39731578
On multiplicative graphs and the product conjecture
We study the following problem: which graphsG have the property that the class of all graphs not admitting a homomorphism intoG is closed under taking the product (conjunction)? Whether all undirected complete graphs have the property is a longstanding open problem due to S. Hedetniemi. We prove that all odd undirected cycles and all prime-power directed cycles have the property. The former result provides the first non-trivial infinite family of undirected graphs known to have the property, and the latter result verifies a conjecture of Nešetřil and Pultr These results allow us (in conjunction with earlier results of Nešetřil and Pultr [17], cf also [7]) to completely characterize all (finite and infinite, directed and undirected) paths and cycles having the property. We also derive the property for a wide class of 3-chromatic graphs studied by Gerards, [5].
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.
16945889
A Proof Theoretic Approach to Operational Semantics
Abstract Proof theory can be applied to the problem of specifying and reasoning about the operational semantics of process calculi. We overview some recent research in which λ-tree syntax is used to encode expressions containing bindings and sequent calculus is used to reason about operational semantics. There are various benefits of this proof theoretic approach for the π-calculus: the treatment of bindings can be captured with no side conditions; bisimulation has a simple and natural specification in which the difference between bound input and bound output is characterized using difference quantifiers; various modal logics for mobility can be specified declaratively; and simple logic programming-like deduction involving subsets of second-order unification provides immediate implementations of symbolic bisimulation. These benefits should extend to other process calculi as well. As partial evidence of this, a simple λ-tree syntax extension to the tyft/tyxt rule format for name-binding and name-passing is possible that allows one to conclude that (open) bisimilarity is a congruence.
100764
Foundational aspects of syntax
A large variety of computing systems, such as compilers, interpreters, static analyzers, and theorem provers, need to manipulate syntactic objects like programs, types, formulas, and proofs. A common characteristic of these syntactic objects is that they contain variable binders, such as quantifiers, scoping operators, and parameters. The presence of binders complicates formal specifications and symbolic processing. Consider, for example, a function definition of the form
3255666
Logic Programming in a Fragment of Intuitionistic Linear Logic
The intuitionistic notion of context is refined by using a fragment of J.-Y. Girard's (Theor. Comput. Sci., vol.50, p.1-102, 1987) linear logic that includes additive and multiplicative conjunction, linear implication, universal quantification, the of course exponential, and the constants for the empty context and for the erasing contexts. It is shown that the logic has a goal-directed interpretation. It is also shown that the nondeterminism that results from the need to split contexts in order to prove a multiplicative conjunction can be handled by viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). Examples taken from theorem proving, natural language parsing, and database programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. >
5180180
Logic programming in a fragment of intuitionistic linear logic
The intuitionistic notion of context is refined by using a fragment of J.-Y. Girard's (Theor. Comput. Sci., vol.50, p.1-102, 1987) linear logic that includes additive and multiplicative conjunction, linear implication, universal quantification, the of course exponential, and the constants for the empty context and for the erasing contexts. It is shown that the logic has a goal-directed interpretation. It is also shown that the nondeterminism that results from the need to split contexts in order to prove a multiplicative conjunction can be handled by viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). Examples taken from theorem proving, natural language parsing, and database programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately.<<ETX>>
7547345
Path Planning Through Time and Space in Dynamic Domains
Realistic robot problems involve navigating the robot through time as well as space. The obstacles that a robot must avoid and the pathways on which it travels are subject to changes throughout time. These changes can occur in a predictable or unpredictable fashion. This paper presents an integrated route planning and spatial representation system that allows paths to be calculated in dynamic domains. The path planner finds the "best route" through a given n-dimensional space. The "best route" is defined as the path through space-time with the best score as determined by a set of user-defined evaluation functions. The algorithm takes into account the following: Capabilities of the robot executing generated plans, traversability of space, and interactions with both predictable and unpredictable dynamic objects. The route planning algorithm is highly parallel and can be run on an SIMD (Single Instruction Multiple Data) machine in 0(p) time, where p is the length of a path. In domains with unpredictable processes, this system may be run in an incremental fashion, allowing the robot to create real-time routes along the optimal path as described by its current information. This paper will discuss: Spatial representation, an SIMD algorithm for route planning in a dynamic domain, and results from an implementation on a traditional computer architecture.