An Analogue-Digital Model of Computation: Turing Machines with Physical Oracles

Abstract

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.

Computability and Complexity of Unconventional Computing Devices

“…Beggs and coauthors (see (Ambaram et al; and references therein) have made a careful analysis of using a range of idealised physical experiments as oracles, in particular, studying the time it takes to interact with the physical device, as a function of the precision of its output. More precision takes more time.…”

“…The kind of physical analogue devices that Ambaram et al (2017) analyse tend to use a unary encoding of the relevant value being accessed via physical measurement, for example, position, or mass, or concentration. So each extra digit of precision has to access an exponentially smaller range of the system being measured.…”

On More or Less Appropriate Notions of ‘Computation’

“…• We do not dogmatically restrict our notion of 'computation' to the currently predominant idea of discrete (binary-digital) computation. Our notion of 'computation' ought to remain 'semantically open' for analog (MacLennan, 2007) as well as hybrid (analog-digital) implementations (Ambaram, Beggs, Costa, Poças, & Tucker, 2017), whereby the 'classical' theories of 'computability', 'programmability' and 'computational complexity' might have to be reconsidered and re-researched for those 'other' implementational possibilities (Beggs, Costa, Poças, & Tucker, 2014;).…”

A Hierarchy for $$ BPP //\log \!\star $$ B P P / / log ⋆ Based on Counting Calls to an Oracle

“…To use the RWE as an oracle, we admit that the probability σ that the particle moves forward, encodes some advice. Unlike scatter machine experiments in [5,11,16], the RWE does not need any parameters to be initialized, i.e., the Turing machine does not provide the oracle with any dyadic rational, it just "pulls the trigger" to start the experiment. We consider both a Turing machine with added RWE oracle, a RW Turing machine, and a fair probabilistic Turing machine with added RWE oracle, a RW fair probabilistic Turing machine For every unknown σ ∈ (0, 1), the time that a particle takes to be absorbed is unbounded.…”

“…It is proved in[12,16] that, for every σ ∈ C3 and for every dyadic rational z, if |σ − z| ≤ 2 −k−5 , then the binary expansions of x and z coincide on the first k bits.…”