Undecidable problem

About: Undecidable problem is a research topic. Over the lifetime, 3135 publications have been published within this topic receiving 71238 citations.

Benefits of Relaxing Punctuality

Abstract: : The most natural, compositional way of modeling real time systems uses a dense domain for time. The satisfiability of real time constraints that are capable of expressing punctual it in this model is, however, known to be undecidable. The authors introduce a temporal language that can constrain the time difference between events only with finite (yet arbitrary) precision and show the resulting logic to be EXPACE-complete. This result allows the authors to develop an algorithm for the verification of timing properties of real time systems with a dense semantics.

Parametric real-time reasoning

Abstract: Traditional approaches to the algorithmic veri cation of real-time systems are limited to checking program correctness with respect to concrete timing properties (e.g., \message delivery within 10 milliseconds"). We address the more realistic and more ambitious problem of deriving symbolic constraints on the timing properties required of real-time systems (e.g., \message delivery within the time it takes to execute two assignment statements"). To model this problem, we introduce parametric timed automata | nite-state machines whose transitions are constrained with parametric timing requirements. The emptiness question for parametric timed automata is central to the veri cation problem. On the negative side, we show that in general this question is undecidable. On the positive side, we provide algorithms for checking the emptiness of restricted classes of parametric timed automata. The practical relevance of these classes is illustrated with several veri cation examples. There remains a gap between the automata classes for which we know that emptiness is decidable and undecidable, respectively, and this gap is related to various hard and open problems of logic and automata theory.

Modular reuse of ontologies: theory and practice

Abstract: In this paper, we propose a set of tasks that are relevant for the modular reuse of ontologies. In order to formalize these tasks as reasoning problems, we introduce the notions of conservative extension, safety and module for a very general class of logic-based ontology languages. We investigate the general properties of and relationships between these notions and study the relationships between the relevant reasoning problems we have previously identified. To study the computability of these problems, we consider, in particular, Description Logics (DLs), which provide the formal underpinning of the W3C Web Ontology Language (OWL), and show that all the problems we consider are undecidable or algorithmically unsolvable for the description logic underlying OWL DL. In order to achieve a practical solution, we identify conditions sufficient for an ontology to reuse a set of symbols "safely"--that is, without changing their meaning. We provide the notion of a safety class, which characterizes any sufficient condition for safety, and identify a family of safety classes-called locality--which enjoys a collection of desirable properties. We use the notion of a safety class to extract modules from ontologies, and we provide various modularization algorithms that are appropriate to the properties of the particular safety class in use. Finally, we show practical benefits of our safety checking and module extraction algorithms.

Unpredictability and undecidability in dynamical systems

Abstract: We show that motion with as few as three degrees of freedom (for instance, a particle moving in a three-dimensional potential) can be equivalent to a Turing machine, and so be capable of universal computation. Such systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos: Even if the initial conditions are known exactly, virtually any question about their long-term dynamics is undecidable.