Kolmogorov complexity and set theoretical representations of integers

Abstract

We reconsider some classical natural semantics of integers (namely iterators of functions, cardinals of sets, index of equivalence relations) in the perspective of Kolmogorov complexity. To each such semantics one can attach a simple representation of integers that we suitably effectivize in order to develop an associated Kolmogorov theory. Such effectivizations are particular instances of a general notion of "self-enumerated system" that we introduce in this paper. Our main result asserts that, with such effectivizations, Kolmogorov theory allows to quantitatively distinguish the underlying semantics. We characterize the families obtained by such effectivizations and prove that the associated Kolmogorov complexities constitute a hierarchy which coincides with that of Kolmogorov complexities defined via jump oracles and/or infinite computations (cf. [6]). This contrasts with the well-known fact that usual Kolmogorov complexity does not depend (up to a constant) on the chosen arithmetic representation of integers, let it be in any base n ≥ 2 or in unary. Also, in a conceptual point of view, our result can be seen as a mean to measure the degree of abstraction of these diverse semantics.

Kolmogorov Complexity in Perspective Part II: Classification, Information Processing and Duality

“…an algorithm as a formal object (namely, an ASM). Gurevich's thesis extends Church-Turing's thesis 23 (at least for sequential algorithms): indeed, Gurevich thesis proves it. More precisely, Church-Turing thesis is about denotational semantics (the diverse computation models which have been imagined are pairwise equivalent: we say they are Turing-complete).…”

“…One of the most fundamental and unprecedented feature of Codd's relational model is the formalization of the notion of query . He founded this notion on a new calculus: relational algebra which is a kind of combinatory logic with operators acting on tables 25 joined together: classical set theoretic operations (union, intersection, cartesian product and projections) and also new operations: 23 Church-Turing thesis states that "Every process or computation which can be done with a machine in a purely mechanical way, i.e. all what is computable with a machine, can be done with a Turing machine " (1936).…”

“…a sort of classification of the less or more abstract nature of different definitions of integers, obtained from the different semantics we consider. We develop this in [23] 48 .…”

Kolmogorov Complexity in Perspective Part I: Information Theory and Randomness

“…This complexity is studied in [1], 2005, by Becher, Figueira, Nies & Picci, and also in our paper [17], 2006.…”

Is Randomness Native to Computer Science? Ten Years After

“…A: Surprise! Ferbus-Zanda & Grigorieff proved in [9] that Church semantics leads to the usual Kolmogorov complexity C. The index semantics leads to that with the first jump ∅ oracular Kolmogorov complexity C ∅ . As for Russell, it leads to something strictly in between.…”