A General Notion of Useful Information

In this paper we introduce a general framework for defining the depth of a sequence with respect to a class of observers. We show that our general framework captures all depth notions introduced in complexity theory so far. We review most such notions, show how they are particular cases of our general depth framework, and review some classical results about the different depth notions.

💡 Research Summary

The paper “A General Notion of Useful Information” proposes a unified, abstract framework for defining the depth of an infinite binary sequence relative to a class of observers. The authors observe that all previously studied depth notions—Bennett’s logical depth, recursive depth, polynomial‑time depth based on distinguishers, predictors, monotone compressors, and finite‑state compressors—share a common pattern: a sequence is considered deep if, for any algorithm from a given class G, there exists a strictly more powerful algorithm (often from a larger class G′) that extracts more “useful information” from the sequence.

To capture this pattern formally, the paper introduces two algorithmic classes G and G′, a performance measure Perf that maps an algorithm and a finite prefix of the sequence to a real number in