The literature on concurrency theory offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proofs of correctness have been developed independently, but have a common underlying structure. This study provides a general view of characteristic formulae that are expressed in terms of logics with a facility for the recursive definition of formulae. It is shown how several examples of characteristic-formula constructions from the literature can be recovered as instances of the proposed general framework, and how the framework can be used to yield novel constructions. The paper also offers general results pertaining to the definition of co-characteristic formulae and of characteristic formulae expressed in terms of infinitary modal logics.

Here we report enrichment from a marine-derived inoculum of a nonphotosynthetic electroactive biofilm that is capable of both consuming electricity (electrotrophy) and producing electricity (electrogenesis) from a single electrode. With the alternation of the electrode potential between −0.4 and 0.0 V SHE every 10 min, alternating anodic and cathodic currents increased in lock step (maximum current density of ±1.4 ± 0.4 A/m 2 in both modes, Coulombic efficiency of ∼98% per charge−discharge cycle), which is consistent with alternating between generation and consumption of energy storage compounds by the biofilm. Cyclic voltammetry exhibited a single sigmoid-shaped feature spanning anodic and cathodic limiting currents centered at −0.15 V SHE , a phenomenon not observed to date for an electroactive biofilm, and square wave voltammetry exhibited reversible peaks at −0.15 and −0.05 V SHE , suggesting the same redox cofactor(s) facilitates electron transport at the biofilm−electrode interface in both modes. Hydrogen and carbon monoxide, known energy and/or carbon sources for cellular metabolism, but no volatile fatty acids, were detected in reactors. Cells and cell clusters were spread across the electrode surface, as seen by confocal microscopy. These results suggest that a single microbial electrochemical biofilm can alternate between storing energy and generating power, furthering the potential applicability of bioelectrochemical systems.

No abstract

No abstract

No abstract