Bisimilarity Congruences for Open Terms and Term Graphs via Tile Logic

The definition of SOS formats ensuring that bisimilarity on closed terms is a congruence has received much attention in the last two decades. For dealing with open terms, the congruence is usually lifted from closed terms by instantiating the free variables in all possible ways; the only alternatives considered in the literature are Larsen and Xinxin's context systems and Rensink's conditional transition systems. We propose an approach based on tile logic, where closed and open terms are managed uniformly, and study the `bisimilarity as congruence' property for several tile formats, accomplishing different concepts of open system.