Structures Computable in Polynomial Time. I

Foundations of Online Structure Theory II: The Operator Approach

“…The main objects in this theory are infinite algebraic structures in which operations and relations are primitive recursive. As we argued in [8], there are natural strong connections of this new theory and the theory of polynomial-time algebraic structures (see also Alaev [1] and Alaev and Selivanov [3]) with applications to automatic structures [9]. Earlier we argued that this kind of punctual structure theory is akin to Turing-Markov computable analysis, in the objects are given effectively.…”

“…Kalimullin, Melnikov and Montalbán (in progress) have recently announced a number of unexpected results connecting relativised primitive recursive presentations with syntax in the spirit of Ash and Knight [5]. Also, as we see in the present paper, an 2 Although the definition above is not restricted to finite languages, we will never consider infinite languages in the paper. 3 Recall that a complete, first-order theory in a computable languadge is decidable if the collection of all Gödel codes of its sentences forms a computable set.…”

“…Polynomial time computable structures. Cenzer and Remmel, Grigorieff, Alaev, and others [25,47,1,2] studied polynomial time presentable structures. We omit the formal definitions, but we note that they are sensitive to how exactly we code the domain.…”

“…Generalisations to polynomial time classes seem to require significant effort in some instances. Alaev [1,2] has recently initiated a research program focused on extending these ideas to polynomial time algebra. Dealing with polynomial time algorithms requires specific techniques and counting combinatorics; this is something we do not have to worry in our more "relaxed" model.…”

“…Also it would seem that although we can incorporate automatic processes in our theories, they are really not general enough for online algorithms in general. Similarly, polynomial time structures such as Cenzer and Remmel, Grigorieff, Alaev, and others [1,2,3,25,47] are rather presentation dependent. Finally, primitive recursive has a nice Church-Turing thesis, in that it models computable processes without unbounded loops.…”