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arXiv2010

Gröbner Bases of Bihomogeneous Ideals generated by Polynomials of Bidegree (1,1): Algorithms and Complexity

Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gröbner bases algorithms seems to be easier than solving homogeneous systems of the same degree. Nevertheless, the reasons of this behaviour are not clear. In this paper, we focus on bilinear systems (i.e. bihomogeneous systems where all equations have bidegree (1,1)). Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gröbner basis computations by using the multihomogeneous structure of those systems. The contributions are theoretical and practical. First, we adapt the classical F5 criterion to avoid reductions to zero which occur when the input is a set of bilinear polynomials. We also prove an explicit form of the Hilbert series of bihomogeneous ideals generated by generic bilinear polynomials and give a new upper bound on the degree of regularity of generic affine bilinear systems. This leads to new complexity bounds for solving bilinear systems. We propose also a variant of the F5 Algorithm dedicated to multihomogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs. Experimental results show that this variant requires less time and memory than the classical homogeneous F5 Algorithm.

cs.SC↗

arXiv2010

Generic design of Chinese remaindering schemes

We propose a generic design for Chinese remainder algorithms. A Chinese remainder computation consists in reconstructing an integer value from its residues modulo non coprime integers. We also propose an efficient linear data structure, a radix ladder, for the intermediate storage and computations. Our design is structured into three main modules: a black box residue computation in charge of computing each residue; a Chinese remaindering controller in charge of launching the computation and of the termination decision; an integer builder in charge of the reconstruction computation. We then show that this design enables many different forms of Chinese remaindering (e.g. deterministic, early terminated, distributed, etc.), easy comparisons between these forms and e.g. user-transparent parallelism at different parallel grains.

cs.SC↗

arXiv2010

Some Algebraic Properties of a Subclass of Finite Normal Form Games

We study the problem of computing all Nash equilibria of a subclass of finite normal form games. With algebraic characterization of the games, we present a method for computing all its Nash equilibria. Further, we present a method for deciding membership to the class of games with its related results. An appendix, containing an example to show working of each of the presented methods, concludes the work.

cs.GT↗

arXiv2010

An in-place truncated Fourier transform and applications to polynomial multiplication

The truncated Fourier transform (TFT) was introduced by van der Hoeven in 2004 as a means of smoothing the "jumps" in running time of the ordinary FFT algorithm that occur at power-of-two input sizes. However, the TFT still introduces these jumps in memory usage. We describe in-place variants of the forward and inverse TFT algorithms, achieving time complexity O(n log n) with only O(1) auxiliary space. As an application, we extend the second author's results on space-restricted FFT-based polynomial multiplication to polynomials of arbitrary degree.

cs.DS↗

arXiv2010

Stability and Bifurcation Analysis of Coupled Fitzhugh-Nagumo Oscillators

Neurons are the central biological objects in understanding how the brain works. The famous Hodgkin-Huxley model, which describes how action potentials of a neuron are initiated and propagated, consists of four coupled nonlinear differential equations. Because these equations are difficult to deal with, there also exist several simplified models, of which many exhibit polynomial-like non-linearity. Examples of such models are the Fitzhugh-Nagumo (FHN) model, the Hindmarsh-Rose (HR) model, the Morris-Lecar (ML) model and the Izhikevich model. In this work, we first prescribe the biologically relevant parameter ranges for the FHN model and subsequently study the dynamical behaviour of coupled neurons on small networks of two or three nodes. To do this, we use a computational real algebraic geometry method called the Discriminant Variety (DV) method to perform the stability and bifurcation analysis of these small networks. A time series analysis of the FHN model can be found elsewhere in related work[15].

q-bio.NC↗

arXiv2010

The Power of Vocabulary: The Case of Cyclotomic Polynomials

We observe that the vocabulary used to construct the "answer" to problems in computer algebra can have a dramatic effect on the computational complexity of solving that problem. We recall a formalization of this observation and explain the classic example of sparse polynomial arithmetic. For this case, we show that it is possible to extend the vocabulary so as reap the benefits of conciseness whilst avoiding the obvious pitfall of repeating the problem statement as the "solution". It is possible to extend the vocabulary either by irreducible cyclotomics or by $x^n-1$: we look at the options and suggest that the pragmatist might opt for both.

cs.SC↗

arXiv2010

Bézout Identities Associated to a Finite Sequence

We consider finite sequences $s\in D^n$ where $D$ is a commutative, unital, integral domain. We prove three sets of identities (possibly with repetitions), each involving $2n$ polynomials associated to $s$. The right-hand side of these identities is a recursively-defined (non-zero) 'product-of-discrepancies'. There are implied iterative algorithms (of quadratic complexity) for the left-hand side coefficients; when the ground domain is factorial, the identities are in effect Bézout identities. We give a number of applications: an algorithm to compute Bézout coefficients over a field; the outputs of the Berlekamp-Massey algorithm; sequences with perfect linear complexity profile; annihilating polynomials which do not vanish at zero and have minimal degree: we simplify and extend an algorithm of Salagean to sequences over $D$. In the Appendix, we give a new proof of a theorem of Imamura and Yoshida on the linear complexity of reverse sequences, initially proved using Hankel matrices over a field and now valid for sequences over a factorial domain.

cs.IT↗

arXiv2010

Gradual sub-lattice reduction and a new complexity for factoring polynomials

We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well.

cs.SC↗

arXiv2010

NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun.

cs.SC↗

arXiv2010

Factorization of Non-Commutative Polynomials

We describe an algorithm for the factorization of non-commutative polynomials over a field. The first sketch of this algorithm appeared in an unpublished manuscript (literally hand written notes) by James H. Davenport more than 20 years ago. This version of the algorithm contains some improvements with respect to the original sketch. An improved version of the algorithm has been fully implemented in the Axiom computer algebra system.

cs.MS↗

arXiv2010

Triangular Decomposition of Semi-algebraic Systems

Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue: semi-algebraic systems. We show that any such system can be decomposed into finitely many {\em regular semi-algebraic systems}. We propose two specifications of such a decomposition and present corresponding algorithms. Under some assumptions, one type of decomposition can be computed in singly exponential time w.r.t.\ the number of variables. We implement our algorithms and the experimental results illustrate their effectiveness.

cs.SC↗

arXiv2010

Stability Analysis of Linear Uncertain Systems via Checking Positivity of Forms on Simplices

In this paper, we mainly study the robust stability of linear continuous systems with parameter uncertainties, a more general kind of uncertainties for system matrices is considered, i.e., entries of system matrices are rational functions of uncertain parameters which are varying in intervals. we present a method which can check the robust Hurwitz stability of such uncertain systems in finite steps. Examples show the efficiency of our approach.

cs.SC↗

arXiv2010

Algorithms for Checking Rational Roots of $b$-functions and their Applications

Bernstein-Sato polynomial of a hypersurface is an important object with numerous applications. It is known, that it is complicated to obtain it computationally, as a number of open questions and challenges indicate. In this paper we propose a family of algorithms called \texttt{checkRoot} for optimized check of whether a given rational number is a root of Bernstein-Sato polynomial and the computations of its multiplicity. This algorithms are used in the new approach to compute the whole global or local Bernstein-Sato polynomial and $b$-function of a holonomic ideal with respect to weights. They are applied in numerous situations, where there is a possibility to compute an upper bound for the polynomial. Namely, it can be achieved by means of embedded resolution, for topologically equivalent singularities or using the formula of A'Campo and spectral numbers. We also present approaches to the logarithmic comparison problem and the intersection homology D-module. Several applications are presented as well as solutions to some challenges which were intractable with the classical methods. One of the main applications consists of computing of a stratification of affine space with the local $b$-function being constant on each stratum. Notably, the algorithm we propose does not employ primary decomposition. Also we apply our results for the computation of Bernstein-Sato polynomials for varieties. The methods from this paper have been implemented in {\sc Singular:Plural} as libraries {\tt dmod.lib} and {\tt bfun.lib}. All the examples from the paper have been computed with this implementation.

math.AG↗

arXiv2010

A Modal Logic for Termgraph Rewriting

We propose a modal logic tailored to describe graph transformations and discuss some of its properties. We focus on a particular class of graphs called termgraphs. They are first-order terms augmented with sharing and cycles. Termgraphs allow one to describe classical data-structures (possibly with pointers) such as doubly-linked lists, circular lists etc. We show how the proposed logic can faithfully describe (i) termgraphs as well as (ii) the application of a termgraph rewrite rule (i.e. matching and replacement) and (iii) the computation of normal forms with respect to a given rewrite system. We also show how the proposed logic, which is more expressive than propositional dynamic logic, can be used to specify shapes of classical data-structures (e.g. binary trees, circular lists etc.).

cs.LO↗

arXiv2010

Protocol indepedence through disjoint encryption under Exclusive-OR

Multi-protocol attacks due to protocol interaction has been a notorious problem for security. Gutman-Thayer proved that they can be prevented by ensuring that encrypted messages are distinguishable across protocols, under a free algebra. In this paper, we prove that a similar suggestion prevents these attacks under commonly used operators such as Exclusive-OR, that induce equational theories, breaking the free algebra assumption.

cs.CR↗

arXiv2010

Disabling equational theories in unification for cryptographic protocol analysis through tagging

In this paper, we show a new tagging scheme for cryptographic protocol messages. Under this tagging, equational theories of operators such as exclusive-or, binary addition etc. are effectively disabled, when terms are unified. We believe that this result has a significant impact on protocol analysis and security, since unification is at the heart of symbolic protocol analysis. Hence, disabling equational theories in unification implies disabling them altogether in protocol analysis for most operators and theories.

cs.CR↗

arXiv2010

A New Proof for the Correctness of F5 (F5-Like) Algorithm

The famous F5 algorithm for computing Gröbner basis was presented by Faugère in 2002 without complete proofs for its correctness. The current authors have simplified the original F5 algorithm into an F5 algorithm in Buchberger's style (F5B algorithm), which is equivalent to original F5 algorithm and may deduce some F5-like versions. In this paper, the F5B algorithm is briefly revisited and a new complete proof for the correctness of F5B algorithm is proposed. This new proof is not limited to homogeneous systems and does not depend on the strategy of selecting critical pairs (i.e. the strategy deciding which critical pair is computed first) such that any strategy could be utilized in F5B (F5) algorithm. From this new proof, we find that the special reduction procedure (F5-reduction) is the key of F5 algorithm, so maintaining this special reduction, various variation algorithms become available. A natural variation of F5 algorithm, which transforms original F5 algorithm to a non-incremental algorithm, is presented and proved in this paper as well. This natural variation has been implemented over the Boolean ring. The two revised criteria in this natural variation are also able to reject almost all unnecessary computations and few polynomials reduce to 0 in most examples.

cs.SC↗

arXiv2010

Syntactic Abstraction of B Models to Generate Tests

In a model-based testing approach as well as for the verification of properties, B models provide an interesting solution. However, for industrial applications, the size of their state space often makes them hard to handle. To reduce the amount of states, an abstraction function can be used, often combining state variable elimination and domain abstractions of the remaining variables. This paper complements previous results, based on domain abstraction for test generation, by adding a preliminary syntactic abstraction phase, based on variable elimination. We define a syntactic transformation that suppresses some variables from a B event model, in addition to a method that chooses relevant variables according to a test purpose. We propose two methods to compute an abstraction A of an initial model M. The first one computes A as a simulation of M, and the second one computes A as a bisimulation of M. The abstraction process produces a finite state system. We apply this abstraction computation to a Model Based Testing process.

cs.LO↗