Webbot

In view of the huge amount of email broadcasted indiscriminately to internet users, it is becoming popular to filter incoming email automatically. A webbot can be a suite of programs that includes such filters. A webbot is faulty (evil) if it filters messages it should let pass, or conversely if it lets pass messages it should filter. The entropy method can be used to quantify the propensity of a webbot for evil as follows.

The state of the system in which we are interested consists, as in the previous two examples, of a pair. This time the set in represents the set of messages input by the webbot and the set out represents the set of messages passed on to the user. We assume that no new messages are introduced by the webbot; in other words out is a subset of in. Letting $D$ denote the set of messages, the state space is

$$X \ =_{\textit{\tiny def}}\ \ \{ (\textit{in}, \textit{out})... ... \subseteq \textit{in} \land \char93 \textit{in} < \infty \}$$

where ${\mathbb{P}} E$ denotes the set of all subsets of set $E$.

To define the level function it is convenient to introduce predicate pass on $D$ defined so that $\textit{pass}(d)$ is true iff message $d$ should pass to the user. Then the set of messages handled incorrectly by the webbot is expressed

$$\textit{in} \, \triangle \, \textit{out} \ =_{\textit{\tiny ... ...tit{in} \mid \textit{pass}(d) \land d \notin \textit{out} \} .$$

Now the level function is simply the size of that set

$$\lambda (\textit{in}, \textit{out}) \ =_{\textit{\tiny def}}\ \ \char93 (\textit{in}\, \triangle\, \textit{out}) .$$

A finer ordering is defined without use of a level function, directly in terms of the state,

$$(\textit{in}_0, \textit{out}_0) \leq (\textit{in}_1, \textit... ...0) \subseteq (\textit{in}_1 \, \triangle \, \textit{out}_1) .$$

In either case we have an entropy structure with respect to which increase in entropy corresponds to evil action of the webbot. The correctly-functioning webbot leaves entropy invariant with $(\textit{in}\, \triangle\, \textit{out}) = \{\,\}$.

In a similar way we can model an eavesdropper able to monitor data passing along a communications medium. The system state is then modeled as the set eve of data eavesdropped and an action on the system is regarded as evil if it increases that set. So, as above, there are two choices of ordering. The first is defined by the level function

$$\lambda (\textit{eve}) \ =_{\textit{\tiny def}}\ \ \char93 \textit{eve}$$

whilst the second pre-order is defined directly in terms of state

$$\textit{eve}_0 \leq \textit{eve}_1 \ =_{\textit{\tiny def}}\ \ \textit{eve}_0 \subseteq \textit{eve}_1 .$$

According to the first (derived) pre-order, increase in entropy corresponds to a greater number of messages being eavesdropped. By the second, increase in entropy corresponds to an increase in the set of eavesdropped messages. But in both cases a 'benign' system leaves entropy invariant with $\textit{eve} = \{\,\}$.