# Indistinguishability Operators Applied to Task Allocation Problems in Multi-Agent Systems
## 3 Preliminaries on Indistinguishability Operators
### An Exponential Response Function and Indistinguishability Operators  
In [13] (see also [11]), an exponential response function was introduced in order to model honey bee division of labor by means of response thresholds. In particular, the exponential response function of an agent taken under consideration was the following. \[P_{exp}(s,\theta)=1-\mathrm{e}^{-\frac{s}{\theta}}, \tag{7}\]  
where \(s\) denotes the intensity of the stimulus for an agent to carry out a task and \(\theta\) is the threshold. Note that, as in the case of response function (1), the probability of engaging task performance is small for \(s<<\theta\), and is close to 1 for \(s>>\theta\). Our final goal of this section is twofold. On one hand, we provide an example of an indistinguishability operator that exhibits a behavior similar to response function (7) and, thus, it could be used in task allocation problems. On the other hand, we are able to retrieve exactly, from the generated indistinguishability operator and through the technique stated in Theorem 2, the response function (7) when it depends on the distance between an agent and a task. Next consider the family of \(t\)-norms \((T_{AA}^{\lambda})_{\lambda\in[0,\infty]}\) introduced by [PERSON] and [PERSON]. Recall, according to [15], that such a family is given as follows:\[T^{\lambda}_{AA}(x,y)=\left\{ \begin{array}{ll}T_{D}(x,y),&\text{if $\lambda=0$;}\\ T_{M}(x,y),&\text{if $\lambda=\infty$;}\\ \mathrm{e}^{-\left((-\log x)^{\lambda}+(-logy)^{\lambda}\right)^{\frac{1}{ \lambda}}},&\text{elsewhere.}\end{array} \right.\]  
Following [15], for each \(\lambda\in]0,\infty[\) we have that \(T^{\lambda}_{AA}\) is continuous and Archimedean. Moreover, an additive generator of \(T^{\lambda}_{AA}\) is given by  
\[f_{T^{\lambda}_{AA}}(x)=(-\log(x))^{\lambda}\]  
for all \(x\in[0,1]\). A straightforward computation shows that the pseudo-inverse \(f^{(-1)}_{T^{\lambda}_{AA}}\) of \(f_{T^{\lambda}_{AA}}\) is given as follows. \[f^{(-1)}_{T^{\lambda}_{AA}}(y)=e^{-\left(y^{\frac{1}{\lambda}}\right)},\]  
for all \(y\in[0,\infty]\) and for each \(\lambda\in]0,\infty[\). In the light of the exposed facts, we are able to introduce the announced indistinguishability operator. To this end, assume that \(d\) is a distance on a nonempty set \(X\) and let \(T^{\lambda}_{AA}\) be an Aczel-Alsina \(t\)-norm for \(\lambda\in]0,\infty[\). Applying Theorem 3, we obtain the \(T^{\lambda}_{AA}\)-indistinguishability operator \(E_{T^{\lambda}_{AA}}\) given by  
\[E_{T^{\lambda}_{AA}}(x,y)=e^{-\left(d(x,y)^{\frac{1}{\lambda}}\right)}\]  
for all \(x,y\in X\). Notice that \(E_{T^{\lambda}_{AA}}\) separates points. Now, following similar arguments to those given in Subsection 4.2, we obtain, for each \(n\in\mathbb{N}\) and \(\theta\in]0,\infty[\), the following indistinguishability operator from the preceding one. \[E_{T^{\frac{1}{\lambda}}_{AA}}(x,y)=e^{-\frac{d(x,y)^{n}}{\theta^{n}}}. \tag{8}\]  
Of course, one can observe that the last operator presented involves the same elements of function (3). Indeed, this operator depends simultaneously on the distance \(d(x,y)\), on a threshold parameter \(\theta\), and it contains the non-linearity constant \(n\). Besides, the nature of this indistinguishability operator is an exponential function as a response function (7). Nevertheless, a slight difference between them must be stressed with the aim of interpreting the operator given in (8) as a response function. It is clear that in (8), \(s\) must be considered as the inverse of the distance with the aim that the indistinguishability operator in (8) can be interpreted as a response function. Indeed, \(s\) must be understood as the inverse of distance in order to preserve the essence of the impact of the stimulus \(s\) in the expression of a response function. Thus, we have that the operator given by (8) acts as response function since it satisfies the following: \(d(x,y)>>\theta\) implies probability response close to \(0\) and \(d(x,y)<<\theta\) returns a probability response close to \(1\).