What can algorithms compute with the help of information provided by an oracle that is a physical system? We have developed a theory that combines Turing machines with experiments that perform physical measurements in which queries are governed by subtle timing protocols and provide the equipment with numerical data with (i) infinite precision, (ii) finite but unbounded precision or (iii) finite but fixed precision. Here, we consider the measurement of physical quantities that are thresholds, whose values are obtained by a sequence of approximate measurements that converge either from above or from below. The thresholds may be authentic physical properties or artefacts of the methods and equipment that performs the measurement. Using a canonical example of a threshold oracle, the broken beam balance for measuring mass, we develop methods to cope with thresholds and classify the computational power in polynomial time of this physical oracle using non-uniform complexity classes. Surprisingly, new complexity classes arise illuminating the influence of the operation of the equipment. All classes break the Turing Barrier.

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Reaction systems are a model of computation inspired by biochemical reactions involving reactants, inhibitors and products from a finite background set. We define a notion of multi-step simulation among reaction systems and derive a classification with respect to the amount of resources (reactants and inhibitors) involved in each reaction. We prove that “simple” reaction systems, having at most one reactant and one inhibitor per reaction, suffice in order to simulate arbitrary systems. Finally, we show that the equivalence relation of mutual simulation induces exactly five linearly ordered classes of reaction systems characterizing well-known subclasses of the functions over Boolean lattices, such as the constant, additive (join-semilattice endomorphisms), monotone, and antitone functions.

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We argue that dynamical systems involving discrete and continuous data can be modelled by Turing machines with oracles that are physical processes. Using the theory introduced in Beggs et al. [2,3], we consider the scope and limits of polynomial time computations by such systems. We propose a general polynomial time Church-Turing Thesis for feasible computations by analogue-digital systems, having the non-uniform complexity class BPP//log* as theoretical upper bound. We show why BPP//log* should be replace P/poly, which was proposed by Siegelmann for neural nets [23,24]. Then we examine whether other sources of hypercomputation can be found in analogue-digital systems besides the oracle itself. We prove that the higher polytime limit P/poly can be attained via non-computable analogue-digital interface protocols.

We introduce an abstract analogue-digital model of computation that couples Turing machines to oracles that are physical processes. Since any oracle has the potential to boost the computational power of a Turing machine, the effect on the power of the Turing machine of adding a physical process raises interesting questions. Do physical processes add significantly to the power of Turing machines; can they break the Turing Barrier? Does the power of the Turing machine vary with different physical processes? Specifically, here, we take a physical oracle to be a physical experiment, controlled by the Turing machine, that measures some physical quantity. There are three protocols of communication between the Turing machine and the oracle that simulate the types of error propagation common to analogue-digital devices, namely: infinite precision, unbounded precision, and fixed precision. These three types of precision introduce three variants of the physical oracle model. On fixing one archetypal experiment, we show to classify the computational power of the three models by establishing the lower and upper bounds. Using new techniques and ideas about timing, we give a complete classification. Corresponding author 4 In the ARNN case, a subsystem of about eleven neurones (variables) performs a measurement of a realvalued weight of the network up to some precision and resumes to a computation with advice simulated by a system of a thousand rational neurones interconnected with integer and a very few rational weights.

This study considers the problem of the extreme behavior exhibited by solutions to Burgers equation subject to stochastic forcing. More specifically, we are interested in the maximum growth achieved by the "enstrophy" (the Sobolev H 1 seminorm of the solution) as a function of the initial enstrophy E 0 , in particular, whether in the stochastic setting this growth is different than in the deterministic case considered by Ayala & Protas (2011). This problem is motivated by questions about the effect of noise on the possible singularity formation in hydrodynamic models. The main quantities of interest in the stochastic problem are the expected value of the enstrophy and the enstrophy of the expected value of the solution. The stochastic Burgers equation is solved numerically with a Monte Carlo sampling approach. By studying solutions obtained for a range of optimal initial data and different noise magnitudes, we reveal different solution behaviors and it is demonstrated that the two quantities always bracket the enstrophy of the deterministic solution. The key finding is that the expected values of the enstrophy exhibit the same power-law dependence on the initial enstrophy E 0 as reported in the deterministic case. This indicates that the stochastic excitation does not increase the extreme enstrophy growth beyond what is already observed in the deterministic case.

We consider computation with real numbers that arise through a process of physical measurement. We have developed a theory in which physical experiments that measure quantities can be used as oracles to algorithms and we have begun to classify the computational power of various forms of experiment using non-uniform complexity classes. Earlier, in Beggs et al. (2014 Reviews of Symbolic Logic7(4) 618–646), we observed that measurement can be viewed as a process of comparing a rational number z – a test quantity – with a real number y – an unknown quantity; each oracle call performs such a comparison. Experiments can then be classified into three categories, that correspond with being able to return test results $$\begin{eqnarray*} z < y\text{ or }z > y\text{ or }\textit{timeout},\\ z < y\text{ or }\textit{timeout},\\ z \neq y\text{ or }\textit{timeout}. \end{eqnarray*} $$ These categories are called two-sided, threshold and vanishing experiments, respectively. The iterative process of comparing generates a real number y. The computational power of two-sided and threshold experiments were analysed in several papers, including Beggs et al. (2008 Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences)464 (2098) 2777–2801), Beggs et al. (2009 Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences)465 (2105) 1453–1465), Beggs et al. (2013a Unconventional Computation and Natural Computation (UCNC 2013), Springer-Verlag 6–18), Beggs et al. (2010b Mathematical Structures in Computer Science20 (06) 1019–1050) and Beggs et al. (2014 Reviews of Symbolic Logic, 7 (4):618-646). In this paper, we attack the subtle problem of measuring physical quantities that vanish in some experimental conditions (e.g., Brewster's angle in optics). We analyse in detail a simple generic vanishing experiment for measuring mass and develop general techniques based on parallel experiments, statistical analysis and timing notions that enable us to prove lower and upper bounds for its computational power in different variants. We end with a comparison of various results for all three forms of experiments and a suitable postulate for computation involving analogue inputs that breaks the Church–Turing barrier.