## Section: New Results

### Foundations of Process Calculi

#### Group Distributed Knowledge.

We introduced spatial constraint systems (scs) as semantic structures for reasoning about spatial and epistemic information in concurrent systems. They have been used to reason about beliefs, lies, and group epistemic behaviour inspired by social networks. They have also been used for proving new results about modal logics and giving semantics to process calculi. In [19] we developed the theory of scs to reason about the distributed information of potentially infinite groups. We characterized the notion of distributed information of a group of agents as the infimum of the set of join-preserving functions that represent the spaces of the agents in the group. We provided an alternative characterization of this notion as the greatest family of join-preserving functions that satisfy certain basic properties. We showed compositionality results for these characterizations and conditions under which information that can be obtained by an infinite group can also be obtained by a finite group. Finally, we provided algorithms that compute the distributive group information of finite groups. Furthermore, in [14] we summarized all the main results we have obtained about scs.

#### Group Polarization.

Social networks can make their users become more radical and isolated in their own ideological circle causing dangerous splits in society in a phenomenon known as group polarization. In [22] we developed a preliminary model for social networks, and a measure of the level of polarization in these social networks, based on Esteban and Ray's classic measure of polarization for economic situations. Our model includes information about each agent's quantitative strength of belief in a proposition of interest and a representation of the strength of each agent's influence on every other agent. We considered how the model changes over time as agents interact and communicate, and included several different options for belief update, including rational belief update and update taking into account irrational responses such as confirmation bias and the backfire effect. Under various scenarios, we considered the evolution of polarization over time, and the implications of these results for real world social networks.

#### Lattice Theory.

Structures involving a lattice and join-endomorphisms on it are ubiquitous in computer science. In [28] we studied the cardinality of the set $J\left(L\right)$ of all join-endomorphisms of a given finite lattice $L$. We showed that the cardinality of $J\left(L\right)$ is sub-exponential, exponential and super-exponential in the size of the lattice for boolean algebras, linear-orders, and arbitrary lattices, respectively. We also studied the following problem: Given a lattice $L$ of size $n$ and a subset $S$ of $J\left(L\right)$ of size $m$, find the greatest lower bound in $J\left(L\right)$ of $S$. This join-endomorphism has meaningful interpretations in epistemic logic, distributed systems, and Aumann structures. We showed that this problem can be solved with worst-case time complexity in $O(n+mlogn)$ for powerset lattices, $O\left(m{n}^{2}\right)$ for lattices of sets, and $O(mn+{n}^{3})$ for arbitrary lattices. The complexity is expressed in terms of the basic binary lattice operations performed by the algorithm.

#### Festschrift Contribution.

In a Festschrift dedicated to Catuscia Palamidessi [26], we presented an article with original solutions to four challenging mathematical puzzles [23]. The first two are concerned with random processes. The first problem can be reduced to computing, for arbitrary large values of $n$, the expected number of iterations of a program that increases a variable at random between 1 and $n$ until exceeds $n$. The second problem can be reduced to determining the probability of reaching a given point after visiting all the others in a circular random walk. The other two problems involve finding optimal winning group strategies in guessing games.