1,139 research outputs found
Near-Optimal Complexity Bounds for Fragments of the Skolem Problem
Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial).
In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds
The Skolem-Bang Theorems in Ordered Fields with an
This paper is concerned with the extent to which the Skolem-Bang theorems in
Diophantine approximations generalise from the standard setting of , where is an ordered field and is an
integer part of . We show that some of these theorems are hold
unconditionally in general case (ordered fields with an integer part). The
remainder results are based on Dirichlet's and Kronecker's theorems. Finally we
extend Dirichlet's theorem to ordered fields with integer part.Comment: 28 page
Controlled Natural Language Processing as Answer Set Programming: an Experiment
Most controlled natural languages (CNLs) are processed with the help of a
pipeline architecture that relies on different software components. We
investigate in this paper in an experimental way how well answer set
programming (ASP) is suited as a unifying framework for parsing a CNL, deriving
a formal representation for the resulting syntax trees, and for reasoning with
that representation. We start from a list of input tokens in ASP notation and
show how this input can be transformed into a syntax tree using an ASP grammar
and then into reified ASP rules in form of a set of facts. These facts are then
processed by an ASP meta-interpreter that allows us to infer new knowledge
Non-principal ultrafilters, program extraction and higher order reverse mathematics
We investigate the strength of the existence of a non-principal ultrafilter
over fragments of higher order arithmetic.
Let U be the statement that a non-principal ultrafilter exists and let
ACA_0^{\omega} be the higher order extension of ACA_0. We show that
ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that
ACA_0^{\omega}+\U is conservative over PA.
Moreover, we provide a program extraction method and show that from a proof
of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U
a realizing term in G\"odel's system T can be extracted. This means that one
can extract a term t, such that A(f,t(f))
- …