We would like to invite everyone to discuss the material that has been presented during virtual conference Computability in Europe (5-9 July 2021).
The entire slideshow (with large sections of text from our paper draft) is available HERE.
We decided to submit it for discussion, because we are now working on a new publication devoted to analog/continuous computations, and all additional critical input, and each additional discussion will be for us very precious.
To encourage you to read the whole slideshow, we put two representative (text) passages below:
Two basic (general) meanings of analogicity
The first meaning, we shall call it AN-C, refers to the concept of continuity. Its essence is the generalisation (broadening) of digital methods in order to make not only discrete (especially binary) but also continuous data processing possible. On a mathematical level, these data correspond to real numbers from a certain continuum (for example, an interval of a form [0,1]), yet on a physical level – certain continuous measurable variables (for example, voltage or electric potentials).
The second meaning, we shall call it AN-E, refers to the concept of analogy. It acknowledges that analog computations are based on natural analogies and consist in the realisation of natural processes which, in the light of defined natural theory (for example physical or biological), correspond to some mathematical operations. Metaphorically speaking, if we want to perform a mathematical operation with the use of a computational system, we should find in nature its natural analogon. It is assumed that such an analogon simply exists in nature and provides the high effectiveness of computations.
In a short comment to this distinction, we would like to add that the meaning of AN-E has, on the one hand, a historical character because the techniques, called analog, which consisted in the use of specific physical processes to specific computations, were applied mainly until the 1960s. On the other hand, it looks ahead to the future – towards computations of a new type that are more and more often called natural (for example, quantum or computations that use DNA).
The meaning of AN-C, by contrast, is more related to mathematical theories of data processing (the theoretical aspect of computations) than to their physical realisations.
The categories AN-C and AN-E are not disjoint, as there are empirical computations that consist in processing continuous quantities. As such, they are AN-E, but also fall into the AN-C category.
Empirical justification of AN-E computations
AN-E computations are closely related to the theories of empirical sciences (e.g., physics or biology). This means that specific computations of this type could neither be specified nor physically implemented without reference to a specific theory of this type.
Typically, such theories are treated as a tool for accurate description of physical reality in terms of mathematical structures and operations. Thus, their cognitive aspect is highlighted.
From the computational point of view (or more precisely: from the implementation one) they can be treated as a basis for realizing certain mathematical operations by means of physical processes described by these operations. With such an approach, a particular theory is treated as something that justifies the physical implementation of certain mathematical-algorithmic operations. It is therefore a justifying theory for a particular type of AN-E computation.
Once again, we invite everyone to discuss our slideshow — Paula Quinon & Paweł Stacewicz.