The Variable Hierarchy for the Games mu-Calculus

Parity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables

Index problems for game automata

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as, respectively, the non-deterministic, alternating, and weak Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognizable by deterministic automata (the alternating variant trivializes). We investigate a wider class of regular languages, recognizable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognize languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is, the alternating index problem does not trivialize any more. Our main contribution is that all three index problems are decidable for languages recognizable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognized by a game automaton

VRCC-3D+: Qualitative spatial and temporal reasoning in 3 dimensions

Qualitative Spatial Reasoning (QSR) has varying applications in Geographic Information Systems (GIS), visual programming language semantics, and digital image analysis. Systems for spatial reasoning over a set of objects have evolved in both expressive power and complexity, but implementations or usages of these systems are not common. This is partially due to the computational complexity of the operations required by the reasoner to make informed decisions about its surroundings. These theoretical systems are designed to focus on certain criteria, including efficiency of computation, ease of human comprehension, and expressive power. Sadly, the implementation of these systems is frequently left as an exercise for the reader. Herein, a new QSR system, VRCC-3D+, is proposed that strives to maximize expressive power while minimizing the complexity of reasoning and computational cost of using the system. This system is an evolution of RCC-3D; the system and implementation are constantly being refined to handle the complexities of the reasoning being performed. The refinements contribute to the accuracy, correctness, and speed of the implementation. To improve the accuracy and correctness of the implementation, a way to dynamically change error tolerance in the system to more accurately reflect what the user sees is designed. A method that improves the speed of determining spatial relationships between objects by using composition tables and decision trees is introduced, and improvements to the system itself are recommended; by streamlining the relation set and enforcing strict rules for the precision of the predicates that determine the relationships between objects. A potential use case and prototype implementation is introduced to further motivate the need for implementations of QSR systems, and show that their use is not precluded by computational complexity. --Abstract, page iv

Abstract Representation of Music: A Type-Based Knowledge Representation Framework

The wholesale efficacy of computer-based music research is contingent on the sharing and reuse of information and analysis methods amongst researchers across the constituent disciplines. However, computer systems for the analysis and manipulation of musical data are generally not interoperable. Knowledge representation has been extensively used in the domain of music to harness the benefits of formal conceptual modelling combined with logic based automated inference. However, the available knowledge representation languages lack sufficient logical expressivity to support sophisticated musicological concepts. In this thesis we present a type-based framework for abstract representation of musical knowledge. The core of the framework is a multiple-hierarchical information model called a constituent structure, which accommodates diverse kinds of musical information. The framework includes a specification logic for expressing formal descriptions of the components of the representation. We give a formal specification for the framework in the Calculus of Inductive Constructions, an expressive logical language which lends itself to the abstract specification of data types and information structures. We give an implementation of our framework using Semantic Web ontologies and JavaScript. The ontologies capture the core structural aspects of the representation, while the JavaScript tools implement the functionality of the abstract specification. We describe how our framework supports three music analysis tasks: pattern search and discovery, paradigmatic analysis and hierarchical set-class analysis, detailing how constituent structures are used to represent both the input and output of these analyses including sophisticated structural annotations. We present a simple demonstrator application, built with the JavaScript tools, which performs simple analysis and visualisation of linked data documents structured by the ontologies. We conclude with a summary of the contributions of the thesis and a discussion of the type-based approach to knowledge representation, as well as a number of avenues for future work in this area

The Variable Hierarchy for the Games mu-Calculus

To appear in the journal Annals of Pure and Applied LogicInternational audienceParity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables

On the separation question for tree languages

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game

Common Knowledge and Interactive Behaviors: A Survey

This paper surveys the notion of common knowledge taken from game theory and computer science. It studies and illustrates more generally the effects of interactive knowledge in economic and social problems. First of all, common knowledge is shown to be a central concept and often a necessary condition for coordination, equilibrium achievement, agreement, and consensus. We present how common knowledge can be practically generated, for example, by particular advertisements or leadership. Secondly, we prove that common knowledge can be harmful, essentially in various cooperation and negotiation problems, and more generally when there are con icts of interest. Finally, in some asymmetric relationships, common knowledge is shown to be preferable for some players, but not for all. The ambiguous welfare effects of higher-order knowledge on interactive behaviors leads us to analyze the role of decentralized communication in order to deal with dynamic or endogenous information structures.Interactive knowledge, common knowledge, information structure, communication.

Appropriate choice of aggregation operators in fuzzy decision support systems

Fuzzy logic provides a mathematical formalism for a unified treatment of vagueness and imprecision that are ever present in decision support and expert systems in many areas. The choice of aggregation operators is crucial to the behavior of the system that is intended to mimic human decision making. This paper discusses how aggregation operators can be selected and adjusted to fit empirical data&mdash;a series of test cases. Both parametric and nonparametric regression are considered and compared. A practical application of the proposed methods to electronic implementation of clinical guidelines is presented<br /