Global View on Reactivity: Switch Graphs and their Logics

Abstract

The notion of reactive graph generalises the one of graph by allowing the base accessibility relation to change when its edges are traversed. Can we represent these more general structures using points and arrows? We prove this can be done by introducing higher order arrows: the switches.The possibility of expressing the dependency of the future states of the accessibility relation on individual transitions by the use of higher-order relations, that is, coding meta-relational concepts by means of relations, strongly suggests the use of modal languages to reason directly about these structures. We introduce a hybrid modal logic for this purpose and prove its completeness.

Reactive Models for Biological Regulatory Networks

“…The first one consists of introducing weighted edges in the reactive network to capture either some form of uncertainty in the cell evolution or describe the consumption of contextual resources. We also intend to resort to the reactive modal logic [10] interpreted over reactive frames to formulate and verify properties of biological regulatory networks.…”

“…Syntactically, such moving structures are represented by switch graphs [10,12], which add higher-level edges to the usual graph structure. These higher-level edges connect basic edges, also called "0-level" edges, which are the ones which are eventually crossed, to other higher-level edges according to the following definition.…”

“…Paths are used to define reactive frames-a semantic model for switch graphs upon which suitable (reactive) modal logics are defined. We will not develop such logics in this paper, the interested reader is referred to [10,12] for an extensive account. Let us end this section, fixing some notation.…”

“…This paper introduces a new qualitative model into the picture-reactive Boolean networks which seems particularly interesting to help in the quest for steady states, a major issue in the area. The model builds on the notion of reactivity [11] which proved successful in both fundamental research in modal logic [10,12,14] and software engineering applications [1,3,9]. Outline.…”

A Note on Reactive Transitions and Reo Connectors

“…This adds to the quest of D. Gabbay and his collaborators for suitable (logic) tools to specify the dynamics of reactivity. We are currently extending our work to switch graphs [13], another sort of reactive frames with an interesting potential for describing coordination patterns, namely in the context of analogues to biological systems -a main topic in the first author's doctoral research [10,11].…”