Rajab Aghamov

Rajab Aghamov, Christel Baier, Joel Ouaknine et al.

Dynamic Bayesian networks (DBNs) are compact graphical representations used to model probabilistic systems where interdependent random variables and their distributions evolve over time. In this paper, we study the verification of the evolution of conditional-independence (CI) propositions against temporal logic specifications. To this end, we consider two specification formalisms over CI propositions: linear temporal logic (LTL), and non-deterministic Büchi automata (NBAs). This problem has two variants. Stochastic CI properties take the given concrete probability distributions into account, while structural CI properties are viewed purely in terms of the graphical structure of the DBN. We show that deciding if a stochastic CI proposition eventually holds is at least as hard as the Skolem problem for linear recurrence sequences, a long-standing open problem in number theory. On the other hand, we show that verifying the evolution of structural CI propositions against LTL and NBA specifications is in PSPACE, and is NP- and coNP-hard. We also identify natural restrictions on the graphical structure of DBNs that make the verification of structural CI properties tractable.

Rajab Aghamov, Andrey Kudinov, Maik Thanh Nguyen et al.

On the product of two neighborhood frames, three natural neighborhood functions can be defined: the horizontal one assigning to a point (x, y) the set of all supersets of the Cartesian product of U and y, where U is a neighborhood of x; the vertical analog; and the product neighborhood function assigning as neighborhoods all supersets of sets Cartesian products of U and V, for neighborhoods U of x and V of y. We define the tri-modal logics Tx+T and Dx+D of classes of full products equipped with all three neighborhood functions of neighborhood frames validating the logic T or D; thereby extending known product results for S4 and D4 to weaker systems. Two interaction principles arise: (sub) = []p -> [1]p & [2]p and (mix) = []p -> [1][2]p & [2][1]p, where the modality [] stands for the product neighborhood function and [1], [2] the horizontal and vertical ones. Namely, we show that Tx+T = T*T*T + (mix) and Dx+D = D*D*D + (mix), where * denotes fusion. Notably, (sub) and (mix) are equivalent over S4*S4*S4 and thus S4*S4*S4 + (mix) axiomatizes the logic of full products of topological spaces.