Algorithmic metatheorems state that if a problem can be described in a certain logic
and the inputs are structured
in a certain way, then the problem can be solved with a certain amount of resources.
As an example, by Courcelle’s Theorem, all monadic second-order (“in a certain logic”) properties of graphs of bounded tree width (“structured in a certain way”) can be solved in linear time (“with a certain amount of resources”). Such theorems have become valuable tools in algorithmics: if a problem happens to have the right structure and can be described in the right logic, they immediately yield a (typically tight) upper bound on the time complexity of the problem. Perhaps even more importantly, several complex algorithms rely on algorithmic metatheorems internally to solve subproblems, which considerably broadens the range of applications of these theorems. This paper is intended as a gentle introduction to the ideas behind algorithmic metatheorems, especially behind some recent results concerning space and circuit classes, and tries to give a flavor of the range of their applications.
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