Predicting zero reductions in Gr\"obner basis computations

Since Buchberger's initial algorithm for computing Gr\"obner bases in 1965 many attempts have been taken to detect zero reductions in advance. Buchberger's Product and Chain criteria may be known the most, especially in the installaton of Gebauer and M\"oller. A relatively new approach are signature-based criteria which were first used in Faug\`ere's F5 algorithm in 2002. For regular input sequences these criteria are known to compute no zero reduction at all. In this paper we give a detailed discussion on zero reductions and the corresponding syzygies. We explain how the different methods to predict them compare to each other and show advantages and drawbacks in theory and practice. With this a new insight into algebraic structures underlying Gr\"obner bases and their computations might be achieved.Comment: 25 pages, 3 figure

Modifying Faug\`ere's F5 Algorithm to ensure termination

The structure of the F5 algorithm to compute Gr\"obner bases makes it very efficient. However, while it is believed to terminate for so-called regular sequences, it is not clear whether it terminates for all inputs. This paper has two major parts. In the first part, we describe in detail the difficulties related to a proof of termination. In the second part, we explore three variants that ensure termination. Two of these have appeared previously only in dissertations, and ensure termination by checking for a Gr\"obner basis using traditional criteria. The third variant, F5+, identifies a degree bound using a distinction between "necessary" and "redundant" critical pairs that follows from the analysis in the first part. Experimental evidence suggests this third approach is the most efficient of the three.Comment: 19 pages, 1 tabl

Resolución de Sistemas de Ecuaciones Polinomiales utilizando las Bases de Gröbner y el Método de Autovalores

El presente trabajo nos permite mostrar una introducción a otra parte de las ecuaciones polinomiales ya que daremos a conocer otro método más general a los tratados comúnmente que nos permitirá la solución de sistemas de ecuaciones polinomiales. En el transcurso de nuestros estudios no percibimos muchos el desarrollo de este método de solución, esperamos que el presente trabajo de un aporte a que estudiantes de nuestra carrera enfaticen más sobre dicho estudio en el Álgebra Abstracta y la Geometría Algebraica. Las herramientas a utilizar en nuestro trabajo son las Bases de Gröbner al ver las variedades de los sistemas de ecuaciones polinomiales, es un interesante método de solución en dichas ecuaciones. Al presentar los ejemplos en nuestro trabajo utilizaremos el sistema computacional CoCoA para la comprobación de los ejercicios presentados. En los ejercicios propuestos hemos sido lo más claro, preciso y coherente para dar a entender los métodos utilizados y ayudar a ser más reflexivas las soluciones para los lectores y estudiantes de la carrera. El método de Autovalores nos permite encontrar las soluciones de un sistema de ecuaciones polinomiales, es decir encontrar los puntos de la variedad de un Ideal generado por los polinomio

Arion: Arithmetization-Oriented Permutation and Hashing from Generalized Triangular Dynamical Systems

In this paper we propose the (keyed) permutation Arion and the hash function ArionHash over $\mathbb{F}_p$ for odd and particularly large primes. The design of Arion is based on the newly introduced Generalized Triangular Dynamical System (GTDS), which provides a new algebraic framework for constructing (keyed) permutation using polynomials over a finite field. At round level Arion is the first design which is instantiated using the new GTDS. We provide extensive security analysis of our construction including algebraic cryptanalysis (e.g. interpolation and Groebner basis attacks) that are particularly decisive in assessing the security of permutations and hash functions over $\mathbb{F}_p$. From a application perspective, ArionHash is aimed for efficient implementation in zkSNARK protocols and Zero-Knowledge proof systems. For this purpose, we exploit that CCZ-equivalence of graphs can lead to a more efficient implementation of Arithmetization-Oriented primitives. We compare the efficiency of ArionHash in R1CS and Plonk settings with other hash functions such as Poseidon, Anemoi and Griffin. For demonstrating the practical efficiency of ArionHash we implemented it with the zkSNARK libraries libsnark and Dusk Network Plonk. Our result shows that ArionHash is significantly faster than Poseidon - a hash function designed for zero-knowledge proof systems. We also found that an aggressive version of ArionHash is considerably faster than Anemoi and Griffin in a practical zkSNARK setting

Qualitative Spatial Reasoning about Relative Orientation --- A Question of Consistency ---

Abstract. After the emergence of Allen s Interval Algebra Qualitative Spatial Reasoning has evolved into a fruitful field of research in artificial intelligence with possible applications in geographic information systems (GIS) and robot navigation Qualitative Spatial Reasoning abstracts from the detailed metric description of space using rich mathematical theories and restricts its language to a finite, often rather small, set of relations that fulfill certain properties. This approach is often deemed to be cognitively adequate . A major question in qualitative spatial reasoning is whether a description of a spatial situation given as a constraint network is consistent. The problem becomes a hard one since the domain of space (often R2 ) is infinite. In contrast many of the interesting problems for constraint satisfaction have a finite domain on which backtracking methods can be used. But because of the infinity of its domains these methods are generally not applicable to Qualitative Spatial Reasoning. Anyhow the method of path consistency or rather its generalization algebraic closure turned out to be helpful to a certain degree for many qualitative spatial calculi. The problem regarding this method is that it depends on the existence of a composition table, and calculating this table is not an easy task. For example the dipole calculus (operating on oriented dipoles) DRAf has 72 base relations and binary composition, hence its composition table has 5184 entries. Finding all these entries by hand is a hard, long and error-prone task. Finding them using a computer is also not easy, since the semantics of DRAf in the Euclidean Plane, its natural domain, rely on non-linear inequalities. This is not a special problem of the DRAf calculus. In fact, all calculi dealing with relative orientation share the property of having semantics based on non-linear inequalities in the Euclidean plane. This not only makes it hard to find a composition table, it also makes it particularly hard to decide consistency for these calculi. As shown in [79] algebraic closure is always just an approximation to consistency for these calculi, but it is the only method that works fast. Methods like Gröbner reasoning can decide consistency for these calculi but only for small constraint networks. Still finding a composition table for DRAf is a fruitful task, since we can use it analyze the properties of composition based reasoning for such a calculus and it is a starting point for the investigation of the quality of the approximation of consistency for this calculus. We utilize a new approach for calculating the composition table for DRAf using condensed semantics, i.e. the domain of the calculus is compressed in such a way that only finitely many possible configurations need to be investigated. In fact, only the configurations need to be investigated that turn out to represent special characteristics for the placement of three lines in the plane. This method turns out to be highly efficient for calculating the composition table of the calculus. Another method of obtaining a composition table is borrowing it via a suitable morphism. Hence, we investigate morphisms between qualitative spatial calculi. Having the composition table is not the end but rather the beginning of the problem. With that table we can compute algebraically closed refinements of constraint networks, but how meaningful is this process? We know that all constraint networks for which such a refinement does not exist are inconsistent, but what about the rest? In fact, they may be consistent or not. If they are all consistent, then we can be happy, since algebraic closure would decide consistency for the calculus at hand. We investigate LR, DRAf and DRAfp and show that for all these calculi algebraic closure does not decide consistency. In fact, for the LR calculus algebraic closure is an extremely bad approximation of consistency. For this calculus we introduce a new method for the approximation of consistency based on triangles, that performs far better than algebraic closure. A major weak spot of the field of Qualitative Spatial Reasoning is the area of applications. It is hard to refute the accusation of qualitative spatial calculi having only few applications so far. As a step into the direction of scrutinizing the applicability of these calculi, we examine the performance of DRA and OPRA in the issue of describing and navigating street networks based on local observations. Especially for OPRA we investigate a factorization of the base relations that is deemed cognitively adequate . Whenever possible we use real-world data in these investigations obtained from OpenStreetMap

Symbolic computation: systems and applications

The article presents an overview of symbolic computation systems, their classification-in-history, the most popular CAS, examples of systems and some of their applications. Symbolics versus numeric, enhancement in mathematics, computing nature of CAS, related projects, networks, references are discussed

Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gr\"obner Bases

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs $G$ and $H$ is at least the product of the domination numbers of $G$ and $H$. Recently Gaar, Krenn, Margulies and Wiegele used the graph class $\mathcal{G}$ of all graphs with $n_\mathcal{G}$ vertices and domination number $k_\mathcal{G}$ and reformulated Vizing's conjecture as the problem that for all graph classes $\mathcal{G}$ and $\mathcal{H}$ the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of $k_\mathcal{G}$, $n_\mathcal{G}$, $k_\mathcal{H}$, and $n_\mathcal{H}$. In this paper, we consider their approach for $k_\mathcal{G} = k_\mathcal{H} = 1$. For this case we are able to derive the unique reduced Gr\"obner basis of the Vizing ideal. Based on this, we deduce the minimum degree $(n_\mathcal{G} + n_\mathcal{H} - 1)/2$ of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes $\mathcal{G}$ and $\mathcal{H}$ with $n_\mathcal{G} + n_\mathcal{H} -1 = d$ for general $d$, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes $\mathcal{G}$ and $\mathcal{H}$ with $k_\mathcal{G}=k_\mathcal{H}=1$ and $n_\mathcal{G} + n_\mathcal{H} \leq 15$.Comment: 36 pages, 2 figure

New Design Techniques for Efficient Arithmetization-Oriented Hash Functions: Anemoi Permutations and Jive Compression Mode

International audienc