88,538 research outputs found
Multiplicity Problems on Algebraic Series and Context-Free Grammars
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
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 -hard, and this
problem for arbitrary number of variables belongs to . 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
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
Given a multi-index sequence , we present a new efficient algorithm
to compute generators of the linear recurrence relations between the terms of
. 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 l\sigma of the Hankel operator
$H$\sigma associated to . 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 \sigma\sigma yields the
structure of the terms $\sigma\alpha N nAK[x 1 ,. .. , xnIHIA$ 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
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
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 is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
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 ,
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
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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