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Ivan Soskov: A Life in Computability

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The Incomputable

Part of the book series: Theory and Applications of Computability ((THEOAPPLCOM))

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Abstract

On May 5, 2013, the Bulgarian logic community lost one of its prominent members—Ivan Nikolaev Soskov. In this paper we shall give a glimpse of his scientific achievements.

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Notes

  1. 1.

    He wrote: “In the academic year 1978/1979, as a supervisor of Ivan Soskov’s master thesis, I had the chance to be a witness of his first research steps in the theory of computability. My role was rather easy thanks to Ivan’s great ingenuity and strong intuition. Several years later, in 1983, Ivan Soskov defended his remarkable PhD dissertation on computability in partial algebraic systems. Although I was indicated as his supervisor again, actually no help of mine was needed at all in the creation of this dissertation. Everything in it, including also the choice of its subject and the form of the presentation, was a deed of Ivan only.”

  2. 2.

    Or with the elements 0, (0, 0), ((0, 0), 0), …, as it is in [22].

  3. 3.

    There are situations when the identification in question could cause problems. This can happen when the authentic natural numbers or at least some of them belong to B. In such situations, it would be appropriate, for instance, to denote the elements 0, (0, 0), (0, (0, 0)), … and their set by \(\bar{0},\bar{ 1},\bar{ 2},\ldots\) and \(\overline{\omega }\), respectively.

  4. 4.

    The iteration defined in the way above is a particular instance of the iteration in so-called iterative combinatory spaces from [32, 33], where an algebraic generalization of a part of the theory of computability is given. The absolute prime computability is actually the instance of the considered generalized notion of computability in a specific kind of iterative combinatory space.

  5. 5.

    The following intuitive description of Friedman’s schemes (applicable also to Ershov’s determinants) is given by Shepherdson in [31]: “Friedman’s effective definitional schemes are essentially definitions by infinitely many cases, where the cases are given by atomic formulae and their negations and the definition has r.e. structure”. As indicated there on p. 458, a result of Gordon (announced in an abstract of his) shows that, over a total structure, the computability by means of such schemes is equivalent to absolute prime computability (a detailed presentation of the result in question can be found in [15]). Unfortunately, Shepherdson’s and Gordon’s papers remained unnoticed by the logicians in Sofia quite a long time (the first publication of Soskov containing a reference to Shepherdson’s paper is [42]; the first reference to Gordon’s result is also in [42], but only his abstract is indicated there). This great delay was due to insufficient international contacts of the Sofia logic group at that time (especially before the several summer schools in mathematical logic organized in Bulgaria from 1983 on). An additional reason for missing these papers is that, at the time in question, the interests of the supervisor of Ivan Soskov’s Ph.D. thesis were mainly in generalization of the theory of computability and in examples of this generalization different from the ones corresponding to the situation from [22].

  6. 6.

    Since computability by means of standard programs with counters is equivalent to prime computability in the case of unary primitive functions, the conclusion from the third of these examples can be made also from the example in [35, 37]. An improvement, however, is the fact that \(\mathfrak{A}_{3}\) is a total structure, whereas the example in [35, 37] makes use of a partial structure.

  7. 7.

    For instance, M can be the Moschovakis’ extension B ∗ from [22] of a set B, and then we can take a = 0, b = (0, 0), J(s, t) = (s, t), D = id{a, b}, L = π, R = δ.

  8. 8.

    In the paper [65] by Alexandra Soskova, and Ivan Soskov a characterization is given of the structures which have effective enumerations.

  9. 9.

    It is known that a replacement of “hyperarithmetical” with “recursively enumerable” in this statement makes it refutable by counter-examples (enumeration reducibility and recursive enumerability being the corresponding replacements of relative hyperarithmeticity and hyperarithmeticity, respectively, in the above definitions).

  10. 10.

    In the original definition, \(J_{e}(A) = K_{A} \oplus \overline{K_{A}}\), where K A  = { x∣x ∈ W x (A)}. This however is enumeration equivalent to the definition we use here.

  11. 11.

    As usual for arbitrary x , x (n) denotes the result of the nth iteration of the jump operation on x .

  12. 12.

    This result was first proven by Bianchini [1].

  13. 13.

    A structure is called trivial if there are finitely many elements, such that every permutation of the domain leaving these elements fixed is an automorphism. For example any complete graph is a trivial structure.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria

    Hristo Ganchev & Dimiter Skordev

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  1. Hristo Ganchev
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  2. Dimiter Skordev
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Correspondence to Hristo Ganchev .

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Editors and Affiliations

  1. School of Mathematics, University of Leeds, Leeds, United Kingdom

    S. Barry Cooper

  2. Dept. of Mathematical Logic & Applications, Sofia University, Sofia, Bulgaria

    Mariya I. Soskova

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Ganchev, H., Skordev, D. (2017). Ivan Soskov: A Life in Computability. In: Cooper, S., Soskova, M. (eds) The Incomputable. Theory and Applications of Computability. Springer, Cham. https://doi.org/10.1007/978-3-319-43669-2_2

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