  Complexity and decidability of logics is a major research area involving a
huge range of different logical systems. This calls for a unified and
systematic approach for the field. We introduce a research program based on an
algebraic approach to complexity classifications of fragments of first-order
logic (FO) and beyond. Our base system GRA, or general relation algebra, is
equiexpressive with FO. It resembles cylindric algebra but employs a finite
signature with only seven different operators. We provide a comprehensive
classification of the decidability and complexity of the systems obtained by
limiting the allowed sets of operators. We also give algebraic
characterizations of the best known decidable fragments of FO. Furthermore, to
move beyond FO, we introduce the notion of a generalized operator and briefly
study related systems.
