What can we conclude about a machine $m$ from that fact that it solves a particular problem $P$?
The computational power of machines / automata / programming languages and similar computational models, is usually studied in these theoretical fields:


Computability (for what can be computed at all)
Complexity (for what can be computed efficiently)
Formal languages (several kinds of automata exist, with different expressiveness)


You will find many textbooks about these topics, as well as university courses (some video lectures could even be available on the 'net)