  What is computable with limited resources? How can we verify the correctness
of computations? How to measure computational power with precision? Despite the
immense scientific and engineering progress in computing, we still have only
partial answers to these questions. In order to make these problems more
precise, we describe an abstract algebraic definition of classical computation,
generalizing traditional models to semigroups. The mathematical abstraction
also allows the investigation of different computing paradigms (e.g. cellular
automata, reversible computing) in the same framework. Here we summarize the
main questions and recent results of the research of finite computation.
