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  • 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

    更新日期:2020-07-15
  • 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

    更新日期:2020-07-14
  • 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

    更新日期:2020-07-13
  • 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

    更新日期:2020-07-07
  • 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 $\mathbb{F}$.

    更新日期:2020-07-02
  • 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

    更新日期:2020-07-02
  • 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

    更新日期:2020-07-02
  • 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

    更新日期:2020-06-26
  • 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

    更新日期:2020-06-25
  • 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

    更新日期:2020-06-24
  • 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

    更新日期:2020-06-23
  • 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

    更新日期:2020-06-19
  • 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

    更新日期:2020-06-19
  • 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

    更新日期:2020-06-18
  • 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

    更新日期:2020-06-18
  • 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

    更新日期:2020-06-16
  • 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

    更新日期:2020-06-12
  • 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

    更新日期:2020-06-11
  • 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.

    更新日期:2020-06-04
  • 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

    更新日期:2020-06-04
  • 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.

    更新日期:2020-06-02
  • 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

    更新日期:2020-06-02
  • 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

    更新日期:2020-06-01
  • 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.

    更新日期:2020-05-29
  • 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

    更新日期:2020-05-24
  • 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

    更新日期:2020-05-22
  • 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

    更新日期:2020-05-20
  • 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

    更新日期:2020-05-19
  • 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

    更新日期:2020-05-18
  • An Algebraic Model For Quorum Systems
    arXiv.cs.SC Pub Date : 2020-05-18
    Alex Pellegrini; Luca Zanolini

    Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing for capturing trust assumptions. A quorum system is a collection of subsets of all processes, called quorums, with the property that each pair of quorums have a non-empty intersection. They can be found at the core of many reliable distributed systems, such as cloud computing platforms, distributed storage systems

    更新日期:2020-05-18
  • Generalizing The Davenport-Mahler-Mignotte Bound -- The Weighted Case
    arXiv.cs.SC Pub Date : 2020-05-16
    Vikram Sharma

    Root separation bounds play an important role as a complexity measure in understanding the behaviour of various algorithms in computational algebra, e.g., root isolation algorithms. A classic result in the univariate setting is the Davenport-Mahler-Mignotte (DMM) bound. One way to state the bound is to consider a directed acyclic graph $(V,E)$ on a subset of roots of a degree $d$ polynomial $f(z) \in

    更新日期:2020-05-16
  • Neural Collaborative Reasoning
    arXiv.cs.SC Pub Date : 2020-05-16
    Hanxiong Chen; Shaoyun Shi; Yunqi Li; Yongfeng Zhang

    Collaborative Filtering (CF) has been an important approach to recommender systems. However, existing CF methods are mostly designed based on the idea of matching, i.e., by learning user and item embeddings from data using shallow or deep models, they try to capture the relevance patterns in data, so that a user embedding can be matched with appropriate item embeddings using designed or learned similarity

    更新日期:2020-05-16
  • The Extended Theory of Trees and Algebraic (Co)datatypes
    arXiv.cs.SC Pub Date : 2020-05-13
    Fabian Zaiser; C. -H. Luke Ong

    The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fr\"uhwirth, we consider an extension of this theory with an additional predicate for finiteness of trees, which is useful for expressing properties about (not just datatypes but also) codatatypes. Based on their work, we present a simplification

    更新日期:2020-05-13
  • A modular extension for a computer algebra system
    arXiv.cs.SC Pub Date : 2020-05-11
    Migran N. Gevorkyan; Anna V. Korolkova; Dmitry S. Kulyabov; Leonid A. Sevastianov

    Computer algebra systems are complex software systems that cover a wide range of scientific and practical problems. However, the absolute coverage cannot be achieved. Often, it is required to create a user extension for an existing computer algebra system. In this case, the extensibility of the system should be taken into account. In this paper, we consider a technology for extending the SymPy computer

    更新日期:2020-05-11
  • On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in $Π\mathbfΣ^*$-field extensions
    arXiv.cs.SC Pub Date : 2020-05-11
    Sergei A. Abramov; Manuel Bronstein; Marko Petkovšek; Carsten Schneider

    We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of $\Pi\Sigma^*$-fields. More generally, we provide a flexible framework for a big class of difference fields that is built by a tower of $\Pi\Sigma^*$-field extensions over a difference field that satisfies

    更新日期:2020-05-11
  • Towards Efficient Normalizers of Primitive Groups
    arXiv.cs.SC Pub Date : 2020-05-11
    Sergio Siccha

    We present the ideas behind an algorithm to compute normalizers of primitive groups with non-regular socle in polynomial time. We highlight a concept we developed called permutation morphisms and present timings for a partial implementation of our algorithm. This article is a collection of results from the author's PhD thesis.

    更新日期:2020-05-11
  • Positional Games and QBF: The Corrective Encoding
    arXiv.cs.SC Pub Date : 2020-05-11
    Valentin Mayer-Eichberger; Abdallah Saffidine

    Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBF) such that a game instance admits a winning strategy for first player if and only if the corresponding formula is true. Our approach improves over previous QBF encodings of games in multiple ways. First, it is

    更新日期:2020-05-11
  • On the complexity of computing integral bases of function fields
    arXiv.cs.SC Pub Date : 2020-05-08
    Simon Abelard

    Let $\mathcal{C}$ be a plane curve given by an equation $f(x,y)=0$ with $f\in K[x][y]$ a monic squarefree polynomial. We study the problem of computing an integral basis of the algebraic function field $K(\mathcal{C})$ and give new complexity bounds for three known algorithms dealing with this problem. For each algorithm, we study its subroutines and, when it is possible, we modify or replace them

    更新日期:2020-05-08
  • Efficient Exact Verification of Binarized Neural Networks
    arXiv.cs.SC Pub Date : 2020-05-07
    Kai Jia; Martin Rinard

    We present a new system, EEV, for verifying binarized neural networks (BNNs). We formulate BNN verification as a Boolean satisfiability problem (SAT) with reified cardinality constraints of the form $y = (x_1 + \cdots + x_n \le b)$, where $x_i$ and $y$ are Boolean variables possibly with negation and $b$ is an integer constant. We also identify two properties, specifically balanced weight sparsity

    更新日期:2020-05-07
  • Algorithmic Averaging for Studying Periodic Orbits of Planar Differential Systems
    arXiv.cs.SC Pub Date : 2020-05-06
    Bo Huang

    One of the main open problems in the qualitative theory of real planar differential systems is the study of limit cycles. In this article, we present an algorithmic approach for detecting how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems via the averaging method. We propose four

    更新日期:2020-05-06
  • Probing the Natural Language Inference Task with Automated Reasoning Tools
    arXiv.cs.SC Pub Date : 2020-05-06
    Zaid Marji; Animesh Nighojkar; John Licato

    The Natural Language Inference (NLI) task is an important task in modern NLP, as it asks a broad question to which many other tasks may be reducible: Given a pair of sentences, does the first entail the second? Although the state-of-the-art on current benchmark datasets for NLI are deep learning-based, it is worthwhile to use other techniques to examine the logical structure of the NLI task. We do

    更新日期:2020-05-06
  • Towards Concise, Machine-discovered Proofs of Gödel's Two Incompleteness Theorems
    arXiv.cs.SC Pub Date : 2020-05-06
    Elijah Malaby; Bradley Dragun; John Licato

    There is an increasing interest in applying recent advances in AI to automated reasoning, as it may provide useful heuristics in reasoning over formalisms in first-order, second-order, or even meta-logics. To facilitate this research, we present MATR, a new framework for automated theorem proving explicitly designed to easily adapt to unusual logics or integrate new reasoning processes. MATR is formalism-agnostic

    更新日期:2020-05-06
  • Learning selection strategies in Buchberger's algorithm
    arXiv.cs.SC Pub Date : 2020-05-05
    Dylan Peifer; Michael Stillman; Daniel Halpern-Leistner

    Studying the set of exact solutions of a system of polynomial equations largely depends on a single iterative algorithm, known as Buchberger's algorithm. Optimized versions of this algorithm are crucial for many computer algebra systems (e.g., Mathematica, Maple, Sage). We introduce a new approach to Buchberger's algorithm that uses reinforcement learning agents to perform S-pair selection, a key step

    更新日期:2020-05-05
  • Subquadratic-Time Algorithms for Normal Bases
    arXiv.cs.SC Pub Date : 2020-05-05
    Mark Giesbrecht; Armin Jamshidpey; Éric Schost

    For any finite Galois field extension $\mathsf{K}/\mathsf{F}$, with Galois group $G = \mathrm{Gal}(\mathsf{K}/\mathsf{F})$, there exists an element $\alpha \in \mathsf{K}$ whose orbit $G\cdot\alpha$ forms an $\mathsf{F}$-basis of $\mathsf{K}$. Such an $\alpha$ is called a normal element and $G\cdot\alpha$ is a normal basis. We introduce a probabilistic algorithm for testing whether a given $\alpha

    更新日期:2020-05-05
  • Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups
    arXiv.cs.SC Pub Date : 2020-05-05
    Tao Zheng

    The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality

    更新日期:2020-05-05
  • It is Time for New Perspectives on How to Fight Bloat in GP
    arXiv.cs.SC Pub Date : 2020-05-01
    Francisco Fernández de Vega; Gustavo Olague; Francisco Chávez; Daniel Lanza; Wolfgang Banzhaf; Erik Goodman

    The present and future of evolutionary algorithms depends on the proper use of modern parallel and distributed computing infrastructures. Although still sequential approaches dominate the landscape, available multi-core, many-core and distributed systems will make users and researchers to more frequently deploy parallel version of the algorithms. In such a scenario, new possibilities arise regarding

    更新日期:2020-05-01
  • Iterative Variable Reordering: Taming Huge System Families
    arXiv.cs.SC Pub Date : 2020-04-28
    Clemens Dubslaff; Andrey Morozov; Christel Baier; Klaus Janschek

    For the verification of systems using model-checking techniques, symbolic representations based on binary decision diagrams (BDDs) often help to tackle the well-known state-space explosion problem. Symbolic BDD-based representations have been also shown to be successful for the analysis of families of systems that arise, e.g., through configurable parameters or following the feature-oriented modeling

    更新日期:2020-04-28
  • An Abstraction-guided Approach to Scalable and Rigorous Floating-Point Error Analysis
    arXiv.cs.SC Pub Date : 2020-04-24
    Arnab Das; Ian Briggs; Ganesh Gopalakrishnan; Sriram Krishnamoorthy

    Automated techniques for rigorous floating-point round-off error analysis are important in areas including formal verification of correctness and precision tuning. Existing tools and techniques, while providing tight bounds, fail to analyze expressions with more than a few hundred operators, thus unable to cover important practical problems. In this work, we present Satire, a new tool that sheds light

    更新日期:2020-04-24
  • CLUE: Exact maximal reduction of kinetic models by constrained lumping of differential equations
    arXiv.cs.SC Pub Date : 2020-04-24
    Alexey Ovchinnikov; Isabel Christina Pérez Verona; Gleb Pogudin; Mirco Tribastone

    Detailed mechanistic models of biological processes can pose significant challenges for analysis and parameter estimations due to the large number of equations used to track the dynamics of all distinct configurations in which each involved biochemical species can be found. Model reduction can help tame such complexity by providing a lower-dimensional model in which each macro-variable can be directly

    更新日期:2020-04-24
  • High performance SIMD modular arithmetic for polynomial evaluation
    arXiv.cs.SC Pub Date : 2020-04-24
    Pierre FortinLIP6; Ambroise FleuryLIFL; François LemaireLIFL; Michael Monagan

    Two essential problems in Computer Algebra, namely polynomial factorization and polynomial greatest common divisor computation, can be efficiently solved thanks to multiple polynomial evaluations in two variables using modular arithmetic. In this article, we focus on the efficient computation of such polynomial evaluations on one single CPU core. We first show how to leverage SIMD computing for modular

    更新日期:2020-04-24
  • GAPS: Generator for Automatic Polynomial Solvers
    arXiv.cs.SC Pub Date : 2020-04-24
    Bo Li; Viktor Larsson

    Minimal problems in computer vision raise the demand of generating efficient automatic solvers for polynomial equation systems. Given a polynomial system repeated with different coefficient instances, the traditional Gr\"obner basis or normal form based solution is very inefficient. Fortunately the Gr\"obner basis of a same polynomial system with different coefficients is found to share consistent

    更新日期:2020-04-24
  • The Imandra Automated Reasoning System (system description)
    arXiv.cs.SC Pub Date : 2020-04-21
    Grant Olney Passmore; Simon Cruanes; Denis Ignatovich; Dave Aitken; Matt Bray; Elijah Kagan; Kostya Kanishev; Ewen Maclean; Nicola Mometto

    We describe Imandra, a modern computational logic theorem prover designed to bridge the gap between decision procedures such as SMT, semi-automatic inductive provers of the Boyer-Moore family like ACL2, and interactive proof assistants for typed higher-order logics. Imandra's logic is computational, based on a pure subset of OCaml in which all functions are terminating, with restrictions on types and

    更新日期:2020-04-23
  • Computing the multilinear factors of lacunary polynomials without heights
    arXiv.cs.SC Pub Date : 2013-11-22
    Arkadev Chattopadhyay; Bruno Grenet; Pascal Koiran; Natacha Portier; Yann Strozecki

    We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$ its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity

    更新日期:2020-04-22
  • Computing all identifiable functions for ODE models
    arXiv.cs.SC Pub Date : 2020-04-16
    Alexey Ovchinnikov; Anand Pillay; Gleb Pogudin; Thomas Scanlon

    Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that are identifiable. The existing algorithms check whether

    更新日期:2020-04-17
  • Efficiently and Effectively Recognizing Toricity of Steady State Varieties
    arXiv.cs.SC Pub Date : 2019-10-09
    Dima Grigoriev; Alexandru Iosif; Hamid Rahkooy; Thomas Sturm; Andreas Weber

    We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals.

    更新日期:2020-04-16
  • A Simple Method for Computing Some Pseudo-Elliptic Integrals in Terms of Elementary Functions
    arXiv.cs.SC Pub Date : 2020-04-10
    Sam Blake

    We introduce a method for computing some pseudo-elliptic integrals in terms of elementary functions. The method is simple and fast in comparison to the algebraic case of the Risch-Trager-Bronstein algorithm. This method can quickly solve many pseudo-elliptic integrals which other well-known computer algebra systems (CAS) either fail, return an answer in terms of special functions, or require more than

    更新日期:2020-04-13
  • New Opportunities for the Formal Proof of Computational Real Geometry?
    arXiv.cs.SC Pub Date : 2020-04-08
    Erika {Á}brahám; James Davenport; Matthew England; Gereon Kremer; Zak Tonks

    The purpose of this paper is to explore the question "to what extent could we produce formal, machine-verifiable, proofs in real algebraic geometry?" The question has been asked before but as yet the leading algorithms for answering such questions have not been formalised. We present a thesis that a new algorithm for ascertaining satisfiability of formulae over the reals via Cylindrical Algebraic Coverings

    更新日期:2020-04-09
  • Resultants over principal Artinian rings
    arXiv.cs.SC Pub Date : 2020-04-07
    Claus Fieker; Tommy Hofmann; Carlo Sircana

    The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified version of the Euclidean algorithm. Using the same strategy, we show how the reduced resultant and a pair of B\'ezout coefficient can be computed. Particular attention

    更新日期:2020-04-08
  • Neural Analogical Matching
    arXiv.cs.SC Pub Date : 2020-04-07
    Maxwell Crouse; Constantine Nakos; Ibrahim Abdelaziz; Kenneth Forbus

    Analogy is core to human cognition. It allows us to solve problems based on prior experience, it governs the way we conceptualize new information, and it even influences our visual perception. The importance of analogy to humans has made it an active area of research in the broader field of artificial intelligence, resulting in data-efficient models that learn and reason in human-like ways. While analogy

    更新日期:2020-04-07
  • Interpolation of Dense and Sparse Rational Functions and other Improvements in $\texttt{FireFly}$
    arXiv.cs.SC Pub Date : 2020-04-03
    Jonas Klappert; Sven Yannick Klein; Fabian Lange

    We present the main improvements and new features in version $\texttt{2.0}$ of the open-source $\texttt{C++}$ library $\texttt{FireFly}$ for the interpolation of rational functions. This includes algorithmic improvements, e.g. a hybrid algorithm for dense and sparse rational functions and an algorithm to identify and remove univariate factors. The new version is applied to a Feynman-integral reduction

    更新日期:2020-04-06
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