Let's start with our General Knowledge Transfer equation.

$F^p(k_i) = r^j_i \Rightarrow G^q(r^j_i) = k_i^{\prime}$

So in other words, if we expose two different persons (p and q) to the same knowledge representation $r^j_i$, we cannot ensure that the knowledge item generated from that representation is the same for the two people.

Under our hypothesis we cannot say anything about the terms $k_i$, they are inside our minds and we cannot observe them yet. But the terms $r^j_i$ are physical and easily comparable... well, maybe note easily in some cases but they are something we can "see".

Equivalent Knowledge

So, based on the observation above we might be able to come up with a way to compare similar knowledge items from two different persons.

Definition. Equivalent Knowledge items
The knowledge items $k_i \in \mathbb{K}^p$ and $k_i^{\prime} \in \mathbb{K}^q$ are said to be equivalent and they are noted as $k_i \sim k_i^{\prime} \iff \exists G^p, G^q \mid G^q(r^j_i) = k_i, G^p(r^j_i) = k_i^{\prime}$

The definition basically says that, given a knowledge representation, for instance a write up on some topic, if two different people reads that write up, they may get different internal representations of what they had read, but they should had understand the same thing.

Well, all of us know that this is not really true. Depending on what a person already knows, on the quality of the knowledge representations (who haven't had a bad teacher ever) or its ambiguity, it might happen that the two persons had actually understood completely different things from the same knowledge representation.

That means we actually need a better definition.

Cross Equivalent Knowledge

The easiest way to verify if two different persons had understood the same thing is actually having an exam. We all know this from our student times (oh that good ol' days). Basically, we ask each person to represent back the knowledge they had acquired using their own words and each ones verifies that that new representation actually matches what was understood in first instance.

Definition. Cross Equivalent Knowledge items
Two equivalent knowledge items $k_i \in \mathbb{K}^p$ and $k_i^{\prime} \in \mathbb{K}^q$ such as $k_i \sim k_i^{\prime}$ are said to be cross equivalent and they are noted as $k_i \eqsim k_i^{\prime} \iff \exists r^k_i, r^l_i \mid F^p(k_i) = r^k_i, F^q(k_i^{\prime}) = r^l_i$ and $G^q(r^l_i) = k_i^{\prime}, G^p(r^k_i) = k_i$

This new definitions cross checks the acquired knowledge by each person exploiting the fact that each knowledge item can be represented in different ways.

Getting closer, but still not yet there.

The two Ladies Tale

Let's introduce Mrs Higgins, a nice British lady that lives in a lovely town in the Kent Count. She bakes delicious meat pies and enjoys playing Bridge while having a cup of tea. A bit far away, in a beautiful town in the Galicia region, a Spanish lady named Señora Carmen cook tasty empanadas and have a very nice time playing brisca with her friends after lunch, while they have a cup of coffee with jotas. Of course, both ladies know that the sun is a star. However let's see what happen when one of them wants to share this knowledge with the other.

So, Señora Carmen, with a serious face, tells Mrs Higgins:

- Sabes?. O Sol e unha estrela?

Mrs Higgins stares at Señora Carmen and says.

- Oh dear, I do not know what those lovely words mean. However I would like to share with you a brilliant concept I had recently learnt. The Sun is indeed a star.

Señora Carmen takes a sip from her coffee with jotas as Mrs Higgins enjoys the aromatic flavor of its British tea. After a while they go back home and do not see them again.

Well, our two ladies are actually expressing the same knowledge item, but the problem is that they do not share a common representation to actually interchange the information. Mrs Higgins can only speak English while Señora Carmen only speaks Galician. Basically the knowledge interchange is not possible because the $G^{Higgins}$ cannot interpret a $r^j_i$ produced by the $F^{Carmen}$ function.

Representation Transforms

What the Two Ladies Tale had teach us is that a person's knowledge representation is not universal. That is bad news for us, and we should try to solve this inconvenience. The obvious solution, at least for the two ladies, is to get a translator who transforms the knowledge representations they produce. With this idea in our minds let's introduce some new definitions.

Definition. Knowledge Representation Transform
A Knowledge Representation Transform noted as $T$ is a function $T:R \rightarrow R \mid T(r^j_i) = r^{j\prime}_i$ such as, $\exists p \in \mathbb{P} \mid G^p (r^j_i) = G^p(r^{j\prime}_i) = k_i$

Then a Knowledge Representation Transform is a function that transforms a knowledge item representation, into a different representation for that same knowledge item, ensuring that, at least, exists one person that can understand both representations and produce the original knowledge item out of the two different knowledge representations. Note that the term $k_i$ in the definition above has to actually be cross equivalent to the element $k_i$ source of the $r^j_i$ and also cross equivalent to the element $r^{j\prime}_i$ that another person will learn.

Now we can define the Identity Representation transformation

Definition. Identity Representation Transformation
The Identity Representation transformation is a Knowledge Representation transformation noted by $T_I$ such as $\forall r^j_i \mid T_I(r^j_i) = r^j_i$

Furthermore, we can now define the Inverse Representation Transformation.

Definition. Inverse Representation Transformation
Given a knowledge representation set $R=\{r^k_i\}$ and a knowledge representation transformation $T \mid T(R) = R^{\prime} = \{r^{j\prime}_i\}$ the inverse representation transformation $T^{-1}$ is defined as the knowledge transformation such as $T^{-1}(R^\prime) = R$.

From the two definitions above we can derive the following property:
Property

Given a knowledge representation transformation $T$ and its inverse $T^{-1}$ then $T \circ T^{-1} = T_I = T^{-1} \circ T$.

Proof:
$T \circ T^{-1}(R^\prime) = T(T^{-1}(R^\prime) = T(R) = R^\prime$
Analogously
$T^{-1} \circ T(R) = T^{-1}(T(R)) = T^{-1}(R^\prime) = R$

Based in this definition, and taking advantage of the function composition properties we can introduce the concept of Equivalent Representation Transformation.

Proposition. Equivalent Knowledge Representation Transform
Given a set of representation transformation $T_0,\cdots T_N$, such as $T_o \circ \cdots \circ T_n (r^j_i) = r^{j\prime}_i$, then the function $T_{eqv} = T_0 \circ \cdots \circ T_N$ such us $T_{eqv}(r^j_i) = r^{j\prime}_i$ is named the equivalent representation transformation for the set of transformation $T_0 \cdots T_N$

The proposition above tell us that, whenever is possible to transform a given representation into another we can always represent such a transformation as a single transformation which is actually the composition of all the required individual transformation.

In the proposition above, the operation $F \circ G$ is a function composition operation representing the operation: $F \circ G(x) = F(G(x))$.

Equivalent Representation Transformation Properties

From our proposition above we may derive a couple of properties for these equivalent representation transformation.

Proposition. Inverse Equivalent Knowledge Representation Transformation
Given a equivalent representation transformation $T_p = T_0 \circ \cdots \circ T_N$, such us $T_p(r^j_i) = s^j_i$, exists $T_p^{-1} = T_n \circ \cdots \circ T_0$ such us $T_p^{-1}(s^j_i) = r^j_i$ and it is called inverse equivalent representation transformation.

Demonstration

Definition. Identity transform
The identity representation transformation, noted as $T_I$, is defined as the transformation such as $\forall r^j_i, T_I(r^j_i) = r^j_i$

Property
Given a equivalent representation transformation $T_p$ and given its inverse transformation $T_p^{-1}$ the composition of this two functions is the identity transform $T_I = T_p \circ T_p^{-1} \iff T_p(T_p^{-1}(s^j_i)) = T_p(r^j_i) = s^j_i$.

Translation Functions

Even when this is not always the case, it makes sense to define a translation function, basically a symmetrical knowledge representation transformation. This function will play the role of human translators, that can translate between, at least, two languages in both directions. So, from this concept:

Definition. Translation Function
Function $I$ is a translation function for person $p \in \mathbb{P} \iff I(s^j_i) = r^j_i, I(r^j_i) = s^j_i \mid G^p(r^j_i) = G^p(s^j_i) = k_i$

According to the definition above, a translation function is able to transform forth and back between two different knowledge representations related to the same knowledge item. Comparing against our translating scenario, these different representations may be the expression of an idea in different languages.

The translation function as is defined has the following property:

Property.
Let $I$ be a translation function between the sets $R=\{r^j_i\}$ and $S = \{s^j_i\}$ such as $\exists G \mid G(r^j_i) = G^(s^j_i) = k_i$ then the inverse translation function of $I$ is $I^{-1} = I \mid I \circ I^{-1} = I \circ I = T_I$.

Proof:

Let $J$ be the inverse translation function verifying that $J(I(R)) = R \Rightarrow J(S) = R = I(S) \Rightarrow J = I$

Based on this definitions, given $n$ translation sets $R_n$, such as $I_n (R_n) = R_{n+1}$. Then $R_0$ is equivalent to $R_n$.

Proof:

$R_n = I_{n-1}(R_{n-1}) = I_{n-1}(I_{n-2}(R_{n-3}))=I_{n-1}(...I_0(R_o)) = I_{n-1} \circ I_{n-2} \ldots \circ I_0(R_0) = I_{eqv}(R_0) = R_n$

Revisiting the Knowledge Equivalence Concept

We had got some results with regards to how to transform knowledge representations. This findings will enable a more accurate definition for the equivalence of knowledge items. However this is still partial as it actually covering the knowledge transfer from one person to another. As we had learn from our two ladies tale, two persons may actually have the same knowledge and they may not be able to interchange that knowledge, invalidating our definition.

In a sense we can say that the knowledge about the sun that our two ladies share can be considered equivalent as far as exists a equivalent knowledge representation transformation that makes our knowledge equivalent definition work. In plain words, we need somebody that translates between English and Galician. If such a person/entity exists then we can conclude, by comparison, that the knowledge is equivalent. If the person/entity does not exits, then, the two knowledge items cannot be compared anyhow as there is no human way to interchange that knowledge.

Said that, our knowledge equivalence definition will become:

Two equivalent knowledge items $k_i \in \mathbb{K}^p$ and $k_i^{\prime} \in \mathbb{K}^q$ such as $k_i \sim k_i{\prime}$ are said to be cross equivalent and they are noted as $k_i \eqsim k_i^{\prime} \iff \exists r^k_i, r^l_i$ and a knowledge representation transformation T such us $ F^p(k_i) = r^k_i, F^q(k_i^{\prime}) = r^l_i$ and $T(r^k_i) = r^l_i, T^{-1} (r^l_i) = r^k_i$ and $G^q(r^l_i) = k_i^{\prime}, G^p(r^k_i) = k_i$

Conclusions

So far we had defined a criteria to compare knowledge items through their knowledge representation. We had also find out that we may need to transform those representations in order to get the knowledge transferred to the other end. And we had learn about the Two Ladies Tale :). To finish up with this second part of this series, let's think, for a second what happen if our $q$ person is actually a computer. For the time being, we had not yet fully succeed on making computers understand natural languages, but we had made some progresses on getting knowledge into computer by other means.

Ontology development, deep learning solutions or symbolic systems (like expert systems) are examples or computer based knowledge representations. To generate those internal computer representations we (well actually the knowledge engineers) are manually implementing these knowledge representation transformation functions, translating human based knowledge representation into computer friendly knowledge representations (like the one listed above). Then the computer can actually produce a primitive human readable output out of those internal representations and, once again, a knowledge representation transformation is in place to get those output into the human brain.

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