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The conventional perspective on Markov chains considers decision problems concerning the probabilities of temporal properties being satisfied by traces of visited states. However, consider the following query made of a stochastic system modelling the weather: given the conditions today, will there be a day with less than 50\% chance of rain? The conventional perspective is ill-equipped to decide such problems regarding the evolution of the initial distribution. The alternate perspective we consider views Markov chains as distribution transformers: the focus is on the sequence of distributions on states at each step, where the evolution is driven by the underlying stochastic transition matrix. More precisely, given an initial distribution vector $\mu$, a stochastic update transition matrix $M$, we ask whether the ensuing sequence of distributions $(\mu, M\mu, M^2\mu, \dots)$ satisfies a given temporal property. This is a special case of the model-checking problem for linear dynamical systems, which is not known to be decidable in full generality. The goal of this article is to delineate the classes of instances for which this problem can be solved, under the assumption that the dynamics is governed by stochastic matrices.

In discrete-time linear dynamical systems (LDSs), a linear map is repeatedly applied to an initial vector yielding a sequence of vectors called the orbit of the system. A weight function assigning weights to the points in the orbit can be used to model quantitative aspects, such as resource consumption, of a system modelled by an LDS. This paper addresses the problems to compute the mean payoff, the total accumulated weight, and the discounted accumulated weight of the orbit under continuous weight functions and polynomial weight functions as a special case. Besides general LDSs, the special cases of stochastic LDSs and of LDSs with bounded orbits are considered. Furthermore, the problem of deciding whether an energy constraint is satisfied by the weighted orbit, i.e., whether the accumulated weight never drops below a given bound, is analysed.

The aim of the paper is to study the topological modal logic of $T_0$ spaces, with the difference modality (for $T_n$, where $n\geq1 $ the corresponding logics were known). We consider propositional modal logic with two modal operators $\square$ and $[\ne]$. $\square$ is interpreted as an interior operator and $[\ne]$ corresponds to the inequality relation. We introduce the logic $S4DT_0$ and show that $S4DT_0$ is the logic of all $T_0$ spaces and has the finite model property.