1440056
Concerning the achromatic number of graphs
Abstract The achromatic number of a graph G is the largest number of colors that can be assigned to the vertices of G so that (i) adjacent vertices are assigned different colors, and (ii) any two different colors are assigned to some pair of adjacent vertices. We study the achromatic number from the point of view of computational complexity. We show that, for each fixed integer n , there is an algorithm which, for an arbitrary graph G , determines in time O (| E ( G )|) whether the achromatic number of G is at least n . In contrast to this, we show that when n is part of the input, the problem is NP-complete even when restricted to bipartite graphs. The complexity of determining the achromatic number of a tree is unknown. We present polynomial algorithms to solve this problem for several classes of trees, and provide upper bounds on the achromatic number of a tree in terms of the maximum degree and the number of edges.
34355186
Achromatic numbers and graph operations
Abstract In this paper we investigate the achromatic number of the disjoint union ofgraphs and the achromatic number of the categorical product of graphs. We obtain the best possible upper and lower bounds for the union of graphs and the best possible lower bound for the product of graphs. Finally we show that the achromatic number of the product of graphs is bounded above.
2634305
Deadlines, Travel Time, and Robot Problem Solving
1 Abstract This paper describes some extensions to the reductionist planning paradigm typified by Sacerdoti's NOAH program. Certain inadequacies of the partial ordering scheme used in NOAH are pointed out and a new architecture is detailed which circumvents these problems. An example from the semi-automated factory domain is used to illustrate features of the new planner. Techniques for eliminating unnecessary travel time by the robot and avoiding backtracking due to deadline failures are discussed and their incorporation in the planner is described. 2 Introduction Most planners since NOAH [8] have represented a plan for a set of tasks as a partial (ie., non-linear) ordering of the steps required for carrying out those tasks. NOAH and its successors (eg., NONLIN [9] and DEVISER [10]) employ a partially ordered network of tasks to avoid early and unnecessary commitment to task orderings. The motivation for this is to eliminate backtrack-ing. However, maintaining a consistent partial order is difficult in domains where the fact that tasks actually take time plays an important role: domains in which deadlines or robot travel time are serious considerations. That a partial order leads unavoidably to a deadline failure usually cannot be discovered until an attempt is made to linearize the ordering. However, failure to notice a deadline violation early will require backtracking later. In addition, planning with a partial order is not well suited to efficiently managing factors like travel or machine running time. It's not difficult to represent that moving from one workstation to another takes time proportional to the distance separating the two workstations. However, generating a plan that eliminates unnecessary travel between workstations requires exploring some of the linearized task orderings. It is not until tasks are completely ordered that the source and destination workstations of each movement can be known and the travel time computed with any accuracy. In this paper we describe an approach to planning that combines the use of a partial order with a method for exploring the possible repercussions of that partial order. This approach has been implemented in the FORBIN planner (First Order RoBot INtender). FORBIN is a planner capable of solving a significantly wider class of problems than any of its predecessors. 2.1 The Factory Domain One problem domain that has been used for exploring our approach to planning is what we refer to as the semi-automated factory. In this domain a mobile robot operator wanders about the …
11019674
A Deterministic Annealing Approach for Parsimonious Design of Piecewise Regression Models
A new learning algorithm is proposed for piecewise regression modeling. It employs the technique of deterministic annealing to design space partition regression functions. While the performance of traditional space partition regression functions such as CART and MARS is limited by a simple tree-structured partition and by a hierarchical approach for design, the deterministic annealing algorithm enables the joint optimization of a more powerful piecewise structure based on a Voronoi partition. The new method is demonstrated to achieve consistent performance improvements over regular CART as well as over its extension to allow arbitrary hyperplane boundaries. Comparison tests, on several benchmark data sets from the regression literature, are provided.
3264852
Reasoning with higher-order abstract syntax in a logical framework
Logical frameworks based on intuitionistic or linear logics with higher-type quantification have been successfully used to give high-level, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that evaluation preserves types. One challenge in developing a framework for such reasoning is that higher-order abstract syntax (HOAS), an elegant and declarative treatment of object-level abstraction and substitution, is difficult to treat in proofs involving induction. In this article, we present a meta-logic that can be used to reason about judgments coded using HOAS; this meta-logic is an extension of a simple intuitionistic logic that admits higher-order quantification over simply typed λ-terms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of definition is a proof-theoretic device that allows certain theories to be treated as "closed" or as defining fixed points. We explore the difficulties of formal meta-theoretic analysis of HOAS encodings by considering encodings of intuitionistic and linear logics, and formally derive the admissibility of cut for important subsets of these logics. We then propose an approach to avoid the apparent trade-off between the benefits of higher-order abstract syntax and the ability to analyze the resulting encodings. We illustrate this approach through examples involving the simple functional and imperative programming languages PCF and PCF:=. We formally derive such properties as unicity of typing, subject reduction, determinacy of evaluation, and the equivalence of transition semantics and natural semantics presentations of evaluation.
22360460
Entropy-constrained tree-structured vector quantizer design
Current methods for the design of pruned or unbalanced tree-structured vector quantizers such as the generalized Breiman-Friedman-Olshen-Stone (GBFOS) algorithm proposed in 1980 are effective, but suffer from several shortcomings. We identify and clarify issues of suboptimality including greedy growing, the suboptimal encoding rule, and the need for time sharing between quantizers to achieve arbitrary rates. We then present the leaf-optimal tree design (LOTD) method which, with a modest increase in design complexity, alters and reoptimizes tree structures obtained from conventional procedures. There are two main advantages over existing methods. First, the optimal entropy-constrained nearest-neighbor rule is used for encoding at the leaves; second, explicit quantizer solutions are obtained at all rates without recourse to time sharing. We show that performance improvement is theoretically guaranteed. Simulation results for image coding demonstrate that close to 1 dB reduction of distortion for a given rate can be achieved by this technique relative to the GBFOS method.
18852816
Higher-Order Logic Programming
In this paper we consider the problem of extending Prolog to include predicate and function variables and typed λ-terms. For this purpose, we use a higher-order logic to describe a generalization to first-order Horn clauses. We show that this extension possesses certain desirable computational properties. Specifically, we show that the familiar operational and least fixpoint semantics can be given to these clauses. A language, λProlog that is based on this generalization is then presented, and several examples of its use are provided. We also discuss an interpreter for this language in which new sources of branching and backtracking must be accommodated. An experimental interpreter has been constructed for the language, and all the examples in this paper have been tested using it.