A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Index theory of one dimensional quantum walks and cellular automata

If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems - namely quantum walks and cellular automata - we make this intuition precise by defining an index, a quantity that measures the "net flow of quantum information" through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S_1, S_2 can be pieced together, in the sense that there is a system S which locally acts like S_1 in one region and like S_2 in some other region, if and only if S_1 and S_2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S_1 into S_2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map S -> ind S is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.Comment: 38 pages. v2: added examples, terminology clarifie

Two-Player Reachability-Price Games on Single-Clock Timed Automata

We study two player reachability-price games on single-clock timed automata. The problem is as follows: given a state of the automaton, determine whether the first player can guarantee reaching one of the designated goal locations. If a goal location can be reached then we also want to compute the optimum price of doing so. Our contribution is twofold. First, we develop a theory of cost functions, which provide a comprehensive methodology for the analysis of this problem. This theory allows us to establish our second contribution, an EXPTIME algorithm for computing the optimum reachability price, which improves the existing 3EXPTIME upper bound.Comment: In Proceedings QAPL 2011, arXiv:1107.074

Model checking embedded system designs

We survey the basic principles behind the application of model checking to controller verification and synthesis. A promising development is the area of guided model checking, in which the state space search strategy of the model checking algorithm can be influenced to visit more interesting sets of states first. In particular, we discuss how model checking can be combined with heuristic cost functions to guide search strategies. Finally, we list a number of current research developments, especially in the area of reachability analysis for optimal control and related issues

Entropy rate of higher-dimensional cellular automata

We introduce the entropy rate of multidimensional cellular automata. This number is invariant under shift-commuting isomorphisms; as opposed to the entropy of such CA, it is always finite. The invariance property and the finiteness of the entropy rate result from basic results about the entropy of partitions of multidimensional cellular automata. We prove several results that show that entropy rate of 2-dimensional automata preserve similar properties of the entropy of one dimensional cellular automata. In particular we establish an inequality which involves the entropy rate, the radius of the cellular automaton and the entropy of the d-dimensional shift. We also compute the entropy rate of permutative bi-dimensional cellular automata and show that the finite value of the entropy rate (like the standard entropy of for one-dimensional CA) depends on the number of permutative sites. Finally we define the topological entropy rate and prove that it is an invariant for topological shift-commuting conjugacy and establish some relations between topological and measure-theoretic entropy rates

Behavioural hybrid process calculus

Process algebra is a theoretical framework for the modelling and analysis of the behaviour of concurrent discrete event systems that has been developed within computer science in past quarter century. It has generated a deeper nderstanding of the nature of concepts such as observable behaviour in the presence of nondeterminism, system composition by interconnection of concurrent component systems, and notions of behavioural equivalence of such systems. It has contributed fundamental concepts such as bisimulation, and has been successfully used in a wide range of problems and practical applications in concurrent systems. We believe that the basic tenets of process algebra are highly compatible with the behavioural approach to dynamical systems. In our contribution we present an extension of classical process algebra that is suitable for the modelling and analysis of continuous and hybrid dynamical systems. It provides a natural framework for the concurrent composition of such systems, and can deal with nondeterministic behaviour that may arise from the occurrence of internal switching events. Standard process algebraic techniques lead to the characterisation of the observable behaviour of such systems as equivalence classes under some suitably adapted notion of bisimulation