Algebraic structures computable without delay

Foundations of Online Structure Theory

“…As was noted in [35], many known proofs from polynomial time structure theory (e.g., [9][10][11]30]) are focused on making the operations and relations on the structure primitive recursive, and then observing that the presentation that we obtain is in fact polynomial-time. Definition 1.1 (Essentially Dedekind [14]).…”

“…It is thus natural to systematically investigate into those structures that admit a presentation with primitive recursive operations, as defined below. Kalimullin, Melnikov, and Ng [35] proposed that an "online" structure must minimally satisfy: 35,51]). A countable structure is fully primitive recursive (fpr) if its domain is N and the operations and predicates of the structure are (uniformly) primitive recursive.…”

Automatic and Polynomial-Time Algebraic Structures

“…What about polynomial-time presentations? It is not hard to show that every (Turing) computable linear ordering admits a polynomial-time presentation [Gri90], and the same can be said about any torsion-free abelian group (essentially [KMN17b], after Downey) and many other structures; see the survey [CR91]. Positive results of this sort suggest that perhaps there could be a general necessary and sufficient condition on a structure to have an automatic or at least a polynomial-time presentation.…”

“…Interestingly enough, initially the authors were not concerned with automatic structures. They were looking at the index set of fully primitive recursive structures [KMN17b]. These are computable structures in which the domain is ω and all operations and relations are primitive recursive.…”

“…Although fully primitive recursive structures do not have to be computationally feasible, the class serves as an abstraction to study polynomial-time presentations in the following sense. Kalimullin, Melnikov and Ng [KMN17b] have observed that eliminating unbounded search from a Turing computable presentation of a structure is often the key step is showing that the structure has a polynomial-time presentation; see [KMN17b,KMN17a,Mel17] for various examples. On the other hand, to show that a structure does not have a polynomial-time presentation it is often easiest to prove it does not even have a fully primitive recursive presentation, as in [KMN17b,KMN17a,CR91,CR95].…”