Digital computers carry out algorithms coded in high level programs. These abstract entities determine what happens at the physical level: they control whether electrons flow through specific transistors at specific times or not, entailing downward causation in both the logical and implementation hierarchies. This paper explores how this is possible in the light of the alleged causal completeness of physics at the bottom level, and highlights the mechanism that enables strong emergence to occur. Although synchronic emergence of higher levels from (...) lower levels is manifestly true, diachronic emergence is generically not the case; indeed we give specific examples where it cannot occur because of the causal effectiveness of higher level variables. (shrink)