High-Level Models of Unconventional Computations by Andrew Schumann & Krzysztof Pancerz

Author:Andrew Schumann & Krzysztof Pancerz
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Differential equation

Initial value

Name

Sum

Product

Inverse

So, we examine streams as dynamical entities, whose behaviour consists of the repeatedly offering of the next element of the stream. Using coinduction streams are defined by specifying their behaviour, and the equality of two streams can be established by proving that they have the same behaviour (in other words, that they are ‘behaviourally’ equivalent).

The main reason for choosing to use streams as the coinductive datatype consists in a possibility to give definitions very close in style to those with self-reference corresponding to

We can try to get non-well-founded probabilities on a non-well-founded algebra of fuzzy subsets that consists of the following [83]: (1) union, intersection, and difference of two non-well-founded fuzzy subsets of ; (2) and . In this case a finitely additive non-well-founded probability measure is a nonnegative set function defined for sets that runs the set V (for example, ) and satisfies the following properties: (1) for all , (2) and , (3) if and are disjoint, then , (4) for all .

This probability measure is called non-well-founded probability . Their main originality is that conditions 3, 4 are independent. As a result, in a probability space some Bayes’ formulas do not hold in the general case.

Suppose that the ordering relation on is defined digit by digit. Then the number is the greatest in . As an example of trivial non-well-founded probability we can introduce the following function defined on p-adic streams by coinduction: for every . Notice that p-adic probabilities used in adelic or p-adic quantum mechanics are particular cases of non-well-founded probabilities.

If we take the p-adic case of non-well-founded probability theory, then we observe essentially new properties of relative frequencies that do not appear on real numbers. For example, consider two attributes and . Suppose that in the first tests the label has realizations, has realizations. According to our intuition, their probabilities should be different, but in real probability theory we obtain: . In 2-adic probability theory we have , because in , , , and

This example shows that in p-adic probability theory there are statistical phenomena for that relative sequences of observed events have non-zero probabilities in the p-adic metric, but do not have positive probabilities in the standard real metric.

We assume that reality is non-well-founded and the Physarum motions can be formalized as some streams defined coinductivelly. Therefore, the slime mould motions can be programmed only within an object-oriented approach that we are going define right now.

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