The Rice-Shapiro theorem in Computable Topology

We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable topological spaces where the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable topological spaces with particular properties from wn--families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class

Fixed Points on Abstract Structures without the Equality Test

In this paper we present a study of definability properties of fixed points of effective operators on abstract structures without the equality test. In particular we prove that Gandy theorem holds for abstract structures. This provides a useful tool for dealing with recursive definitions using Sigma-formulas. One of the applications of Gandy theorem in the case of the reals without the equality test is that it allows us to define universal Sigma-predicates. It leads to a topological characterisation of Sigma-relations on |R

Making big steps in trajectories

We consider the solution of initial value problems within the context of hybrid systems and emphasise the use of high precision approximations (in software for exact real arithmetic). We propose a novel algorithm for the computation of trajectories up to the area where discontinuous jumps appear, applicable for holomorphic flow functions. Examples with a prototypical implementation illustrate that the algorithm might provide results with higher precision than well-known ODE solvers at a similar computation time

An Upper Bound on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems

In this paper we study Pfaffian hybrid systems which were first introduced in [8]. Pfaffian hybrid systems are a sub-class of o-minimal hybrid systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. The existence of finite bisimulations for ominimal hybrid systems has been shown by several authors (see e.g. [1, 9]). The next natural question to investigate is how the sizes of such bisimulations can be bounded. First step in this direction was done in [8] where a double exponential upper bound was shown for Pfaffian hybrid systems. In this paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of Pfaffian hybrid systems on which the exponential bound is attained. 1

Towards Computability over Effectively Enumerable Topological Spaces

In this paper we study different approaches to computability over effectively enumerable topological spaces. We introduce and investigate the notions of computable function, strongly-computable function and weakly-computable function. Under natural assumptions on effectively enumerable topological spaces the notions of computability and weakly-computability coincide

CICADA Collection

This volume is CICADA Collection which contains contributions developed by researches working on CICADA Project, The University of Manchester. CICADA Project creates a warm and fruitful atmosphere for research collaboration in many aries of Mathematics, Computer Science, Engineering including hybrid and dynamical systems, verification of safety critical systems, human robotics, model reduction and high dimensional systems, max-pus algebra, stochastic hybrid systems, analysis of adaptive systems and control. This volume presents examples and software which have been developed and used by CICADA community