The effect of negative feedback loops on the dynamics of Boolean networks

Feedback loops in a dynamic network play an important role in determining the dynamics of that network. Through a computational study, in this paper we show that networks with fewer independent negative feedback loops tend to exhibit more regular behavior than those with more negative loops. To be precise, we study the relationship between the number of independent feedback loops and the number and length of the limit cycles in the phase space of dynamic Boolean networks. We show that, as the number of independent negative feedback loops increases, the number (length) of limit cycles tends to decrease (increase). These conclusions are consistent with the fact, for certain natural biological networks, that they on the one hand exhibit generally regular behavior and on the other hand show less negative feedback loops than randomized networks with the same numbers of nodes and connectivity

A representation theorem for integral rigs and its applications to residuated lattices

We prove that every integral rig in Sets is (functorially) the rig of global sections of a sheaf of really local integral rigs. We also show that this representation result may be lifted to residuated integral rigs and then restricted to varieties of these. In particular, as a corollary, we obtain a representation theorem for pre-linear residuated join-semilattices in terms of totally ordered fibers. The restriction of this result to the level of MV-algebras coincides with the Dubuc-Poveda representation theorem.Comment: Manuscript submitted for publicatio

Picturing counting reductions with the ZH-calculus

Counting the solutions to Boolean formulae defines the problem #SAT, which is complete for the complexity class #P. We use the ZH-calculus, a universal and complete graphical language for linear maps which naturally encodes counting problems in terms of diagrams, to give graphical reductions from #SAT to several related counting problems. Some of these graphical reductions, like to #2SAT, are substantially simpler than known reductions via the matrix permanent. Additionally, our approach allows us to consider the case of counting solutions modulo an integer on equal footing. Finally, since the ZH-calculus was originally introduced to reason about quantum computing, we show that the problem of evaluating ZH-diagrams in the fragment corresponding to the Clifford+T gateset, is in $FP^{\#P}$. Our results show that graphical calculi represent an intuitive and useful framework for reasoning about counting problems