88,538 research outputs found

    Multiplicity Problems on Algebraic Series and Context-Free Grammars

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    In this paper we obtain complexity bounds for computational problems on algebraic power series over several commuting variables. The power series are specified by systems of polynomial equations: a formalism closely related to weighted context-free grammars. We focus on three problems -- decide whether a given algebraic series is identically zero, determine whether all but finitely many coefficients are zero, and compute the coefficient of a specific monomial. We relate these questions to well-known computational problems on arithmetic circuits and thereby show that all three problems lie in the counting hierarchy. Our main result improves the best known complexity bound on deciding zeroness of an algebraic series. This problem is known to lie in PSPACE by reduction to the decision problem for the existential fragment of the theory of real closed fields. Here we show that the problem lies in the counting hierarchy by reduction to the problem of computing the degree of a polynomial given by an arithmetic circuit. As a corollary we obtain new complexity bounds on multiplicity equivalence of context-free grammars restricted to a bounded language, language inclusion of a nondeterministic finite automaton in an unambiguous context-free grammar, and language inclusion of a non-deterministic context-free grammar in an unambiguous finite automaton.Comment: full technical report of a LICS'23 pape

    Tropical differential equations

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    Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed. We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is NPNP-hard, and this problem for arbitrary number of variables belongs to NPNP. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series

    Weakly-Unambiguous Parikh Automata and Their Link to Holonomic Series

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    We investigate the connection between properties of formal languages and properties of their generating series, with a focus on the class of holonomic power series. We first prove a strong version of a conjecture by Castiglione and Massazza: weakly-unambiguous Parikh automata are equivalent to unambiguous two-way reversal bounded counter machines, and their multivariate generating series are holonomic. We then show that the converse is not true: we construct a language whose generating series is algebraic (thus holonomic), but which is inherently weakly-ambiguous as a Parikh automata language. Finally, we prove an effective decidability result for the inclusion problem for weakly-unambiguous Parikh automata, and provide an upper-bound on its complexity

    Fast algorithm for border bases of Artinian Gorenstein algebras

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    Given a multi-index sequence σ\sigma, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of σ\sigma. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne lII\sigma of the Hankel operator $H$\sigma associated to σ\sigma. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra AA\sigmaassociatedtothesequence associated to the sequence \sigma yields the structure of the terms $\sigma\alphaforall for all α\alpha \in N n.Thisstructureisexplicitlygivenbyaborderbasisof. This structure is explicitly given by a border basis of Aσ\sigma,whichispresentedasaquotientofthepolynomialring, which is presented as a quotient of the polynomial ring K[x 1 ,. .. , xn]bythekernel] by the kernel Iσ\sigmaoftheHankeloperator of the Hankel operator Hσ\sigma.Thealgorithmprovidesgeneratorsof. The algorithm provides generators of Iσ\sigmaconstitutingaborderbasis,pairwiseorthogonalbasesof constituting a border basis, pairwise orthogonal bases of Aσ\sigma$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm

    Algebraicity and transcendence of power series: combinatorial and computational aspects

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    DoctoralFrom ancient times, mathematicians are interested in the following question: is a given real number "algebraic" (that is, a root of a nonzero univariate polynomial with rational number coefficients), or is it "transcendental"? Although almost all real numbers are transcendental, it is notoriously difficult to actually prove, or disprove, the transcendence of a given constant. More recently, and especially with the advent of computers, different related questions arose: What is the "complexity" of a real number? How fast can one compute the first digits, or one single digit, of a (computable) real number? Can digits of algebraic numbers be computed faster than those of (computable) transcendental numbers? In this series of lectures, we will consider the (simpler) functional analogues of these questions: given a formal power series with rational number coefficients, decide whether it is algebraic (root of a nontrivial bivariate polynomial) or transcendental, and determine how fast can one compute its coefficients? We will first motivate these questions by presenting some examples of algebraic power series coming from combinatorics, with a focus on enumeration of lattice walks. Then we will discuss several methods that allow to discover and prove the nature (algebraic or transcendental) of a generating function, with an emphasis on an experimental mathematics approach combined with algorithmic methods such as Guess'n'Prove and Creative Telescoping. Finally, we will overview efficient algorithms for various operations on algebraic power series, including the computation of one or several selected terms

    Discovering the roots: Uniform closure results for algebraic classes under factoring

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    Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots rr is small but the multiplicities are exponentially large. Our method sets up a linear system in rr unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,,xn)f(x_1,\ldots,x_n) of size ss we prove that each factor has size at most a polynomial in: ss and the degree of the squarefree part of ff. Consequently, if f1f_1 is a 2Ω(n)2^{\Omega(n)}-hard polynomial then any nonzero multiple ifiei\prod_{i} f_i^{e_i} is equally hard for arbitrary positive eie_i's, assuming that ideg(fi)\sum_i \text{deg}(f_i) is at most 2O(n)2^{O(n)}. It is an old open question whether the class of poly(nn)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial ff of degree nO(1)n^{O(1)} and formula (resp. ABP) size nO(logn)n^{O(\log n)} we can find a similar size formula (resp. ABP) factor in randomized poly(nlognn^{\log n})-time. Consequently, if determinant requires nΩ(logn)n^{\Omega(\log n)} size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation τ\tau, f(τx)f(\tau\overline{x}) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

    Faster relaxed multiplication

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    In previous work, we have introduced several fast algorithms for relaxed power series multiplication (also known under the name on-line multiplication) up till a given order n. The fastest currently known algorithm works over an effective base field K with sufficiently many 2^p-th roots of unity and has algebraic time complexity O(n log n exp (2 sqrt (log 2 log log n))). In this note, we will generalize this algorithm to the cases when K is replaced by an effective ring of positive characteristic or by an effective ring of characteristic zero, which is also torsion-free as a Z-module and comes with an additional algorithm for partial division by integers. We will also present an asymptotically faster algorithm for relaxed multiplication of p-adic numbers
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