This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator @math for aggregate programs, independently proposed This operator allows us to closely tie the computational complexity of the answer set checking and answer sets existence problems to the cost of checking a solution of the aggregates in the program. Finally, we relate the semantics described by the operator to other proposals for logic programming with aggregates. To appear in Theory and Practice of Logic Programming (TPLP).
This notion of unfolding derives from the work on unfolding of intensional sets @cite_8 , and has been independently described in @cite_5 .
@cite_5 @cite_8
2004142419 2126998703 1783384750 2059566060
Abstract Two conditions on a collection of simple orders - unimodality and straightness - are necessary but not jointly sufficient for unidimensional unfolding representations. From the analysis of these conditions, a polynomial time algorithm is derived for the testing of unidimensionality and for the construction of a representation when one exists. Here, we propose a planning method for knotting unknotting of deformable linear objects. First, we propose a topological description of the state of a linear object. Second, transitions between these states are defined by introducing four basic operations. Then, possible sequences of crossing state transitions, i.e. possible manipulation processes, can be generated once the initial and the objective states are given. Third, a method for determining grasping points and their directions of movement is proposed to realize derived manipulation processes. Our proposed method indicated that it is theoretically possible for any knotting manipulation of a linear object placed on a table to be realized by a one-handed robot with three translational DOF and one rotational DOF. Furthermore, criteria for evaluation of generated plans are introduced to reduce the candidates of manipulation plans. Fourth, a planning method for tying knots tightly is established because they fulfill their fixing function by tightening them. Finally, we report knotting unknotting manipulation performed by a vision-guided system to demonstrate the usefulness of our approach. We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the outer dynamics' along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the scattering map'. We find pseudo-orbits of the scattering map that keep advancing in some privileged direction. Then we use the recurrence property of the inner dynamics', restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known two-dynamics' approach, the method we present relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. We include several applications, such as bridging large gaps in a priori unstable models in any dimension, and establishing diffusion in cases when the inner dynamics is a non-twist map. We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative. For example, we compute the width of the largest downward spike of a given length. Through the prism of these results, we find that the extrema of the free field under entropic repulsion (minus its mean) and those of the unconditioned free field are identical. Finally, when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two-dimensional simple random walk on the discrete torus.