18,934 research outputs found
On equations over sets of integers
Systems of equations with sets of integers as unknowns are considered. It is
shown that the class of sets representable by unique solutions of equations
using the operations of union and addition S+T=\makeset{m+n}{m \in S, \: n \in
T} and with ultimately periodic constants is exactly the class of
hyper-arithmetical sets. Equations using addition only can represent every
hyper-arithmetical set under a simple encoding. All hyper-arithmetical sets can
also be represented by equations over sets of natural numbers equipped with
union, addition and subtraction S \dotminus T=\makeset{m-n}{m \in S, \: n \in
T, \: m \geqslant n}. Testing whether a given system has a solution is
-complete for each model. These results, in particular, settle the
expressive power of the most general types of language equations, as well as
equations over subsets of free groups.Comment: 12 apges, 0 figure
S-parts of values of univariate polynomials, binary forms and decomposable forms at integral points
Let be a finite set of primes. The -part of a non-zero integer
is the largest positive divisor of that is composed of primes from .
In 2013, Gross and Vincent proved that if is a polynomial with integer
coefficients and with at least two roots in the complex numbers, then for every
integer at which is non-zero, we have (*) , where and are effectively computable and . Their proof
uses Baker-type estimates for linear forms in complex logarithms of algebraic
numbers. As an easy application of the -adic Thue-Siegel-Roth theorem we
show that if has degree and no multiple roots, then an
inequality such as (*) holds for all , provided we do not require
effectivity of . Further, we show that such an inequality does not hold
anymore with and sufficiently small . In addition we prove a density
result, giving for every an asymptotic estimate with the right
order of magnitude for the number of integers with absolute value at most
such that has -part at least . The result of
Gross and Vincent, as well as the other results mentioned above, are
generalized to values of binary forms and decomposable forms at integral
points. Our main tools are Baker type estimates for linear forms in complex and
-adic logarithms, the -adic Subspace Theorem of Schmidt and Schlickewei,
and a recent general lattice point counting result of Barroero and Widmer.Comment: 42 page
Sparse solutions of linear Diophantine equations
We present structural results on solutions to the Diophantine system
,
with the smallest number of non-zero entries. Our tools are algebraic and
number theoretic in nature and include Siegel's Lemma, generating functions,
and commutative algebra. These results have some interesting consequences in
discrete optimization
Tree Regular Model Checking for Lattice-Based Automata
Tree Regular Model Checking (TRMC) is the name of a family of techniques for
analyzing infinite-state systems in which states are represented by terms, and
sets of states by Tree Automata (TA). The central problem in TRMC is to decide
whether a set of bad states is reachable. The problem of computing a TA
representing (an over- approximation of) the set of reachable states is
undecidable, but efficient solutions based on completion or iteration of tree
transducers exist. Unfortunately, the TRMC framework is unable to efficiently
capture both the complex structure of a system and of some of its features. As
an example, for JAVA programs, the structure of a term is mainly exploited to
capture the structure of a state of the system. On the counter part, integers
of the java programs have to be encoded with Peano numbers, which means that
any algebraic operation is potentially represented by thousands of applications
of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an
extended version of tree automata whose leaves are equipped with lattices. LTAs
allow us to represent possibly infinite sets of interpreted terms. Such terms
are capable to represent complex domains and related operations in an efficient
manner. We also extend classical Boolean operations to LTAs. Finally, as a
major contribution, we introduce a new completion-based algorithm for computing
the possibly infinite set of reachable interpreted terms in a finite amount of
time.Comment: Technical repor
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