A Note on the Use of Sum in the Logic of Proofs

The Logic of Proofs~LP, introduced by Artemov, encodes the same reasoning as the modal logic~S4 using proofs explicitly present in the language. In particular, Artemov showed that three operations on proofs (application~$\cdot$, positive introspection~!, and sum~+) are sufficient to mimic provability concealed in S4~modality. While the first two operations go back to G{\"o}del, the exact role of~+ remained somewhat unclear. In particular, it was not known whether the other two operations are sufficient by themselves. We provide a positive answer to this question under a very weak restriction on the axiomatization of LP.

High-density electrode array for imaging in vitro electrophysiological activity

The development of a high-density active microelectrode array for in vitro electrophysiology is reported. Based on the Active Pixel Sensor (APS) concept, the array integrates 4096 gold microelectrodes (electrode separation 20 microm) on a surface of 2.5 mmx2.5 mm as well as a high-speed random addressing logic allowing the sequential selection of the measuring pixels. Following the electrical characterization in a phosphate solution, the functional evaluation has been carried out by recording the spontaneous electrical activity of neonatal rat cardiomyocytes. Signals with amplitudes from 130 microVp-p to 300 microVp-p could be recorded from different pixels. The results demonstrate the suitability of the APS concept for developing a new generation of high-resolution extracellular recording devices for in vitro electrophysiology.