Let G be a graph on n vertices with adjacency matrix A, and let 1 be the all-ones vector. We call G controllable if the set of vectors 1, A1,..., An-11 spans the whole space Rn. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs.

Homomorphisms between relational structures play a central role in finite model theory, constraint satisfaction and database theory. A central theme in quantum computation is to show how quantum resources can be used to gain advantage in information processing tasks. In particular, non-local games have been used to exhibit quantum advantage in boolean constraint satisfaction, and to obtain quantum versions of graph invariants such as the chromatic number. We show how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad on the (classical) category of relational structures and homomorphisms. We show a general connection between these notions and state-independent quantum realizations of strong contextuality in the Abramsky-Brandenburger formulation of contextuality. We use these results to exhibit a wide range of examples of contextuality-powered quantum advantage, and to unify several apparently diverse strands of previous work.

No model can be considered effective if fundamentally it is more complicated than what it’s trying to represent. However, extreme simplification may potentially overlook important non-primary features, or even neglect the possibility to represent ambiguous or unclear observations. Therefore, achieving a balance between parsimonious and detailed models is of utmost importance for science and engineering. Over the past few decades, random Boolean networks (RBNs) (Kauffman, 1969) have become popular models for genetic regulatory networks. This popularity is associated with the fact that RBNs are very general models. No functionality or structure is particularly assumed when constructing them. However, the Boolean idealization has been constantly criticized based on the assumption that constraining the variables of the model to have only two possible values (0 and 1) entails a loss of dynamical information in the analysis of real gene expression data.