1. Introduction

2. What is knowledge to us?

3. A case study of knowledge across selected disciplines

3.4. What is knowledge to Information Theorists?

3.4.1. Intuition

Shannon’s (1948) theory of information is about the transmission of information in a channel from point to point [56]. Based on Shannon, the greater the surprise, the more entropy produced [57]. If entropy is a measure of information, redundancy in Shannon’s theory may reduce entropy when it helps to communicate across a noisy room, where noise is “negative information”. Assuming independence and identical (i.i.d) behavior between the letters in the English alphabet plus one blank space, entropy= $-{\sum }_1^{27}\frac{1}{27}log(1/27)=-log(1/27)=4.75$ (where $1/27\approx .037$). However, more realistically, based on the frequencies of the English letters used (e.g., considering the letters e and z, where the letter ‘e’ has a frequency of 0.12702, and the letter ‘z’ has 0.00074), the entropy is reduced; taking digrams (e.g., “en” as in “spoken”) and trigrams (e.g., “ent” as in “enlightenment”) into account, the average entropy reduces even further, possibly categorized as disembodied language (Shannon recognized this in part as the redundancy in languages). Moreover, in experiments by Shannon with humans, Shannon found the entropy to range as low as approximately 1 bit (e.g., [58]). For example, Sliwa [59] found that members of a team use a shorthand to communicate with each other, first calling to each other to gain an awareness that couples their brains, increasing the power of their collective, allowing a member to be aware for action before the signal to act is sent.

3.4.2. Details

For a discrete random variable X with N outcomes, with outcomes $x_i;i=1,...,N$, we define the surprise [60] (or self-information,) $s\left(x_i\right)$, of an outcome as

$$s\left(x_i\right)=-logp\left(x_i\right),$$

where logarithms are base two (to ensure that bits are the units of measurement) and $p\left(x_i\right):=P(X=x_i)$. This definition is quite intuitive because if an outcome always happens, that is $p\left(x\right)=1$, then $s\left(x\right)=0$, which means that there is no surprise when x occurs. On the other hand, if an outcome never occurs, that is $p\left(x\right)=0$, then we would have infinite surprise when it “did” occur. There are other nice characteristic of $s\left(x\right)$ that we will not go into that support its definition as given above. Shannon never explicitly called out $s\left(x_i\right)$ in his initial papers, but like most of information theory, Shannon laid the groundwork that those who followed were able to fill in the details to make Shannon’s work mathematically rigorous (e.g., [61,62]; also see the discussions and references in [63]).

The symbol, $s\left(X\right)$ is itself a random variable and its expectation is called the entropy of $H\left(X\right)$. That is,

$$H\left(X\right):=E\left(s\left(X\right)\right)-\sum _i^{N}p\left(x_i\right)logp\left(x_i\right).$$

This is a measure of the information of a random variable (actually the mean information).

For example, if you have a unfair coin which is biased towards heads, you are not getting much information when you flip it and a head appears. The case of maximum entropy comes from a fair coin. If u is the probability of a head, $1-u$ the probability of a tail, then the entropy of the coin flip is (logarithms base two):

$$H\left(U\right)=-ulogu-(1-u)log(1-u).$$

$H\left(U\right)$ is maximized when it is a fair coin, resulting in an entropy of 1 bit.

Figure 1. Probability on the x-axis and the entropy $H\left(U\right)$ on the vertical axis.

Figure 1. Probability on the x-axis and the entropy $H\left(U\right)$ on the vertical axis.

With this background, Shannon’s [56] theory of information went on to measure how much information can be sent through a noisy communication channel.

The random variable X is the input to a channel, the random variable Y is the output. For simplicity, in this article, we assume that the channel is memoryless and stationary (but since we are going through a channel, the units change to bits/symbol). Therefore, the channel is given by a matrix, $M=m_{i,j}$, such that the probability of Y is conditioned on X:

$$m_{i,j}=P(Y=y_j|X=x_i).$$

Given random variables V and W, we also have the conditional entropy (assume that probabilites are well behaved and that we do not divide by zero):

$$\begin{array}{cc}\hfill H\left(V\right|W)& =-\sum _{i}p\left(w_i\right)H\left(V\right|W=w_i)\hfill \\ \hfill & =-\sum _{i}p\left(w_i\right)\sum _{j}p\left(v_i\right|w_j)logp\left(v_i\right|w_j)\hfill \\ \hfill & =-\sum _{i,j}p(v_i,w_j)log\frac{p(v_i,w_j)}{p\left(w_j\right)}.\hfill \end{array}$$

Shannon next introduced the (symmetric) mutual information, $I(X,Y)$:

$$I(X,Y):=H\left(X\right)-H\left(X\right|Y)=H(Y)-H(Y\left|X\right),$$

with units in bits/symbol or the equivalent bits/channel-use.

Given a distribution for the input $X,$ and with with the channel matrix M, in terms of conditional probabilities, we calculate the distribution on the output Y to find $I(X,Y)$. Being allowed to vary the distribution on X leads us to the definition of channel capacity, C:

$$C=\underset{p\left(x_i\right)}{max}I(X,Y),$$

where the maximum exists, and is in units of bits/symbol.

The novelty of the above expression is that, from Shannon this is the upper bound for asymptotically error free communication [56, Theorem 11].

If symbols take different amounts of time to transit a channel, the formula for capacity is replaced by, $C_t$, with units in bits/time:

$$C_t=\underset{p\left(x_i\right)}{max}\frac{I(X,Y)}{E\left(T\right)},$$

where $E\left(T\right)$ is the mean time for a symbol to go through the channel ( see[56, App. 4] and [63,64,65]).

3.4.3. Information Theory is a natural theory

Did Shannon devise something that fit reality, or is it reality? From a simplistic channel analysis, we show that Shannon’s information theory represents a natural phenomenon.

We assume in the article that we have a discrete memoryless communication channel with a binary input and a binary output; next, the channel matrix, $M=\left(m_{i,j}\right):=P(Y=j|X=i)$, is assumed stationary, becoming:

$$M=\left(\begin{array}{cc}P(Y=0|X=0)& P(Y=1|X=0)\\ P(Y=0|X=1)& P(Y=1|X=1)\end{array}\right)=:\left(\begin{array}{cc}a& \overline{a}\\ c& \overline{c}\end{array}\right)$$

(2)

Please note that $(a,c)\in [0,1]\times [0,1]$. If we assume that $a\ne c$, the need of a useless channel (with zero capacity) disappears [61, p. 52], since the output has no way of probabilistically distinguishing what was the input symbol.

Figure 2. The noisy channel diagram.

Figure 2. The noisy channel diagram.

To make this situation simple, we call it a $(2,2)$ channel. The input to the $(2,2)$ channel is the random variable X, such that $P(X=0)=x,P(X=1)=1-x=:\overline{x}$ (i.e., not x). The output is denoted by the random variable Y. Then M and X completely determine Y since

$$\begin{array}{ccc}\hfill P(Y=0)& =& P(Y=0|X=0)·P(X=0)+P(Y=0|X=1)·P(X=1)\hfill \\ & =& ax+c\overline{x}=(a-c)x+c.\hfill \end{array}$$

We simply denote this channel as $(a,c)$. Note that we freely use [67] as a reference throughout, and $lnx$ is log base e (i.e., ${log}_e\left(x\right)$).6

The base e binary entropy function,

$$h_e:[0,1]\to {\mathbb{R}}^{+},$$

is given by

$$h_e\left(x\right):=-xln\left(x\right)-(1-x)ln(1-x)$$

and the base 2 binary entropy function is

$$h\left(x\right):=-xlog\left(x\right)-(1-x)log(1-x).$$

We define

$$\beta :=\frac{h_e\left(a\right)-h_e\left(c\right)}{a-c}.$$

It can be shown (see [66, Eq. 5], [61, 3.3], 67]) that the capacity (in bits per symbol) of this channel is

$$C(a,c)=\frac{(c-1)h\left(a\right)+(1-a)H\left(c\right)}{a-c}+log\left(1+e^{\beta }\right),$$

and with the input probability $P(X=0)=x$ that achieves a capacity of

$$\chi =\frac{1}{a-c}\left(\frac{1}{1+e^{\beta }}-c\right).$$

Theorem 1.

(see Rumsey [68]) For the $(2,2)$ channel above, we have that

$$\frac{1}{e}<\chi <1-\frac{1}{e}.$$

This result is quite remarkable. Shannon used the base 2 logarithm to obtain bits, but it seems that the basis for information is actually found with the natural logarithm, $lnx$. It is counter-intuitive [69] that, even with the probability of one symbol being sent, the probabilities of the input symbols are still constrained to be within $(1/e,1-\frac{1}{e})$.

Perhaps our binary digital concept of information using primarily base 2 is an approximation to the true information of the universe based on Euler’s number e?

4. Discussion

5. Conclusions

“Interdependence means that important behaviors will be highly correlated. However, the evidence for complementarity is scarce."

Abbreviations

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