Structure and Complexity in Planning with Unary Operators

Unary operator domains -- i.e., domains in which operators have a single effect -- arise naturally in many control problems. In its most general form, the problem of STRIPS planning in unary operator domains is known to be as hard as the general STRIPS planning problem -- both are PSPACE-complete. However, unary operator domains induce a natural structure, called the domain's causal graph. This graph relates between the preconditions and effect of each domain operator. Causal graphs were exploited by Williams and Nayak in order to analyze plan generation for one of the controllers in NASA's Deep-Space One spacecraft. There, they utilized the fact that when this graph is acyclic, a serialization ordering over any subgoal can be obtained quickly. In this paper we conduct a comprehensive study of the relationship between the structure of a domain's causal graph and the complexity of planning in this domain. On the positive side, we show that a non-trivial polynomial time plan generation algorithm exists for domains whose causal graph induces a polytree with a constant bound on its node indegree. On the negative side, we show that even plan existence is hard when the graph is a directed-path singly connected DAG. More generally, we show that the number of paths in the causal graph is closely related to the complexity of planning in the associated domain. Finally we relate our results to the question of complexity of planning with serializable subgoals

A Formal Framework for Speedup Learning from Problems and Solutions

Speedup learning seeks to improve the computational efficiency of problem solving with experience. In this paper, we develop a formal framework for learning efficient problem solving from random problems and their solutions. We apply this framework to two different representations of learned knowledge, namely control rules and macro-operators, and prove theorems that identify sufficient conditions for learning in each representation. Our proofs are constructive in that they are accompanied with learning algorithms. Our framework captures both empirical and explanation-based speedup learning in a unified fashion. We illustrate our framework with implementations in two domains: symbolic integration and Eight Puzzle. This work integrates many strands of experimental and theoretical work in machine learning, including empirical learning of control rules, macro-operator learning, Explanation-Based Learning (EBL), and Probably Approximately Correct (PAC) Learning.Comment: See http://www.jair.org/ for any accompanying file

The problem of time and gauge invariance in the quantization of cosmological models. I. Canonical quantization methods

The paper is the first of two parts of a work reviewing some approaches to the problem of time in quantum cosmology, which were put forward last decade, and which demonstrated their relation to the problems of reparametrization and gauge invariance of quantum gravity. In the present part we remind basic features of quantum geometrodynamics and minisuperspace cosmological models, and discuss fundamental problems of the Wheeler - DeWitt theory. Various attempts to find a solution to the problem of time are considered in the framework of the canonical approach. Possible solutions to the problem are investigated making use of minisuperspace models, that is, systems with a finite number of degrees of freedom. At the same time, in the last section of the paper we expand our consideration beyond the minisuperspace approximation and briefly review promising ideas by Brown and Kuchar, who propose that dust interacting only gravitationally can be used for time measuring, and the unitary approach by Barvinsky and collaborators. The latter approach admits both the canonical and path integral formulations and anticipates the consideration of recent developments in the path integral approach in the second part of our work.Comment: 16 pages, to be published in Grav. Cosmo

Byzantine Approximate Agreement on Graphs

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures

Canonical Quantization of (2+1)-Dimensional Gravity

We consider the quantum dynamics of both open and closed two- dimensional universes with ``wormholes'' and particles. The wave function is given as a sum of freely propagating amplitudes, emitted from a network of mapping class images of the initial state. Interference between these amplitudes gives non-trivial scattering effects, formally analogous to the optical diffraction by a multidimensional grating; the ``bright lines'' correspond to the most probable geometries.Comment: 22 pages, Mexico preprint ICN-UNAM-93-1

Path integral for minisuperspaces and its relation with non equivalent canonical quantizations

The relation between a recently proposed path integral for minisuperspaces and different canonical quantizations is established. The step of the procedure where a choice between non equivalent theories is made is identified. Coordinates avoiding such a choice are found for a class of homogeneous cosmologies.Comment: 11 pages, to appear in Physics Letters