# Extensional Information Articulation from the Universe

^{*}

## Abstract

**:**

## 1. Introduction

His approach is based on two main notions: an object and information about object. In his theory, properties of objects are represented (and denoted) by subsets of those objects that has this property, i.e., each property is a subset of all objects and each subset is a property of objects. It is the classical set-theoretical approach to the concept of property [1].

- present a model of the process of extensional information articulation as a generalized extensive measurement;
- provide a detailed example of a step of the extensional informational articulation process—a model of attribute creation.

## 2. Information and IGUS

## 3. Extensive Measurement

- Suppose that there is a collection I of things or events in the real world (in which all things/events are identifiable by the measurer), and let the elements (i.e., things or events) in the collection I be countable. (Countability and identifiability enable to make n-copy of an element, later.)
- A comparator m is introduced as a measuring instrument. It can compare an element in I with another one. That is, the comparator m shows that whether the compared elements are indistinguishable or distinguishable from the comparator’s viewpoint (that is, in the aspect of the attribute under consideration). Introduce an equivalence relation ∼ . Elements a and b in I have a relation a ∼ b if and only if the comparator m shows that they are indistinguishable, and under this equivalence relation ∼ , all the elements in I can be partitioned into equivalence classes X = {α, β, γ, . . .}. Note that the equivalence class consists of real physical tokens (e.g., apples). There can be a collection of comparators M for a measurer and the relation ∼ is relative to each comparator m ∈ M.
- Assume that one can define a physical relation ≿ that has a total order, and a physical operation for concatenation (or composition) that is associative and strictly increasing. “Physical” here means that the relation and the operation are presented as physical phenomena in the real world. From any pair of the equivalence classes α and β, choose any elements a and b, respectively. Equivalence classes α and β have a relation α ≿ β if and only if the comparator shows a distinctive state: “when a is on the right pan and b is on the left pan, the right pan sinks (or they are balanced)”, for example. Choosing the elements a, b and c respectively from the any equivalence classes α, β, and γ, the concatenation is defined as: the concatenation of equivalent classes α and β, α β, is a physical state in which a and b are on the same pan; and α β = γ if and only if c is on the other pan and they are balanced. By an abuse of notation, we also write α ∼ β if and only if α≿ β and α ≾ β holds.
- Choose an equivalence class α from X. For any positive integer n, define nα inductively as follows: 1α = α, (n + 1)α =(nα) α. This nα is called n-copy of α. A copy of α can be found operationally. As for the weight, for instance, put x
_{1}∈ α on a pan and find x_{2}∈ α which is balanced with x_{1}. Now what is being considered is “the attribute, weight”, hence x_{2}can be anything, say, a pile of sand, for example. In this way, infinitely many copies of the weight of x_{1}, x_{3},x_{4},..., can be made in theory. Then one can define 2α as a unity of x_{1}and x_{2}, 3α as a unity of x_{1}, x_{2}, and x_{3}, and so on. - The Archimedes’ condition is required between the elements in X. The Archimedes’ condition is: for any pair of equivalent classes α and β, there exists a positive number n such that nα≻ β. Here α ≻ β means α≿ β and not α ∼ β. Though this condition has issues logically (because it cannot be described in first order predicate logics) and practically (because it implies infinite operations), it has great values to construct the theory.
- Corresponding to the above physical situations, a mathematical model, i.e., a formal system is set up. On the mathematical set A, introduce a totally ordered algebraic relation ≿
_{m}that is derived from the corresponding relation in I, and a connective and strictly increasing operation ◦_{m}that is also derived from the corresponding operation in I. (Thus, they have operational meaning in the real physical world.) - Introduce a measure, φ: A → ℝ as follows: (I) Select an element u from X as a unit, that is, φ(u)=1. (II) Construct φ satisfying following two conditions: (II-i) φ(nx)= nφ(x), (II-ii) If and only if x ≿ y, φ(x) ≥φ(y) holds. The precision of φ can be arbitrarily increased by the countability and the Archimedes’ condition on the elements of I. Suppose that x is not equivalent to the unit u. If x ≻ u, then the Archimedes’ condition assures there exists a positive integer k such that (k +1)u ≿ x ≿ ku. By this and (II-i),(II-ii) above, φ((k +1)u)= k +1 >φ(x) ≥ k = φ(ku) holds. Thus, φ(x) is obtained within the error 1. Here an error is defined as the upper bound of the difference between the true value and the measured value. Then, taking 2-copy of the each side of the equation above, (2k + 2)u ≿2x ≿2ku. With the Archimedes’ condition, this means (2k + 2)u ≿2x ≿(2k + 1)u or (2k + 1)u ≻2x ≿2ku holds. Applying the function φ and simplifying the equations with the conditions (II-i),(II-ii) yields k +1 ≥φ(x) ≥ kx + or k + >φ(x) ≥ k. Thus φ(x) is obtained within the error . Thus, by considering the virtual n-copy on the mathematical model A, the function φ can be obtained with arbitrary precision.

## 4. Extensional Information Articulation as Generalized Measurement

#### 4.1. Review of the extensive measurement

- Introduction of a physical apparatus, i.e., a measuring instrument (e.g., equal arm beam for the weight), representing the physical intervention(s): The measuring instrument is introduced by the third person (the measurer) and observed by her/him. This naturally means that the measurer can identify and manipulate the instrument.
- Identification of target objects by a third person (the measurer): In a measurement, the target objects are identified by the measurer beforehand.
- Identification of required physical operations and their structures: Relations such as equivalence relation and operations such as concatenation are introduced physically, in other words, they are introduced with an operational semantics.
- Introduction of n-copy: Beyond actual physical tokens, multiple copies/alternatives for (potential) repetitive operations are modeled to be obtained physically.
- Establishment of maps between physical operations and mathematical operations and relations.
- Establishment of the measure, that is, the map from the mathematical structure representing the physical situation to the real number.
- The Archimedes’ condition: Universality of the measurement even for the potential measurements in the future is theoretically secured by the Archimedes’ condition (and the concatenation operation). The Archimedes’ condition theoretically assures the arbitrary precision of the measurement. Moreover, with the concatenate operation, it theoretically secures the universality of the measurement, in the sense that it enables to measure beyond a particular physical situations specified (by the actual physical tokens) as a measurement.

#### 4.2. Implicit Assumptions in Extensive Measurement

- Identities of objects in I are secured by the measurer. That is, the measurer is assumed to be able to identify the objects to be measured. In addition, the identities of objects are almost stable. That is, in most cases, the identity of each object under consideration is kept during the measurement (at least).
- The comparator m (a part of the universe) is identified by the measurer. That is, an attribute corresponding to a comparator can be identified by the measurer. Moreover, the comparator can be manipulated by the measurer. Manipulation here means, (1) putting the object(s) into the situation to be measured by the comparator, (2) capturing the information on the object(s) the comparator displays. That is, the physical interactions between the comparator and the measurer are distinguished (and utilized) by the measurer.
- The measurer has abilities to construct the suitable representations (sets for collection of objects, equivalence relations, real numbers for measured values, and so on).

#### 4.3. Requirements for the IGUS

- An IGUS is assumed to be able to directly indicate an FoU by a certain physical interaction i. Only during interaction, the IGUS can indicate a specific FoU just as it is. When the interaction is lost, the IGUS cannot identify the same FoU (if there is no further information). Using a term in linguistics, diachronically indistinguishable interactions are assumed to work for indicating FoUs. Such kind of interactions will be called as the indicative interactions in this article. The most distinguishable character of the indicative interactions may be their durability. Durability is essentially relative. This durability is related to the similarity detections by the IGUS (see the next item). The “source” of the indicative interaction should be the IGUS. In this sense, metaphorically, the indicative interaction can be regarded as an “active” interaction of an IGUS, and the physical implementation of such an interaction is often called as an actuator of the IGUS. An IGUS may have representations of each FoU paired with the indicative interaction i. Note that this representation is rather loose. If the IGUS activate the same interaction i, the exactness of the indication is governed by chance. That is, the IGUS cannot “specify” the FoU by using the indicative interaction alone.
- Another particular physical interaction p is assumed. An FoU a picked out by an indicative interaction i is compared with another FoU b picked out in the same way (during i is kept). The interaction p tells the IGUS that, from the viewpoint of the IGUS, the compared FoUs are whether indistinguishable or not. Thus this interaction p detects a similarity between a and b. Using a term in linguistics, synchronically indistinguishable interactions are assumed to work for detecting the similarities among FoUs. Such kind of interactions will be called as the comparing interactions in this article. Most of the case, this interaction can be regarded as, metaphorically speaking, a “passive” interaction of an IGUS, and the physical implementation of such an interaction is often called as a sensor of the IGUS.
- And memory is required to store the effects of interactions (physical difference made by another part of the universe), keep them, and utilize them later if necessary.

#### 4.4. A Model of Extensional Information Articulation Process

- As already mentioned, there is no a priori thing or object. An IGUS may indicate an FoU by an indicative interaction. An FoU is, as it were, proto-information. A collection of FoUs, I, may be made physically or within the IGUS (as representation of a collection of physical FoUs).
- Comparator (i.e., Sensor) for indifferences m ∈ M is introduced. It is a pair of the indicative interaction and the comparing interaction. The comparator can compare an FoU with another one. That is, the comparator m shows that the compared FoUs are whether indistinguishable or distinguishable from the comparator’s viewpoint. This distinction induces an equivalence relation ∼ . In addition, the FoUs are partitioned into equivalence classes, tentatively. (Recall that an FoU is not identified by the IGUS at this stage. It is only indicated during the indicative interaction, and at best, has a representation loosely related to the actual physical FoU.) There may be a collection of comparators M for an IGUS and the relation ∼ is relative to each comparator m 2 M.
- Assume that one can define a physical relation R between the (tentative) equivalence classesof FoUs. Suffix for the comparator isreplaced/generalized by the set of comparators because acombination of the comparators is commonly used for articulating one piece of information.
- Characterize the physical relation R by the physical operations/interactions on the equivalenceclasses of FoUs, if possible.
- Try to deduce a universal structure constructed physically from the equivalence class of FoUs and R . If this step is not successful, the universal structure is inquired in the formal side, i.e., in the mathematical model (step 7).
- Corresponding the above physical situations, relate the equivalence classes of the FoU to a set X in the mathematical model. The set X must have the structure reflecting the physical relation R (Thus, they have operational meaning in the real physical world. )
- Introduce the homomorphisms between X and another (known) mathematical structure K, which represents the articulated information. Deduce a universal structure from the K. The universal structure may be a model of the information that is captured (and to be utilized) by the IGUS.

## 5. Attributes as Limit

- The IGUS has collections of FoUs each of which forms an equivalence class. Each of the equivalence relations is induced by certain symmetry of the physical interaction (comparing interaction) of the IGUS;
- The inclusion relation between the collections of FoUs is considered as the relation R in the step 3;
- The IGUS has enough physical capabilities to meet the requirements for implementing the model described in the following.

#### 5.1. Universality

_{t}. Between the equivalence classes A, B in S

_{t}, there exists a natural inclusion relation ⊆: A ⊆ B if and only if for any a ∈ A, a ∈ B. Here, a is a representation of FoU in the IGUS. It is natural to simply think that a certain uniformity of FoUs, that is, an attribute is represented by each equivalence class (as Chechkin). However, it has some drawbacks as mentioned in the Section 1.

_{t}, which is denoted as D

_{t}, and the sequence of them. D

_{t}is the down-set of S

_{t}if, whenever X ∈ D

_{t}and Y ∈ S

_{t}and Y ⊆ X, we have Y ∈ D

_{t}. Recall that X, Y here are equivalence classes of FoUs.

_{t}can be considered to be virtually the same as E

_{t}, in the context of this paper.

_{0}= ∅,D

_{1},D

_{2},..., which incrementally increases according as the IGUS has more experiences, that is, more interactions with the universe. The universality is extracted from this virtually infinite sequence.

_{t}}

_{t∈T}and appropriately defined functions f

_{tu}: D

_{u}→ D

_{t}, one can construct a mathematical structure called a projective system (D

_{t}, (f

_{tu})

_{t}

_{≤}

_{u}). f

_{tu}can be defined as:

_{tu}: D

_{t}'→ D

_{u}is the inclusion map preserving the top and the bottom in D

_{t}, and the symbol ⋁denotes the least upper bound.

_{t},f

_{tu})

_{t,u∈T}, a set D and a family of maps (p

_{t}: D → D

_{t})

_{t}

_{∈}

_{T}are given, and for any t, u ∈ T satisfying t ≤u, if p

_{t}= f

_{tu}

_{◦}p

_{u}, then there exists a unique map ρ

_{D}: D → D

_{t}that satisfies p

_{t}= π

_{t}

_{◦ }ρ

_{D}for any t ∈ T where π

_{t}is the projection map for t-th component of D

_{t}. This is a realization of universality in a mathematical model.

_{t}, the limit remains the same. In this sense, the attributes modeled as this structural limit can be said to be universal, and this corresponds to the mathematical universality mentioned above.

_{t}, i.e., (D

_{s})

_{s}

_{≤}

_{t}

_{∈}

_{T}can be considered to be a finite approximation of the projective limit D

_{t}. By construction from the family of down-sets, the finite sequence up to D

_{t}can be easily retrieved from D

_{t}, hence D

_{t}can be identified with (D

_{s})

_{ s}

_{≤}

_{t}

_{∈}

_{T}as a finite approximation of D

_{t}.

_{t}is an approximate value of the ratio of the circumference of a circle to its diameter, π, that is, D

_{0}=3, D

_{1}=3.1, D

_{2}=3.14, D

_{3 }=3.141,.... Then the projective limit is the value π. The universal value π is represented extensionally as

#### 5.2. Specifiability

_{t}, that is, the projection of an s-attribute in D

_{t}. These approximations can clearly be handled extensionally. Moreover, there might be a physical implementation of the limit, which provides extensionality of the attributes directly.

#### 5.3. Galois Connection

## 6. Discussion and Conclusions

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Yoshitake, M.; Saruwatari, Y. Extensional Information Articulation from the Universe. *Information* **2012**, *3*, 644-660.
https://doi.org/10.3390/info3040644

**AMA Style**

Yoshitake M, Saruwatari Y. Extensional Information Articulation from the Universe. *Information*. 2012; 3(4):644-660.
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**Chicago/Turabian Style**

Yoshitake, Makoto, and Yasufumi Saruwatari. 2012. "Extensional Information Articulation from the Universe" *Information* 3, no. 4: 644-660.
https://doi.org/10.3390/info3040644