Computability

About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.

Deterministic computations in time-varying graphs: Broadcasting under unstructured mobility

Abstract: Most highly dynamic infrastructure-less networks have in common that the assumption of connectivity does not necessarily hold at a given instant. Still, communication routes can be available between any pair of nodes over time and space. These networks (variously called delay-tolerant, disruptive-tolerant, challenged) are naturally modeled as time-varying graphs (or evolving graphs), where the existence of an edge is a function of time. In this paper we study deterministic computations under unstructured mobility, that is when the edges of the graph appear infinitely often but without any (known) pattern. In particular, we focus on the problem of broadcasting with termination detection. We explore the problem with respect to three possible metrics: the date of message arrival (foremost), the time spent doing the broadcast (fastest), and the number of hops used by the broadcast (shortest). We prove that the solvability and complexity of this problem vary with the metric considered, as well as with the type of knowledge a priori available to the entities. These results draw a complete computability map for this problem when mobility is unstructured.

Real Number Computability and Domain Theory

Abstract: We present the different constructive definitions of real number that can be found in the literature. Using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them. This definition of computability turns out to be equivalent to other definitions given in the literature using different methods. Domain theory is a useful tool to study higher order computability on real numbers. An interesting connection between Scott topology and the standard topologies on the real line and on the space of continuous functions on reals is stated. An important result in this paper is the proof that every computable functional on real numbers is continuous w.r.t. the compact open topology on the function space.

Instruction sequence processing operators

Abstract: Instruction sequence is a key concept in practice, but it has as yet not come prominently into the picture in theoretical circles. This paper concerns instruction sequences, the behaviours produced by them under execution, the interaction between these behaviours and components of the execution environment, and two issues relating to computability theory. Positioning Turing’s result regarding the undecidability of the halting problem as a result about programs rather than machines, and taking instruction sequences as programs, we analyse the autosolvability requirement that a program of a certain kind must solve the halting problem for all programs of that kind. We present novel results concerning this autosolvability requirement. The analysis is streamlined by using the notion of a functional unit, which is an abstract state-based model of a machine. In the case where the behaviours exhibited by a component of an execution environment can be viewed as the behaviours of a machine in its different states, the behaviours concerned are completely determined by a functional unit. The above-mentioned analysis involves functional units whose possible states represent the possible contents of the tapes of Turing machines with a particular tape alphabet. We also investigate functional units whose possible states are the natural numbers. This investigation yields a novel computability result, viz. the existence of a universal computable functional unit for natural numbers.

What Will Be Eventually True of Polynomial Hybrid Automata

Abstract: Hybrid automata have been introduced in both control engineering and computer science as a formal model for the dynamics of hybrid discrete-continuous systems. While computability issues concerning safety properties have been extensively studied, liveness properties have remained largely uninvestigated. In this article, we investigate decidability of state recurrence and of progress properties.First, we show that state recurrence and progress are in general undecidable for polynomial hybrid automata. Then, we demonstrate that they are closely related for hybrid automata subject to a simple model of noise, even though these automata are infinite-state systems. Based on this, we augment a semi-decision procedure for recurrence with a semidecision method for length-boundedness of paths in such a way that we obtain an automatic verification method for progress properties of linear and polynomial hybrid automata that may only fail on pathological, practically uninteresting cases. These cases are such that satisfaction of the desired progress property crucially depends on the complete absence of noise, a situation unlikely to occur in real hybrid systems.