This paper deals with logics of programs. The objective is to formalize a notion of program description, and to give both plausible (semantic) and effective (syntactic) criteria for the notion of truth of a description. A novel feature of this treatment is the development of the mathematics underlying Floyd-Hoare axiom systems independently of such systems. Other directions that such research might take are considered.

This paper adopts the latter strategy in order to pursue the complexity of sorting

. In this paper we consider lower bounds for external-memory computational geometry problems. We find that it is not quite clear which model of computation to use when considering such problems. As an attempt of providing a model, we define the external memory Turing machine model, and we derive lower bounds for a number of problems, including the element distinctness problem, in this model. For these lower bounds we make the standard assumption that records are indivisible. Waiving the indivisibility assumption we show how to beat the lower bound for element distinctness. As an alternative model, we briefly discuss an external-memory version of the algebraic computation tree. 1. Introduction The Input/Output (or just I/O) communication between fast internal memory and slower external storage is the bottleneck in many large-scale computations. The significance of this bottleneck is increasing as internal computation gets faster, and as parallel computation gains popularity. Currently,...

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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computer-aided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areas-convex hulls, intersections, searching, proximity, and combinatorial optimizations-are discussed. Seven algorithmic techniques incremental construction, plane-sweep, locus, divide-andconquer, geometric transformation, prune-and-search, and dynamization-are each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.

We develop lower bounds on the number of primitive operations required to solve several fundamental problems in computational geometry. For example, given a set of points in the plane, are any three colinear? Given a set of points and lines, does any point lie on a line? These and similar questions arise as subproblems or special cases of a large number of more complicated geometric problems, including point location, range searching, motion planning, collision detection, ray shooting, and hidden surface removal. Previously these problems were studied only in general models of computation, but known techniques for these models are too weak to prove useful results. Our approach is to consider, for each problem, a more specialized model of computation that is still rich enough to describe all known algorit...

OF THE DISSERTATION The Complexity of Querying Indefinite Information: Defined Relations, Recursion and Linear Order by Ronald van der Meyden, Ph.D. Dissertation Director: L.T. McCarty This dissertation studies the computational complexity of answering queries in logical databases containing indefinite information arising from two sources: facts stated in terms of defined relations, and incomplete information about linearly ordered domains. First, we consider databases consisting of (1) a DATALOG program and (2) a description of the world in terms of the predicates defined by the program as well as the basic predicates. The query processing problem in such databases is related to issues in database theory, including view updates and DATALOG optimization, and also to the Artificial Intelligence problems of reasoning in circumscribed theories and sceptical abductive reasoning. If the program is non-recursive, the meaning of the database can be represented by Clark's Predicate Completion,...

This paper examines the lower-bounds of worst-case complexity measures of ray-shooting algorithms. It demonstrates that ray-shooting requires at least logarithmic time and discusses the strategies how to design such optimal algorithms. It also examines the lower-bounds of storage complexity of logarithmic-time algorithms and concludes that logarithmic time has very high price in terms of required storage.