Communication

We consider a subsystem of Cyberspace that models two computers in communication. Data are transmitted from sender to receiver through an intervening medium $M$ by communication action $C$. Action $C$ may result in corrupt data being received, in practice because medium $M$ is imperfect. Let us see how an entropy structure may be defined in order to quantify the amount of corruption achieved -- i.e. evil perpetrated -- by $C$.

Let $D$ denote the set of data under consideration and let $D^\ast$ denote the set of all finite sequences of elements of $D$. The state of the system consists of a pair $(s,r)$ of sequences of data, having the same length. Sequence $s$ represents the data transmitted so far and sequence $r$ represents the data received so far. Both sequences accumulate as time evolves and reflect the order in which data are handled. They have the same length because we choose to describe the system only after a datum has been received but before the next has been transmitted (since we here have no interest in observing the state of the system whilst data are in transit). Thus the set of system states is

$$X \ =_{\textit{\tiny def}}\ \ \{(s,r) : D^\ast \times D^\ast \mid \char93 s = \char93 r \}$$

where $\char93 t$ denotes the size (i.e. length) of finite sequence $t$.
The communication action $C$ is represented relationally as follows. If datum $d$ is communicated successfully then

$$(s,r)~C~(s . d,r . d)$$

where $t . d$ is the sequence obtained by placing item $d$ at the end of sequence $t$. However if datum $d$ is corrupted to $d'$ then

$$(s,r)~C~(s . d,r . d') .$$

The pre-order we choose on $X$ is defined using the level function $\lambda : X \fun {\mathbb{N}}$ which is itself defined to reflect the number of messages corrupted

$$\lambda (s,r) \ =_{\textit{\tiny def}}\ \ \char93 \{ j\ \mid\ r_j \neq s_j \}$$

where $r_j$ and $s_j$ denote the $j$th elements of sequences $r$ and $s$ respectively and $\char93 S$ denotes the size (i.e. cardinality) of finite set $S$.
With respect to the pre-order defined by equation (1) we have now defined an entropy structure $(X,\leq,\sim)$ in which, if action $C$ corrupts data, it increases entropy and so may be viewed as being evil. If $C$ corrupts no data it leaves entropy invariant and so is viewed as benign.

Communications protocols are designed in layers of abstraction. When corrupt data are detected at a certain layer the next-highest layer regards them as having been lost and hence must cope with their retransmission. So we now consider the medium $M$, and hence action $C$, to be capable of losing data (but not of corrupting it).

The state of the system is similar with the exception that the sequence $r$ of received data is a subsequence of $s$, the sequence of transmitted data:

$$X \ =_{\textit{\tiny def}}\ \ \{(s,r) : D^\ast \times D^\ast \mid r \lhd s \} ,$$

where $r \lhd s$ means that $r$ is a subsequence of $s$. The action of $C$ can result in either accurate transmission

$$(s,r)~C~(s . d,r . d)$$

or loss

$$(s,r)~C~(s . d,r) .$$

The level function is chosen to measure the number of lost items

$$\lambda (s,r) \ =_{\textit{\tiny def}}\ \ \char93 s - \char93 r .$$

With respect to the pre-order defined by equation (1) an action which loses more data increases entropy more and so is more evil. A medium which loses no data leaves entropy invariant.