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Courcelle's theorem and its adaptations to cliquewidth have shaped the fieldof exact parameterized algorithms and are widely considered the archetype ofalgorithmic meta-theorems. In the past decade, there has been growing interestin developing parameterized approximation algorithms for problems which are notcaptured by Courcelle's theorem and, in particular, are considered notfixed-parameter tractable under the associated widths.  We develop a generalization of Courcelle's theorem that yields efficientapproximation schemes for any problem that can be captured by an expanded logicwe call Blocked CMSO, capable of making logical statements about the sizes ofset variables via so-called weight comparisons. The logic controls weightcomparisons via the quantifier-alternation depth of the involved variables,allowing full comparisons for zero-alternation variables and limitedcomparisons for one-alternation variables. We show that the developed frameworkthreads the very needle of tractability: on one hand it can describe a broadrange of approximable problems, while on the other hand we show that therestrictions of our logic cannot be relaxed under well-established complexityassumptions.  The running time of our approximation scheme is polynomial in$1/\varepsilon$, allowing us to fully interpolate between faster approximatealgorithms and slower exact algorithms. This provides a unified framework toexplain the tractability landscape of graph problems parameterized by treewidthand cliquewidth, as well as classical non-graph problems such as Subset Sum andKnapsack.

Approximate Evaluation of Quantitative Second Order Queries