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Ker-I Ko and the Study of Resource-Bounded Kolmogorov Complexity

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Complexity and Approximation

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12000))

Abstract

Ker-I Ko was among the first people to recognize the importance of resource-bounded Kolmogorov complexity as a tool for better understanding the structure of complexity classes. In this brief informal reminiscence, I review the milieu of the early 1980’s that caused an up-welling of interest in resource-bounded Kolmogorov complexity, and then I discuss some more recent work that sheds additional light on the questions related to Kolmogorov complexity that Ko grappled with in the 1980’s and 1990’s.

In particular, I include a detailed discussion of Ko’s work on the question of whether it is \({\mathsf{NP}}\)-hard to determine the time-bounded Kolmogorov complexity of a given string. This problem is closely connected with the Minimum Circuit Size Problem (\({\mathsf{MCSP}}\)), which is central to several contemporary investigations in computational complexity theory.

Supported in part by NSF Grants CCF-1514164 and CCF-1909216.

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Notes

  1. 1.

    If the reader is not familiar with Kolmogorov complexity, then we recommend some excellent books on this topic [25, 44].

  2. 2.

    Levin was Kolmogorov’s student, but he did not receive his Ph.D. until after he emigrated to the US, and Albert Meyer was his advisor at MIT. The circumstances around Levin being denied his Ph.D. in Moscow are described in the excellent article by Trakhtenbrot [59].

  3. 3.

    This result also appears as Exercise 13.20 in what was probably the most popular complexity theory textbook for the early 1980’s [33], which credits Levin for that result, but not for what is now called the Cook-Levin theorem.

  4. 4.

    In [1], in addition to Levin, Adleman also credits Meyer and McCreight [46] with developing similar ideas. I have been unable to detect any close similarity, although the final paragraph of [46] states “Our results are closely related to more general definitions of randomness proposed by Kolmogorov, Martin-Löf, and Chaitin” [and here the relevant literature is cited, before continuing] “A detailed discussion must be postponed because of space limitations” [and here Meyer and McCreight include a citation to a letter from the vice-president of Academic Press (which presumably communicated the space limitations to the authors).] Indeed, Meyer and McCreight were interested in when a decidable (and therefore very non-random) set can be said to “look random” and thereby deserve to be called pseudorandom. We will return to this topic later in the paper.

  5. 5.

    During the review and revision phase of preparing this paper, I was given a paper that settles this question! Ilango, Loff, and Oliveira have now shown that the “circuit” version of this problem (which they call \({\mathsf{Partial}\hbox {-}\mathsf{MCSP}}\)) is \({\mathsf{NP}}\)-complete [35]. For additional discussion of this result and how it contrasts with Ko’s work [38], see [4].

  6. 6.

    In particular, this is the problem that Trakhtenbrot calls “Task 5” in [59].

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    Eric Allender

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    Ding-Zhu Du

  2. University of Massachusetts Lowell, Lowell, MA, USA

    Jie Wang

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Allender, E. (2020). Ker-I Ko and the Study of Resource-Bounded Kolmogorov Complexity. In: Du, DZ., Wang, J. (eds) Complexity and Approximation. Lecture Notes in Computer Science(), vol 12000. Springer, Cham. https://doi.org/10.1007/978-3-030-41672-0_2

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