On Transition Systems and Non‐well‐founded Sets

We consider monadic second order logic (MSO) and the modal μ-calculus (Lμ) over transition systems (Kripke structures). It is well known that every class of transition systems which is definable by a sentence of Lμ is definable by a sentence of MSO as well. It will be shown that the converse is also true for an important fragment of MSO: every class of transition systems which is MSO-definable and which is closed under bisimulation - i.e., the sentence does not distinguish between bisimilar models - is also Lμ-definable. Hence we obtain the following expressive completeness result: the bisimulation invariant fragment of MSO and Lμ are equivalent. The result was proved by David Janin and Igor Walukiewicz. Our presentation is based on their article [91]. The main step is the development of automata-based characterizations of Lμ over arbitrary transition systems and of MSO over transition trees (see also Chapter 16). It turns out that there is a general notion of automaton subsuming both characterizations, so we obtain a common ground to compare these two logics. Moreover we need the notion of the ω-unravelling for a transition system, on the one hand to obtain a bisimilar transition tree and on the other hand to increase the possibilities of choosing successors.

Bisimilar linear systems

The area of coalgebra has emerged within theoretical computer science with a unifying claim: to be the mathematics of computational dynamics. It combines ideas from the theory of dynamical systems and from the theory of state-based computation. Although still in its infancy, it is an active area of research that generates wide interest. Written by one of the founders of the field, this book acts as the first mature and accessible introduction to coalgebra. It provides clear mathematical explanations, with many examples and exercises involving deterministic and non-deterministic automata, transition systems, streams, Markov chains and weighted automata. The theory is expressed in the language of category theory, which provides the right abstraction to make the similarity and duality between algebra and coalgebra explicit, and which the reader is introduced to in a hands-on manner. The book will be useful to mathematicians and (theoretical) computer scientists and will also be of interest to mathematical physicists, biologists and economists.

Preface: Volume 59, Issue 4

In formal design verification, successful model checking is typically preceded by a laborious manual process of constructing design abstractions. We present a methodology for partially—and in some cases, fully—bypassing the abstraction process. For this purpose, we provide to the designer abstraction operators which, if used judiciously in the description of a design, structure the corresponding state space hierarchically. This structure can then be exploited by verification tools, and makes possible the automatic and exhaustive exploration of state spaces that would otherwise be out of scope for existing model checkers.Specifically, we present the following contributions: - A temporal abstraction operator that aggregates transitions and hides intermediate steps. Mathematically, our abstraction operator is a function that maps a flat transition system into a two-level hierarchy where each atomic upper-level transition expands into an entire lower-level transition system. For example, an arithmetic operation may expand into a sequence of bit operations. - A BDD-based algorithm for the symbolic exploration of multi-level hierarchies of transition systems. The algorithm traverses a level-n transition by expanding the corresponding level-(n − 1) transition system on-the-fly. The level-n successors of a state are determined by computing a level-(n − 1) reach set, which is then immediately released from memory. In this fashion, we can exhaustively explore hierarchically structured state spaces whose flat counterparts cause memory overflows. - We experimentally demonstrate the efficiency of our method with three examples—a multiplier, a cache coherence protocol, and a multiprocessor system. In the first two examples, we obtain significant improvements in run times and peak BDD sizes over traditional state-space search. The third example cannot be model checked at all using conventional methods (without manual abstractions), but can be analyzed fully automatically using transition hierarchies. KeywordsModel CheckTransition RelationParallel CompositionReactive ModuleReachability AnalysisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vedānta is one of the oldest philosophical systems. While there are many detailed commentaries on Vedānta, there are very few mathematical descriptions of the different concepts developed there. This article shows how ideas from theoretical computer science can be used to explain Vedānta. The standard ideas of transition systems and modal logic are used to develop a formal description for the different ideas in Vedānta. The generality of the formalism is illustrated via a number of examples including saṃsāra, Patañjali’s Yogasūtras, karma, the three avasthās from the Māṇḍūkya Upaniṣad and the key difference between advaita and dvaita in relation to mokṣa.

Narrowing is a universal unification procedure for equational theories defined by a canonical term rewriting system. In its original form it is extremely inefficient. Therefore, many optimizations have been proposed during the last years. In this paper, we introduce a new narrowing strategy, called LSE narrowing. LSE narrowing is complete for arbitrary canonical systems. It is optimal in the sense that two different LSE narrowing derivations cannot generate the same narrowing substitution.

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

Preface: Volume 11

It is proved that if an anisotropic medium has an open set of singular directions, then this medium has two slowness surfaces that completely coincide. The coinciding slowness surfaces form one double singular slowness surface. The corresponding anisotropic medium is an elliptical orthorhombic (ORT) medium with equal stiffness coefficients c44=c55=c66 rotated to an arbitrary coordinate system. Based on the representation of the Christoffel matrix as a uniaxial tensor and considering that the elements of the Christoffel matrix are quadratic forms in the components of the slowness vector, a system of homogeneous polynomial equations was derived. Then, the identical equalities between homogeneous polynomials are replaced by the equalities between their coefficients. As a result, a new system of equations is obtained, the solution of which is the values of the reduced (density normalized) stiffness coefficients in a medium with a singular surface. Conditions for the positive definite of the obtained stiffness matrix are studied. For the defined medium, the Christoffel equations and equations of group velocity surfaces are derived. The orthogonal rotation matrix that transforms the medium with a singular surface into an elliptic ORT medium in the canonical coordinate system is determined. In the canonical coordinate system, the slowness surfaces S1 and S2 waves coincide and are given by a sphere with a radius . The slowness surface of qP waves in the canonical coordinate system is an ellipsoid with semi-axes , , . The polarization vectors of S1 and S2 waves can be arbitrarily selected in the plane orthogonal to the polarization vector of the qP wave. However, the qP wave polarization vector can be significantly different from the wave vector. This feature should be taken into account in the joint processing and modelling of S and qP waves. The results are illustrated in one example of an elliptical ORT medium.

An extension to action systems is presented facilitating the modeling of continuous behavior in the discrete domain. The original action system formalism has been developed by Back et al. in order to describe parallel and distributed computations of discrete systems, i.e. systems with discrete state space and discrete control. In order to cope with hybrid systems, i.e. systems with continuous evolution and discrete control, two extensions have been proposed: hybrid action systems and continuous action systems. Both use differential equations (relations) to describe continuous evolution. Our version of action systems takes an alternative approach by adding a level of abstraction: continuous behavior is modeled by Qualitative Differential Equations that are the preferred choice when it comes to specifying abstract and possibly non-deterministic requirements of continuous behavior. Because their solutions are transition systems, all evolutions in our qualitative action systems are discrete.Based on hybrid action systems, we develop a new theory of qualitative action systems and discuss how we have applied such models in the context of automated test-case generation for hybrid systems.KeywordsAction SystemHybrid SystemDiscrete ActionHybrid AutomatonConformance TestingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

On the Continuity of Effective Multifunctions

This chapter presents the results of a fine-grained grammatical analysis of live commentaries of football (soccer) games, and so adds to a growing body of literature concerned with the linguistics of sport and, therein, particularly with football. More specifically, two subtly different types of football commentary are compared via a linguistic analysis: U.K.-produced live radio and television commentaries. The linguistic inquiry here explores how the experiences reported in these commentaries are encoded grammatically in the language used. As such, a detailed and large-scale analysis of the commentary data is conducted using Systemic Functional Linguistics, specifically a Transitivity analysis. In Systemic Functional theory where language is seen as inherently related to social context and culture, such an analysis can tell us a lot about the data contextually. The paper starts by introducing the Transitivity system within Halliday’s Systemic Functional Linguistics. This is followed by an introduction to the data analysed, explaining the contextual attributes and offering some preliminary linguistic characteristics of the component parts of the dataset. The body of the chapter presents the results of the analysis of Transitivity undertaken on the commentary data, divided into three parts to reflect three pertinent trends consequent from the data analysis.

Constraint Satisfaction Problems (CSPs) are natural computational problems that appear in many areas of theoretical computer science. Exploring which CSPs are solvable in polynomial time and which are NP-hard reveals a surprising link with central questions in universal algebra. This monograph presents a self-contained introduction to the universal-algebraic approach to complexity classification, treating both finite and infinite-domain CSPs. It includes the required background from logic and combinatorics, particularly model theory and Ramsey theory, and explains the recently discovered link between Ramsey theory and topological dynamics and its implications for CSPs. The book will be of interest to graduate students and researchers in theoretical computer science and to mathematicians in logic, combinatorics, and dynamics who wish to learn about the applications of their work in complexity theory.

In chapter 4 we have generalized finite automata to a wider class of structures, finite transition systems, and we have extended the run method to extend the notion of behavior. Via subset construction we showed this larger class of structures to still have periodic behavior only. Such results are useful; they provide additional freedom to describe periodic sets, that is, automata behavior. Stated differently, to show that an event is periodic it suffices to show it to be definable by a transition system. We made use of this freedom in section 4.4, to show that regular events are periodic. In this chapter we will find a large extension of finite transition systems to structures we call regular canonical systems. To extend the notion of behavior adequately, we will have to abandon the run idea. Its place will be taken by the notion of a formal deduction (or proof), which plays a central role in modern logic and linguistics.