It seems to me that I am rewriting Turing’s paper in a new perspective. The whole first chapter is to reclaim Turing’s idea, but in a way to discuss its necessity but not its completeness.

The set of Turing’s abilities is the subset of the total abilities.

Distinguishability is the most fundamental ability of intelligence.

What if we don’t have the ability to distinguish any information?

The universe will be uniformly indistinguishable. And there will be no basis to do anything. All the signals are the same. There will be no sense of space and time. A rock has no intelligence, since it doesn’t has the ability to distinguish.

What do we mean when we talk about distinguishability?

This is essentially as same as distinct symbols as elements of a set. For any set of symbols, we have a relation or so to speak an ability to return true or false based on given information. Formally:

$$\mathbf{Identical} (A, B)$$

This is the relation of any two objects, presented as the most expanded form. It’s true if A and B is identical, false otherwise.

What’s special about distinguishability?

Though distinguishability seems intuitive and have no need to talk about, yet it is the first thing we may consider at all intellectual procedures and one of the most powerful tools we may have.

For an iPhone X, you can’t use it to detect whether a human female is recently pregnant or not. Also the blood pressure is an indistinguishable information for an iPhone X because of the limitation of its sensor, yet it can detect the heart rate, or distinguish the difference between heart rates, using optical information it gathers.

In general relativity, you can’t distinguish from an elevator falls freely and a space elevator accelerates at the same rate as gravitational acceleration. In Turing test, the core idea is the indistinguishability of a machine and a child. For some movies or games, one of their goals is to make computer graphics indistinguishable from the reality to human viewers.

In the solving process, the intelligent agent might ask first what distinguishable objects at hand. The limitation of any intelligent agents is also determined by its ability to distinguish.

The distinguishability is not a property of the objects, but what the intelligent agents determine. The agents might determine two objects is identical based on its sensor. So in reality, any objects in the real world is somehow distinguishable, but for the machines things are identical to some degree, we may discuss this in detail in the following abstraction chapter.

The reference or identifier or encoding of distinguishable objects.

To the solver or the agent, they may have different way to store the distinguishable objects. In the following discussion, we use the form of $\langle G\rangle$ to denote the encoding of an object G which it can distinguish. Some common identifier systems are IP address, phone number, languages and emojis.

For the intelligent agents, it has to have this encoding system to represent the different objects. So the requirements of the agents is to have a systematic way to produce new distinguishable symbols, ideally infinite many symbols, but infinite is an unclear here. We may say for the purpose of solving its problems in its lifetime, the agents have to produce enough identifiers for the purpose of solving.

Most of the time, when we solve, we only care its distinguishability to some degree.

Once we can distinguish, we can abstract all other information. We can use any symbols system to represent as long as they have enough number of distinguishable symbols. Let’s say we have a combination problem for three people, it doesn’t matter whether we have three people or we have A B C or we have three shopping malls, as long as they are three distinguishable objects. For this same problem, we may have three eggs, three new notebooks, somehow they can be distinguished, but to the agent or a human solver, it can be regarded as three indistinguishable objects.

So the ability to be distinguished or not is the most fundamental information we can get. If we strip this ability away, we will have no information.

That is to say, if we talk abstractly, the first thing we can talk or the first care we can add to an object is its distinguishability.

We may present the empty object, all we know about it is that it has the boundary to distinguishable from the environment all other information is unknown to us. Then we introduce more objects, all we may known for those objects is whether they are distinguishable or not. That’s to say for a given universe of objects, we have the relation or the ability to return true or false:

$$\forall x\in \Gamma. \forall y\in \Gamma.(\mathbf{Identical}(x,y) \vee \lnot \mathbf{Identical}(x,y))$$

As a matter of fact, all we need is that $\mathbf{Identical}$ has the range of all possible combination of x and y. We don’t care their contents, we only care based on its content, whether we can distinguish them or not.

If there are some information that can be only sensible by human agents, we are doomed.

Core ideas to discuss:

1. The necessary of having the ability to distinguish.
2. The necessary of having the ability to represent the distinguishable objects.
3. Distinguishability is the lowest level of abstract computation.
4. To remember, so represent.
5. And every computation can be represented by this distinguishability computation.

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What’s the fundamental principle about distinguishability?

Here we present the first principle of this book:

For any indistinguishable object(s), after any indistinguishable computational transformations, it can’t have distinguishable results.

$$\mathbf{Identical}(x, y) \rightarrow \lnot \exists \mathbf{T}.\mathbf{T}(x)\neq \mathbf{T}(y)$$

The above principle also gives us following property:

Two objects are distinguishable if there exists a computational transformation, under which yield different results for these two objects.

$$(\exists \mathbf{T}.\mathbf{T}(x)\neq \mathbf{T}(y) )\rightarrow \lnot \mathbf{Identical}(x, y)$$