Foundations of Online Structure Theory

Abstract

The survey contains a detailed discussion of methods and results in the new emerging area of online “punctual” structure theory. We also state several open problems.

Punctual Categoricity and Universality

“…This solves a problem left open in [9]; see also [2]. The proof of the theorem is not merely a generalization of the proof from [9] since it relies on a new strategy; it is also combinatorially more intricate.…”

“…Although a primitive recursive algorithm does not have to be computationally feasible, it serves as a useful abstraction which unites most common complexity classes of interest. In fact, as discussed in [2,22], very often eliminating unbounded search is the crucial step in turning a general Turing computable algebraic procedure into, say, a polynomial time or a polylogspace one; see, for example, [5,6,7,16]. A nontrivial illustration of this phenomenon is the recent solution [3] to a problem of Khouissainov and Nerode on the characterization of automatic structures ( [24], Question 4.9).…”

“…Therefore, the above-mentioned totality phenomenon is still within the reach of this framework; see the recent paper [21] of Kalimullin, Melnikov, and Montalbán for more on totality, relativization, and definability in primitive recursive algebra. We cite the recent surveys [2,11,27] for a detailed exposition of the framework and its connections with computable structure theory, Weihrauch reducibility, infinite games on structures, incremental reducibility in computer science, complexity of real functions, and feasible combinatorics.…”

A Note on Computable Embeddings for Ordinals and Their Reverses

“…We note that except the reductions ≤ c and ≤ tc , there are many other approaches to comparing computability-theoretic complexity of classes of structures. These approaches include: transferring degree spectra and other algorithmic properties [13], Σ-reducibility [8,17], computable functors [11,16], Borel functors [12], primitive recursive functors [1,7], etc.…”

“…By monotonicity, (3) implies that Γ (N ) |= b n < d n . Now, again by monotonicity, (1) and 2 Let n = 2k for some k ≥ 1. Then Γ n works so that, for any input A, it outputs k disjoint copies of Γ 2 (A).…”