Nice labeling problem for event structures: a counterexample

In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ admits a labeling with at most $f(n)$ labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barth\'elemy and Constantin in 1993

Hybrid VCSPs with crisp and conservative valued templates

A constraint satisfaction problem (CSP) is a problem of computing a homomorphism ${\bf R} \rightarrow {\bf \Gamma}$ between two relational structures. Analyzing its complexity has been a very fruitful research direction, especially for fixed template CSPs, denoted $CSP({\bf \Gamma})$, in which the right side structure ${\bf \Gamma}$ is fixed and the left side structure ${\bf R}$ is unconstrained. Recently, the hybrid setting, written $CSP_{\mathcal{H}}({\bf \Gamma})$, where both sides are restricted simultaneously, attracted some attention. It assumes that ${\bf R}$ is taken from a class of relational structures $\mathcal{H}$ that additionally is closed under inverse homomorphisms. The last property allows to exploit algebraic tools that have been developed for fixed template CSPs. The key concept that connects hybrid CSPs with fixed-template CSPs is the so called "lifted language". Namely, this is a constraint language ${\bf \Gamma}_{{\bf R}}$ that can be constructed from an input ${\bf R}$. The tractability of that language for any input ${\bf R}\in\mathcal{H}$ is a necessary condition for the tractability of the hybrid problem. In the first part we investigate templates ${\bf \Gamma}$ for which the latter condition is not only necessary, but also is sufficient. We call such templates ${\bf \Gamma}$ widely tractable. For this purpose, we construct from ${\bf \Gamma}$ a new finite relational structure ${\bf \Gamma}'$ and define $\mathcal{H}_0$ as a class of structures homomorphic to ${\bf \Gamma}'$. We prove that wide tractability is equivalent to the tractability of $CSP_{\mathcal{H}_0}({\bf \Gamma})$. Our proof is based on the key observation that ${\bf R}$ is homomorphic to ${\bf \Gamma}'$ if and only if the core of ${\bf \Gamma}_{{\bf R}}$ is preserved by a Siggers polymorphism. Analogous result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706

Variability Abstraction and Refinement for Game-Based Lifted Model Checking of Full CTL

One of the most promising approaches to fighting the configuration space explosion problem in lifted model checking are variability abstractions. In this work, we define a novel game-based approach for variability-specific abstraction and refinement for lifted model checking of the full CTL, interpreted over 3-valued semantics. We propose a direct algorithm for solving a 3-valued (abstract) lifted model checking game. In case the result of model checking an abstract variability model is indefinite, we suggest a new notion of refinement, which eliminates indefinite results. This provides an iterative incremental variability-specific abstraction and refinement framework, where refinement is applied only where indefinite results exist and definite results from previous iterations are reused. The practicality of this approach is demonstrated on several variability models

Harmonic Shears and Numerical Conformal Mappings

In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLAB implementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.Comment: 15 pages, 6 figure

Generalizing Boolean Satisfiability II: Theory

This is the second of three planned papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying ZAP, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the Davis-Putnam-Logemann-Loveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method. The third paper in this series discusses ZAPs implementation and presents experimental performance results

Towards rule-based visual programming of generic visual systems

This paper illustrates how the diagram programming language DiaPlan can be used to program visual systems. DiaPlan is a visual rule-based language that is founded on the computational model of graph transformation. The language supports object-oriented programming since its graphs are hierarchically structured. Typing allows the shape of these graphs to be specified recursively in order to increase program security. Thanks to its genericity, DiaPlan allows to implement systems that represent and manipulate data in arbitrary diagram notations. The environment for the language exploits the diagram editor generator DiaGen for providing genericity, and for implementing its user interface and type checker.Comment: 15 pages, 16 figures contribution to the First International Workshop on Rule-Based Programming (RULE'2000), September 19, 2000, Montreal, Canad