Rewriting logic extends to concurrent systems with state changes the body of theory developed within the algebraic semantics approach. It is both a foundational tool and the kernel language of several implementation efforts (Cafe, ELAN, Mande). Tile logic extends (unconditional) rewriting logic since it takes into account state changes with side effects and synchronization. It is especially useful for defining compositional models of computation of reactive systems, coordination languages, mobile calculi, and causal and located concurrent systems. In this paper, the two logics are defined and compared using a recently developed algebraic specification methodology, membership equational logic. Given a theory T, the rewriting logic of T is the free monoidal S-category, and the tile logic of T is the free monoidal double category, both generated by T. An extended version of monoidal 2-categories, called g VH-categories, is also defined, able to include in an appropriate sense the structure of monoidal double categories. We show that 2VH-categories correspond to an extended version of rewriting logic, which is able to embed tile logic, and which can be implemented in the basic version of rewriting logic using suitable internal strategies. These strategies can be significantly simpler when the theory is uniforra. A uniform theory is provided in the paper for CCS, and it is conjectured that uniform theories exist for most process algebras.