision constraints. We present a global solution to the optimal IK problem for a general serial 7DOF manipulator with revolute joints and a quadratic polynomial objective function. We show that the kinematic constraints due to rotations can all be generated by second-degree

  • Groebner basis structure of ideal interpolation
    arXiv.cs.SC Pub Date : 2020-07-23
    Yihe Gong; Xue Jiang

    We study the relationship between certain Groebner bases for zero dimensional ideals, and the interpolation condition functionals of ideal interpolation. Ideal interpolation is defined by a linear idempotent projector whose kernel is a polynomial ideal. In this paper, we propose the notion of "reverse" complete reduced basis. Based on the notion, we present a fast algorithm to compute the reduced Groebner

  • On Algorithmic Estimation of Analytic Complexity for Polynomial Solutions to Hypergeometric Systems
    arXiv.cs.SC Pub Date : 2020-07-18
    Vitaly A. Krasikov

    The paper deals with the analytic complexity of solutions to bivariate holonomic hypergeometric systems of the Horn type. We obtain estimates on the analytic complexity of Puiseux polynomial solutions to the hypergeometric systems defined by zonotopes. We also propose algorithms of the analytic complexity estimation for polynomials.

  • Interpretable Control by Reinforcement Learning
    arXiv.cs.SC Pub Date : 2020-07-20
    Daniel Hein; Steffen Limmer; Thomas A. Runkler

    In this paper, three recently introduced reinforcement learning (RL) methods are used to generate human-interpretable policies for the cart-pole balancing benchmark. The novel RL methods learn human-interpretable policies in the form of compact fuzzy controllers and simple algebraic equations. The representations as well as the achieved control performances are compared with two classical controller

  • Computing stable resultant-based minimal solvers by hiding a variable
    arXiv.cs.SC Pub Date : 2020-07-17
    Snehal Bhayani; Zuzana Kukelova; Janne Heikkilä

    Many computer vision applications require robust and efficient estimation of camera geometry. The robust estimation is usually based on solving camera geometry problems from a minimal number of input data measurements, i.e., solving minimal problems, in a RANSAC-style framework. Minimal problems often result in complex systems of polynomial equations. The existing state-of-the-art methods for solving

  • Computing regular meromorphic differential forms via Saito's logarithmic residues
    arXiv.cs.SC Pub Date : 2020-07-20
    Katsusuke Nabeshima; Shinichi Tajima

    Logarithmic differential forms and logarithmic residues associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. An effective method is introduced for computing logarithmic residues. A relation between logarithmic differential forms and the Brieskorn formula on Gauss-Manin connection are discussed. Some examples are also given for illustration

  • Bit-Slicing the Hilbert Space: Scaling Up Accurate Quantum Circuit Simulation to a New Level
    arXiv.cs.SC Pub Date : 2020-07-18
    Yuan-Hung Tsai; Jie-Hong R. Jiang; Chiao-Shan Jhang

    Quantum computing is greatly advanced in recent years and is expected to transform the computation paradigm in the near future. Quantum circuit simulation plays a key role in the toolchain for the development of quantum hardware and software systems. However, due to the enormous Hilbert space of quantum states, simulating quantum circuits with classical computers is extremely challenging despite notable

  • On the Complexity of Binomialization for Polynomial Differential Equations
    arXiv.cs.SC Pub Date : 2020-07-17
    Mathieu HemeryLifeware; François FagesLifeware; Sylvain SolimanLifeware

    Chemical reaction networks (CRNs) are a standard formalism used in chemistry and biology to reason about the dynamics of molecular interaction networks. In their interpretation by ordinary differential equations, CRNs provide a Turing-complete model of analog computation , in the sense that any computable function over the reals can be computed by a finite number of molecular species with a continuous

  • Algorithmic applications of the corestriction of central simple algebras
    arXiv.cs.SC Pub Date : 2020-07-14
    Tímea Csahók; Péter Kutas; Gergely Zábrádi

    Let $L$ be a separable quadratic extension of either $\mathbb{Q}$ or $\mathbb{F}_q(t)$. We propose efficient algorithms for finding isomorphisms between quaternion algebras over $L$. Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple $L$-algebra. In order to obtain efficient algorithms in the characteristic 2 case, we propose an algorithm for finding

  • Learning Reasoning Strategies in End-to-End Differentiable Proving
    arXiv.cs.SC Pub Date : 2020-07-13
    Pasquale Minervini; Sebastian Riedel; Pontus Stenetorp; Edward Grefenstette; Tim Rocktäschel

    Attempts to render deep learning models interpretable, data-efficient, and robust have seen some success through hybridisation with rule-based systems, for example, in Neural Theorem Provers (NTPs). These neuro-symbolic models can induce interpretable rules and learn representations from data via back-propagation, while providing logical explanations for their predictions. However, they are restricted

  • ACORNS: An Easy-To-Use Code Generator for Gradients and Hessians
    arXiv.cs.SC Pub Date : 2020-07-09
    Deshana Desai; Etai Shuchatowitz; Zhongshi Jiang; Teseo Schneider; Daniele Panozzo

    The computation of first and second-order derivatives is a staple in many computing applications, ranging from machine learning to scientific computing. We propose an algorithm to automatically differentiate algorithms written in a subset of C99 code and its efficient implementation as a Python script. We demonstrate that our algorithm enables automatic, reliable, and efficient differentiation of common

  • A Family of Denominator Bounds for First Order Linear Recurrence Systems
    arXiv.cs.SC Pub Date : 2020-07-06
    Mark van Hoeij; Moulay Barkatou; Johannes Middeke

    For linear recurrence systems, the problem of finding rational solutions is reduced to the problem of computing polynomial solutions by computing a content bound or a denominator bound. There are several bounds in the literature. The sharpest bound leads to polynomial solutions of lower degrees, but this advantage need not compensate for the time spent on computing that bound. To strike the best balance

  • Error Correcting Codes, finding polynomials of bounded degree agreeing on a dense fraction of a set of points
    arXiv.cs.SC Pub Date : 2020-06-29
    Priyank Deshpande

    Here we present some revised arguments to a randomized algorithm proposed by Sudan to find the polynomials of bounded degree agreeing on a dense fraction of a set of points in $\mathbb{F}^{2}$ for some field $\mathbb{F}$.

  • Improvement on Extrapolation of Species Abundance Distribution Across Scales from Moments Across Scales
    arXiv.cs.SC Pub Date : 2020-07-01
    Saeid Alirezazadeh; Khadijeh Alibabaei

    Raw moments are used as a way to estimate species abundance distribution. The almost linear pattern of the log transformation of raw moments across scales allow us to extrapolate species abundance distribution for larger areas. However, results may produce errors. Some of these errors are due to computational complexity, fittings of patterns, binning methods, and so on. We provide some methods to reduce

  • Learning an arbitrary mixture of two multinomial logits
    arXiv.cs.SC Pub Date : 2020-07-01
    Wenpin Tang

    In this paper, we consider mixtures of multinomial logistic models (MNL), which are known to $\epsilon$-approximate any random utility model. Despite its long history and broad use, rigorous results are only available for learning a uniform mixture of two MNLs. Continuing this line of research, we study the problem of learning an arbitrary mixture of two MNLs. We show that the identifiability of the

  • Machine learning the real discriminant locus
    arXiv.cs.SC Pub Date : 2020-06-24
    Edgar A. Bernal; Jonathan D. Hauenstein; Dhagash Mehta; Margaret H. Regan; Tingting Tang

    Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a different number of real solutions, the parameter space is decomposed into regions whose

  • Noetherian operators and primary decomposition
    arXiv.cs.SC Pub Date : 2020-06-24
    Justin Chen; Marc Härkönen; Robert Krone; Anton Leykin

    Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic

  • Quantum Runge-Lenz Vector and the Hydrogen Atom, the hidden SO(4) symmetry using Computer Algebra
    arXiv.cs.SC Pub Date : 2020-06-22
    Pascal Szriftgiser; Edgardo S. Cheb-Terrab

    Pauli first noticed the hidden SO(4) symmetry for the Hydrogen atom in the early stages of quantum mechanics [1]. Departing from that symmetry, one can recover the spectrum of a spinless hydrogen atom and the degeneracy of its states without explicitly solving Schr\"odinger's equation [2]. In this paper, we derive that SO(4) symmetry and spectrum using a computer algebra system (CAS). While this problem

  • Neuro-Symbolic Visual Reasoning: Disentangling "Visual" from "Reasoning"
    arXiv.cs.SC Pub Date : 2020-06-20
    Saeed Amizadeh; Hamid Palangi; Oleksandr Polozov; Yichen Huang; Kazuhito Koishida

    Visual reasoning tasks such as visual question answering (VQA) require an interplay of visual perception with reasoning about the question semantics grounded in perception. Various benchmarks for reasoning across language and vision like VQA, VCR and more recently GQA for compositional question answering facilitate scientific progress from perception models to visual reasoning. However, recent advances

  • A unified framework for equivalences in social networks
    arXiv.cs.SC Pub Date : 2020-06-18
    Nina Otter; Mason A. Porter

    A key concern in network analysis is the study of social positions and roles of actors in a network. The notion of "position" refers to an equivalence class of nodes that have similar ties to other nodes, whereas a "role" is an equivalence class of compound relations that connect the same pairs of nodes. An open question in network science is whether it is possible to simultaneously perform role and

  • Toric Eigenvalue Methods for Solving Sparse Polynomial Systems
    arXiv.cs.SC Pub Date : 2020-06-18
    Matías R. Bender; Simon Telen

    We consider the problem of computing homogeneous coordinates of points in a zero-dimensional subscheme of a compact toric variety $X$. Our starting point is a homogeneous ideal $I$ in the Cox ring of $X$, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse

  • Logic, Probability and Action: A Situation Calculus Perspective
    arXiv.cs.SC Pub Date : 2020-06-17
    Vaishak Belle

    The unification of logic and probability is a long-standing concern in AI, and more generally, in the philosophy of science. In essence, logic provides an easy way to specify properties that must hold in every possible world, and probability allows us to further quantify the weight and ratio of the worlds that must satisfy a property. To that end, numerous developments have been undertaken, culminating

  • On the Complexity of Solving Generic Over-determined Bilinear Systems
    arXiv.cs.SC Pub Date : 2020-06-16
    John B. Baena; Daniel Cabarcas; Javier Verbel

    In this paper, we study the complexity of solving generic over-determined bilinear systems over a finite field $\mathbb{F}$. Given a generic bilinear sequence $B \in \mathbb{F}[\mathbf{x},\mathbf{y}]$, with respect to a partition of variables $\mathbf{x}$, $\mathbf{y}$, we show that, the solutions of the system $B= \mathbf{0}$ can be efficiently found on the $\mathbb{F}[\mathbf{y}]$-module generated

  • Computing Igusa's local zeta function of univariates in deterministic polynomial-time
    arXiv.cs.SC Pub Date : 2020-06-16
    Ashish Dwivedi; Nitin Saxena

    Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression

  • Pointer Data Structure Synthesis from Answer Set Programming Specifications
    arXiv.cs.SC Pub Date : 2020-06-12
    Sarat Chandra Varanasi; Neeraj Mittal; Gopal Gupta

    We develop an inductive proof-technique to generate imperative programs for pointer data structures from behavioural specifications expressed in the Answer Set Programming (ASP) formalism. ASP is a non-monotonic logic based formalism that employs negation-as-failure which helps emulate the human thought process, allowing domain experts to model desired system behaviour succinctly. We argue in this

  • Walsh functions, scrambled $(0,m,s)$-nets, and negative covariance: applying symbolic computation to quasi-Monte Carlo integration
    arXiv.cs.SC Pub Date : 2020-06-11
    Jaspar Wiart; Elaine Wong

    We investigate base $b$ Walsh functions for which the variance of the integral estimator based on a scrambled $(0,m,s)$-net in base $b$ is less than or equal to that of the Monte-Carlo estimator based on the same number of points. First we compute the Walsh decomposition for the joint probability density function of two distinct points randomly chosen from a scrambled $(t,m,s)$-net in base $b$ in terms

  • Decomposable sparse polynomial systems
    arXiv.cs.SC Pub Date : 2020-06-04
    Taylor Brysiewicz; Jose Israel Rodriguez; Frank Sottile; Thomas Yahl

    The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this package may be used to exploit this structure, with examples.

  • Analogical Proportions
    arXiv.cs.SC Pub Date : 2020-06-04
    Christian Antić

    Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper contributes to the foundations of artificial general intelligence by introducing an abstract algebraic framework of analogical proportions of the form `$a$ is to $b$ what $c$ is to $d$' in the general setting

  • Characterizing an Analogical Concept Memory for Newellian Cognitive Architectures
    arXiv.cs.SC Pub Date : 2020-06-02
    Shiwali Mohan; Matt Klenk; Matthew Shreve; Kent Evans; Aaron Ang; John Maxwell

    We propose a new long-term declarative memory for Soar that leverages the computational models of analogical reasoning and generalization. We situate our research in interactive task learning (ITL) and embodied language processing (ELP). We demonstrate that the learning methods implemented in the proposed memory can quickly learn a diverse types of novel concepts that are useful in task execution.

  • Good pivots for small sparse matrices
    arXiv.cs.SC Pub Date : 2020-06-02
    Manuel Kauers; Jakob Moosbauer

    For sparse matrices up to size $8 \times 8$, we determine optimal choices for pivot selection in Gaussian elimination. It turns out that they are slightly better than the pivots chosen by a popular pivot selection strategy, so there is some room for improvement. We then create a pivot selection strategy using machine learning and find that it indeed leads to a small improvement compared to the classical

  • Nonlinear observability algorithms with known and unknown inputs: analysis and implementation
    arXiv.cs.SC Pub Date : 2020-06-01
    Nerea Martínez; Alejandro F. Villaverde

    The observability of a dynamical system is affected by the presence of external inputs, either known (such as control actions) or unknown (disturbances). Inputs of unknown magnitude are especially detrimental for observability, and they also complicate its analysis. Hence the availability of computational tools capable of analysing the observability of nonlinear systems with unknown inputs has been

  • First Neural Conjecturing Datasets and Experiments
    arXiv.cs.SC Pub Date : 2020-05-29
    Josef Urban; Jan Jakubův

    We describe several datasets and first experiments with creating conjectures by neural methods. The datasets are based on the Mizar Mathematical Library processed in several forms and the problems extracted from it by the MPTP system and proved by the E prover using the ENIGMA guidance. The conjecturing experiments use the Transformer architecture and in particular its GPT-2 implementation.

  • miniKanren as a Tool for Symbolic Computation in Python
    arXiv.cs.SC Pub Date : 2020-05-24
    Brandon T. Willard

    In this article, we give a brief overview of the current state and future potential of symbolic computation within the Python statistical modeling and machine learning community. We detail the use of miniKanren as an underlying framework for term rewriting and symbolic mathematics, as well as its ability to orchestrate the use of existing Python libraries per . We also discuss the relevance and potential

  • A machine learning based software pipeline to pick the variable ordering for algorithms with polynomial inputs
    arXiv.cs.SC Pub Date : 2020-05-22
    Dorian Florescu; Matthew England

    We are interested in the application of Machine Learning (ML) technology to improve mathematical software. It may seem that the probabilistic nature of ML tools would invalidate the exact results prized by such software, however, the algorithms which underpin the software often come with a range of choices which are good candidates for ML application. We refer to choices which have no effect on the

  • Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric
    arXiv.cs.SC Pub Date : 2020-05-20
    Hannes Bartz; Thomas Jerkovits; Sven Puchinger; Johan Rosenkilde

    We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by reducing the core of the underlying computational problems of the decoders to one common tool: computing left and right

  • Pegasus: Sound Continuous Invariant Generation
    arXiv.cs.SC Pub Date : 2020-05-19
    Andrew Sogokon; Stefan Mitsch; Yong Kiam Tan; Katherine Cordwell; André Platzer

    Continuous invariants are an important component in deductive verification of hybrid and continuous systems. Just like discrete invariants are used to reason about correctness in discrete systems without unrolling their loops forever, continuous invariants are used to reason about differential equations without having to solve them. Automatic generation of continuous invariants remains one of the biggest

  • Applying Genetic Programming to Improve Interpretability in Machine Learning Models
    arXiv.cs.SC Pub Date : 2020-05-18
    Leonardo Augusto Ferreira; Frederico Gadelha Guimarães; Rodrigo Silva

    Explainable Artificial Intelligence (or xAI) has become an important research topic in the fields of Machine Learning and Deep Learning. In this paper, we propose a Genetic Programming (GP) based approach, named Genetic Programming Explainer (GPX), to the problem of explaining decisions computed by AI systems. The method generates a noise set located in the neighborhood of the point of interest, whose

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