Punctual Categoricity and Universality

Abstract

We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is ${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is $0''$. We also prove that, with a bit of work, the latter result can be pushed beyond $\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability.

Punctual Categoricity Relative to a Computable Oracle

“…The proof of Theorem 4 relies on the relatively complex "pressing strategy" introduced in [12] and then explained in much detail in [11,17]. The strategy is combinatorially quite involved, but if the reader is familiar with the strategy they should have no problem understanding the proof.…”

“…We explain only the idea. For details, see [11,12,17]. The main purpose of this strategy is to "press" the opponent's punctual structure to reveal the needed segment of his structure within a precomputed number of stages.…”

“…Here "feasible" can be interpreted in many different ways, e.g, automatic, polynomial-time, primitive recursive, or punctual (to be defined). The general intuition is that, when compared to computably categorical structures, feasibly categorical structures tend to be even more rare (but see [11,12] for artificial counterexamples). For instance, a structure is automatically categorical iff it is finite [13].…”