59,443 research outputs found
Decision problems concerning sets of equations
This thesis is about "decision problems concerning properties of sets of equations". If L is a first-order language with equality and if P is a property of sets of L-equations, then "the decision problem of P in L" is the problem of the existence or not of an algorithm, which enables us to decide whether, given a set Sigma of L-equations, Sigma has the property P or not. If such an algorithm exists, P is decidable in L. Otherwise, it is undecidable in L. After surveying the work that has been done in the field, we present a new method for proving the undecidability of a property P, for finite sets of L-equations. As an application, we establish the undecidability of some basic model-theoretical properties, for finite sets of equations of non-trivial languages. Then, we prove the non-existence of an algorithm for deciding whether a field is finite and, as a corollary, we derive the undecidability of certain properties, for recursive sets of equations of infinite non-trivial languages. Finally, we consider trivial languages, and we prove that a number of properties, undecidable in languages with higher complexity, are decidable in them.<p
Set Unification
The unification problem in algebras capable of describing sets has been
tackled, directly or indirectly, by many researchers and it finds important
applications in various research areas--e.g., deductive databases, theorem
proving, static analysis, rapid software prototyping. The various solutions
proposed are spread across a large literature. In this paper we provide a
uniform presentation of unification of sets, formalizing it at the level of set
theory. We address the problem of deciding existence of solutions at an
abstract level. This provides also the ability to classify different types of
set unification problems. Unification algorithms are uniformly proposed to
solve the unification problem in each of such classes.
The algorithms presented are partly drawn from the literature--and properly
revisited and analyzed--and partly novel proposals. In particular, we present a
new goal-driven algorithm for general ACI1 unification and a new simpler
algorithm for general (Ab)(Cl) unification.Comment: 58 pages, 9 figures, 1 table. To appear in Theory and Practice of
Logic Programming (TPLP
On the complexity of finding and counting solution-free sets of integers
Given a linear equation , a set of integers is
-free if does not contain any `non-trivial' solutions to
. This notion incorporates many central topics in combinatorial
number theory such as sum-free and progression-free sets. In this paper we
initiate the study of (parameterised) complexity questions involving
-free sets of integers. The main questions we consider involve
deciding whether a finite set of integers has an -free subset
of a given size, and counting all such -free subsets. We also
raise a number of open problems.Comment: 27 page
Advances and applications of automata on words and trees : abstracts collection
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
Branch-and-lift algorithm for deterministic global optimization in nonlinear optimal control
This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced, which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example. © 2013 Springer Science+Business Media New York
Mutually Unbiased Bases and Semi-definite Programming
A complex Hilbert space of dimension six supports at least three but not more
than seven mutually unbiased bases. Two computer-aided analytical methods to
tighten these bounds are reviewed, based on a discretization of parameter space
and on Grobner bases. A third algorithmic approach is presented: the
non-existence of more than three mutually unbiased bases in composite
dimensions can be decided by a global optimization method known as semidefinite
programming. The method is used to confirm that the spectral matrix cannot be
part of a complete set of seven mutually unbiased bases in dimension six.Comment: 11 pages
- …