Arthur M. Farley

Minimum broadcast graphs

Let a graph G = (V, E) represent a communication network. The set V of vertices corresponds to the members of the network, and the set E of edges corresponds to the communication lines connecting pairs of members. Suppose that a member originates a message which is to be communicated to all other members of the network. This is to be accomplished as quickly as possible by a series of calls placed over lines of the network. We adopt the constraints that (i) each call requires one unit of time, (ii) a member can participate in only one call per unit of time, and (iii) a member can only call an adjacent member. We refer to this one-to-all communication process as broadcasting. Consider the following problem: given a connected graph G and a meSSage originator, vertex u, what is the minimum number of time units required to complete broadcasting from vertex u? We define the broadcast time of u vertex u, b(u), to equal this minimum time. It is easy to see that for any vertex u in a connected graph G with n vertices, b(u) a [ log, nl , since during each time unit the number of informed vertices can at most double. We define the broadcast time of u graph G, 6(G), to equal the maximum broadcast time of any vertex u in G, i.e., b(G) = max {6(u) 1 u E V(G)}. For the complete graph K, with n > 2 vertices, b(K,) = [ log, nl , yet K, is not minimal with respect to this property. Th’at is, we can remove several edges from & and still have a granh G such that b(G) = [ log, nl . We define a minimal broadcast graph to be a graph G with n vertices such that b(G) = [log, nl but for every proper spanning subgraph G’c G, b( G’) > [log* 4 l We define the broadcast function B(n) to equal the minimum number of edges in any minimal broadcast graph on n vertices. A minimum broadcast graph is a minimal broadcast graph on n vertices having B(n) edges. Frolm the point of view of applications, minimum broadcast graphs represent the cheapest possible cammunication networks (in terms of number of lines) in which broadcasting can be accomplished from any vertex as fast as theoretically possible.

Minimum‐time line broadcast networks

Broadcasting refers to the process of message dissemination in a communication network whereby a message, originated by one member, becomes known to all members. Line broadcasting assumes that members may “switch-through” any number of calls during a time unit. An algorithm is presented which produces a calling schedule completing line broadcasting in minimum time from any member of any tree. A discussion of three types of broadcasting and their associated minimum time networks is included.

Burden of proof in legal argumentation

We present a computational model of dialectical argumentation that could serve as a basis for studying elements of legal reasoning. Argumentation is well-suited to decisionmaking in the legal domain, where knowledge is incomplete, uncertain, and inconsistent, We model an argument both as information structure, i.e., argument units connecting claims with supporting data, and as dialectical process, i.e., an alternating series of moves made by opposing sides. Inspired by the legal domain, our model includes burden of proof as a key element, indicating the level of support that must be achieved by a particular side to an argument. Burden of proof acts as a move filter and termination criterion during argumentation and determines the eventual winner. We demonstrate our model by considering two examples that have been discussed previously in the artificial intelligence and legal reasoning literature. INTRODUCTION As the artificial intelligence (AI) and legal reasoning communities are well aware, most decisions are reached against a background of incomplete, uncertain, and inconsistent knowledge (i.e., weak theory domains; Porter, et. al., 1990). The most widely used AI methods for reasoning under uncertainty either rely on an absence of outright contradictions (e. g., probabilistic reasoning; Pearl, 1987) or are unable to support motivated decision-making in the face of inconsistent information (e.g., default reasoning; Ginsberg, 1987). Both solutions put the problem of deciding what to believe outside their respective domains of discourse. Choosing the proposition with highest Permission to copywithoutfee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appea, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republisb, requires a fee andlor specific permission.

A model of argumentation and its application to legal reasoning

We present a computational model of dialectical argumentation that could serve as a basis for legal reasoning. The legal domain is an instance of a domain in which knowledge is incomplete, uncertain, and inconsistent. Argumentation is well suited for reasoning in such weak theory domains. We model argument both as information structure, i.e., argument units connecting claims with supporting data, and as dialectical process, i.e., an alternating series of moves by opposing sides. Our model includes burden of proof as a key element, indicating what level of support must be achieved by one side to win the argument. Burden of proof acts as move filter, turntaking mechanism, and termination criterion, eventually determining the winner of an argument. Our model has been implemented in a computer program. We demonstrate the model by considering program output for two examples previously discussed in the artificial intelligence and legal reasoning literature.

Networks immune to isolated failures

The notion of isolated failure immune (IFI) networks is introduced. A network is an IFI network if and only if all message transfers between operating sites can be completed as long as all site and line failures are isolated. Failures occurring at any time are isolated if they are pairwise isolated. A pair of site failures is isolated if the down sites are not neighbors. A pair of line failures is isolated if the lines are not incident to a common site. A site and a line failure are isolated if the down line is not incident to a neighbor of the down site. We show that 2-trees are minimum IFI networks. An algorithm is described which adds lines to an arbitrary tree network to produce an IFI network (a 2-tree). The algorithm also determines routing tables which allow a simple calling protocol to complete all message transfers between operative sites under isolated failures. Specializations and generalizations of our notion of IFI networks are discussed and issues for future research are proposed.

Spanners and message distribution in networks

We investigate the applicability of spanners as substructures that offer both low cost and low delay for broadcast and multicast. A k-spanner has the potential to offer lower delay than shared multicast trees because it limits the distance between any two nodes in the network to a multiplicative factor k of the shortest-path distance. Using simulation over random topologies, we compare k-spanners to single-source minimum-distance spanning trees and show that a k-spanner can have similar cost and lower delay. We illustrate that by varying the value of k a spanner can be made to gradually favor cost over delay. These results indicate the spanner can reduce the cost of flooding, which is a basic mechanism used in multicast routing protocols.

Computation of the center and diameter of outerplanar graphs

Abstract The center of a graph is the set of vertices with minimum eccentricity. An outerplanar graph is a planar segmentation of a polygon. We define a notion of edge eccentricities for the edges of an outerplanar graph. We present an algorithm which efficiently computes these edge eccentricities. Knowledge of the edge eccentricities allows subsequent linear time computation of the center and diameter of outerplanar graphs. The computation of edge eccentricities is shown to require linear time for certain subclasses of outerplanar graphs.

SELF-REPAIRING NETWORKS

In this paper, we provide a specification for a class of communication networks that are immune to single site failures, not only maintaining the ability to transfer messages between operable sites but doing so with no additional delay (i.e. no increase in length of communication path). We call these networks self-repairing and refer to their underlying topologies as self-repairing graphs. Our specification includes a constructive characterization of a class of minimal self-repairing graphs and an algorithmic determination of associated routing tables that can be used by a simple message transfer procedure to realize the desired immune behavior.

Qualitative reasoning in economics

Abstract We present a scheme for qualitative reasoning about economic systems. The fundamental element of our scheme is the market, a homeostatic entity that adjusts to restore equilibrium after perturbation. Complex theories are represented in terms of multiple markets, with their interaction realized by variables of one market being parameters to others. We define a simulation process, making it possible to reason about multiple-market models. Comparing final with initial values of market variables implements comparative statics. We define a process of backward reasoning from changes in variables as desired effects to parameter perturbations as possible causes. Our notions are illustrated by consideration of Keynesian IS-LM and general macroeconomic models.

Networks immune to isolated line failures

A network is immune to a set of failures if all message transfers between operative sites can be completed in the presence of such failures. A set of line failures is isolated if no two failing lines are incident to the same site. Several classes of isolated line failure immune (ILFI) networks are defined, including a class with fewest lines for a given number of sites. An algorithm is presented which turns an arbitrary tree into one of these minimum ILFI networks and computes routing tables for the new network.