Turing machines can be efficiently simulated by the General Purpose Analog Computer

The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and at a computational complexity level modulo polynomial reductions. However, the situation is less clear in what concerns models of computation using real numbers, and no analog of the Church-Turing thesis exists for this case. Recently it was shown that some models of computation with real numbers were equivalent from a computability perspective. In particular it was shown that Shannon's General Purpose Analog Computer (GPAC) is equivalent to Computable Analysis. However, little is known about what happens at a computational complexity level. In this paper we shed some light on the connections between this two models, from a computational complexity level, by showing that, modulo polynomial reductions, computations of Turing machines can be simulated by GPACs, without the need of using more (space) resources than those used in the original Turing computation, as long as we are talking about bounded computations. In other words, computations done by the GPAC are as space-efficient as computations done in the context of Computable Analysis

Polynomial differential equations compute all real computable functions on computable compact intervals

In the last decade, the eld of analog computation has experienced renewed interest. In particular, there have been several attempts to un- derstand which relations exist between the many models of analog com- putation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equiva- lent models of analog computation. Using this approach, it has been shown recently that the Gamma func- tion becomes computable by a GPAC [1]. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions over compact intervals can be de ned by such models

Parapatric distribution and sexual competition between two tick species, [i]Amblyomma variegatum[/i] and [i]A. hebraeum[/i] (Acari, Ixodidae), in Mozambique

[b]Background[/b]: [i]Amblyomma variegatum[/i] and [i]A. hebraeum[/i] are two ticks of veterinary and human health importance in south-east Africa. In Zimbabwe they occupy parapatric (marginally overlapping and juxtaposed) distributions. Understanding the mechanisms behind this parapatry is essential for predicting the spatio-temporal dynamics of Amblyomma spp. and the impacts of associated diseases. It has been hypothesized that exclusive competition between these species results from competition at the levels of male signal reception (attraction-aggregation-attachment pheromones) or sexual competition for mates. This hypothesis predicts that the parapatry described in Zimbabwe could also be present in other countries in the region. [br/][b]Methods[/b]: To explore this competitive exclusion hypothesis we conducted field surveys at the two species' range limits in Mozambique to identify areas of sympatry (overlapping areas) and to study potential interactions (communicative and reproductive interference effects) in those areas. At sympatric sites, hetero-specific mating pairs were collected and inter-specific attractiveness/repellent effects acting at long and short distances were assessed by analyzing species co-occurrences on co-infested herds and co-infested hosts.[br/] [b]Results[/b]: Co-occurrences of both species at sampling sites were infrequent and localized in areas where both tick and host densities were low. At sympatric sites, high percentages of individuals of both species shared attachment sites on hosts and inter-specific mating rates were high. Although cross-mating rates were not significantly different for[i] A. variegatum[/i] and [i]A. hebraeum[/i] females, attraction towards hetero-specific males was greater for [i]A. hebraeum[/i] females than for A. variegatum females and we observed small asymmetrical repellent effects between males at attachment sites.[br/][b]Conclusions[/b]: Our observations suggest near-symmetrical reproductive interference between [i]A. variegatum[/i] and [i]A. hebraeum[/i], despite between-species differences in the strength of reproductive isolation barriers acting at the aggregation, fixation and partner contact levels. Theoretical models predict that sexual competition coupled with hybrid inviability, greatly reduces the probability of one species becoming established in an otherwise suitable location when the other species is already established. This mechanism can explain why the parapatric boundary in Mozambique has formed within an area of low tick densities and relatively infrequent host-mediated dispersal events

Computational bounds on polynomial differential equations

In this paper we study from a computational perspective some prop-erties of the solutions of polynomial ordinary di erential equations. We consider elementary (in the sense of Analysis) discrete-time dynam-ical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary diferential equations with coe cients in Q[ ]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines. We also apply the previous methods to show that the problem of de-termining whether the maximal interval of defnition of an initial-value problem defned with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56. Combined with earlier results on the computability of solutions of poly-nomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines

Towards an Axiomatization of Simple Analog Algorithms

International audienceWe propose a formalization of analog algorithms, extending the framework of abstract state machines to continuous-time models of computation

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

Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

Boundedness of the domain of definition is undecidable for polynomial odes

Consider the initial-value problem with computable parameters dx dt = p(t, x) x(t0) = x0, where p : Rn+1 ! Rn is a vector of polynomials and (t0, x0) 2 Rn+1. We show that the problem of determining whether the maximal interval of definition of this initial-value problem is bounded or not is in general undecidable