Sophistication (complexity theory)

In algorithmic information theory, sophistication is a measure of complexity related to algorithmic entropy.

When K is the Kolmogorov complexity and c is a constant, the sophistication of x can be defined as ^[1]

${\displaystyle \operatorname {Soph} _{c}(x):=\inf\{\operatorname {K} (S):x\in S\land \operatorname {K} (x\mid S)\geq \log _{2}(|S|)-c\land |S|\in \mathbb {N} _{+}\}.}$

The constant c is called significance. The S variable ranges over finite sets.

Intuitively, sophistication measures the complexity of a set of which the object is a "generic" member.

See also

• Logical depth

References

1. Mota, Francisco; Aaronson, Scott; Antunes, Luís; Souto, André (2013). "Sophistication as Randomness Deficiency" (PDF). Descriptional Complexity of Formal Systems. Lecture Notes in Computer Science. Vol. 8031. pp. 172–181. doi:10.1007/978-3-642-39310-5_17. ISBN   978-3-642-39309-9.

Further reading

• Koppel, Moshe (1995). Herken, Rolf (ed.). "Structure". The Universal Turing Machine (2nd Ed.). Springer-Verlag New York, Inc.: 403–419. ISBN   3-211-82637-8.
• Antunes, Luís; Fortnow, Lance (August 30, 2007). "Sophistication Revisited" (PDF). Theory of Computing Systems. 45: 150–161. doi:10.1007/s00224-007-9095-5. S2CID   2020289.
• Luís, Antunes; Bauwens, Bruno; Souto, André; Teixeira, Andreia (2013). "Sophistication vs Logical Depth". arXiv: 1304.8046 [cs.IT].

External links

• The First Law of Complexodynamics