Dr. Işık Barış Fidaner
Depending on the degree of their potentiality, there are four main types of one-sided coins:
Logically most basic coin is the Yay coin as the full potentiality p ≈ 1: It gives the outcome. It’s an unfragmented potentiality. It gives zero or low information.
Nay coin is the absence of the potentiality p ≈ 0: It does not give the outcome. It’s an infinitely fragmented potentiality. It gives zero or low information.
May coin is a partial potentiality p ≈ 1/e ≈ 37%: It may give the outcome. It’s uncertain. It’s a finitely fragmented potentiality. It gives positive information.
Yaay coin is an over-potentiality p > 1. It gives the outcome with surplus-certainty. It’s a defragmented potentiality. It gives negative information.
When the potentialities of a set of one-sided coins are linked by an event of tossing, then they are called an linked set of one-sided coins, or an ensemble of one-sided coins. For example, two May coins with p = 1/2 can be linked by an event of tossing.
Informations from an ensemble of one-sided coins can be combined to give a sum of information. For example, an ensemble of two May coins with p = 1/2 can be combined to get the information for the event of tossing. This would simulate a regular two-sided coin.
In an ensemble of one-sided coins, the certainty of Yaay coins can cancel the uncertainty of May coins. In a May coin, p = 1/e ≈ 37% gives +1/e information, which is the maximum possible uncertainty for a May coin. In a Yaay coin, p ≈ 132% gives -1/e information, a certainty that can cancel the maximum uncertainty for a May coin. To recall the example from [1], four times the information for p = 1/2 is +2log2 and the information for p = 2 is -2log2, i.e. the uncertainty of four May coins with p = 1/2 can be canceled by the certainty of a single Yaay coin with p = 2.
Note the special meaning of certainty & uncertainty here. The conventional idea says that something is either certain or uncertain and never both, i.e. it says that certainty and uncertainty are exclusive and never happen together. In an ensemble of one-sided coins, however, the potentialities of the coins are linked to a single event of tossing, and therefore their certainties and uncertainties may occur together and even cancel each other: There is a logarithmic balance between certainty and uncertainty, as exemplified by the numbers above. Moreover, the conventional idea says that a certainty can equally be of a No or a Yes. In an ensemble of one-sided coins, however, certainty is always on the side of Yes. No never brings any certainty; it’s just a lack of uncertainty.
One can merge two equal-sized ensembles of coins by asserting that their sets of potentialities represent one and the same set of potentialities. When two ensembles are merged, their matching coins are merged one by one. In merging two coins, the potentialities p and q of the coins are averaged like s = (p + q)/2. When merging individual coins, (1) Nay coins can cancel Yay & Yaay coins into May coins, adding to uncertainty, (2) Yaay coins can cancel May coins into Yay coins, reducing uncertainty.
Nay | May | Yay | Yaay | |
Nay | Nay | May | May | May |
May | May | May | May | Yay |
Yay | May | May | Yay | Yaay |
Yaay | May | Yay | Yaay | Yaay |
Table 1: Example coin mergings
Low information in a merged ensemble is interpreted as a correlation between the two merged ensembles. The basis of correlation is the correlation to a simple potentiality, to a Yay coin:
— Nay coin is not correlated to a Yay coin (positive info).
— May coin has only a slight correlation to a Yay coin (less positive info).
— Yay coin has complete correlation to a Yay coin (zero info).
— Yaay coin has extra correlation to a Yay coin (negative info).
***
A one-sided coin yields Heads.
A Yay coin is a one-sided coin that directly represents the event of yielding Heads. The most common proverbial meaning of “one-sided coin” is a Yay coin that always gives the same outcome.
A Nay coin is a one-sided coin that represents the negation of the event of yielding Heads. Nay coin is a coin without an outcome, which makes it a somewhat impossible entity. The second most common proverbial meaning of “one-sided coin” refers to such an impossible entity.
A May coin is a one-sided coin that represents the failure of the event of yielding Heads: There is an uncertainty about the event. In other words, its actualization is fragmented.
A Yaay coin is a one-sided coin that represents the insistence of the event of yielding Heads: There is a surplus-certainty about the event.
[1] Lack and Excess in Entropy Agglomeration
[2] Entropy as a statistical test of over/under-representation
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