1,195 research outputs found
Langford sequences and a product of digraphs
Skolem and Langford sequences and their many generalizations have
applications in numerous areas. The -product is a generalization of
the direct product of digraphs. In this paper we use the -product
and super edge-magic digraphs to construct an exponential number of Langford
sequences with certain order and defect. We also apply this procedure to
extended Skolem sequences.Comment: 10 pages, 6 figures, to appear in European Journal of Combinatoric
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
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))
The Continuous Skolem-Pisot Problem: On the Complexity of Reachability for Linear Ordinary Differential Equations
We study decidability and complexity questions related to a continuous
analogue of the Skolem-Pisot problem concerning the zeros and nonnegativity of
a linear recurrent sequence. In particular, we show that the continuous version
of the nonnegativity problem is NP-hard in general and we show that the
presence of a zero is decidable for several subcases, including instances of
depth two or less, although the decidability in general is left open. The
problems may also be stated as reachability problems related to real zeros of
exponential polynomials or solutions to initial value problems of linear
differential equations, which are interesting problems in their own right.Comment: 14 pages, no figur
On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
The complexity of independence-friendly fixpoint logic
Abstract. We study the complexity of model-checking for the fixpoint extension of Hintikka and Sandu’s independence-friendly logic. We show that this logic captures ExpTime; and by embedding PFP, we show that its combined complexity is ExpSpace-hard, and moreover the logic includes second order logic (on finite structures).
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