Enumeration Reducibility and Computable Structure Theory

In classical computability theory the main underlying structure is that of the natural numbers or equivalently a structure consisting of some constructive objects, such as words in a finite alphabet. In the 1960’s computability theorists saw it as a challenge to extend the notion of computable to arbitrary structure. The resulting subfield of computability theory is commonly referred to as computability on abstract structures. One approach towards this is the theory of computability in admissible sets of the hereditarily finite superstructure HF(A) over a structure A. The development of computability on ordinals was initiated by Kreisel and Sacks [43, 42], who investigated computability notions on the first incomputable ordinal, and then further developed by Kripke and Platek [44, 58] on arbitrary admissible ordinals and by Barwise [6], who considered admissible sets with urelements. The notion of Σ-definability onHF(A), introduced and studied by Ershov [16, 17] and his students Goncharov, Morozov, Puzarenko, Stukachev, Korovina, etc., is a model of nondeterministic computability on A. A survey of results on HF-computability and on abstract computability based on the notion of Σ-definability can be found in [18, 95]. Montague [53] took a model theoretic approach to generalized computability theory, considering computability as definability in higher order logics. The approach towards abstract computability that ultimately lead to the results discussed in this article starts with searching for ways in which one can identify abstract computability on a structure internally. Let A be an arbitrary abstract structure. There are many different internal ways to define a class of functions that can be considered as the analog of classical computable functions. Different models of computation on A give rise to different classes of computable functions: PC(A) denotes the functions that are prime computable in A, introduced by Moschovakis [54]. REDS(A) is the set of functions computable by means of recursively enumerable definitional schemes, introduced by Friedman and Shepherdson [21, 65]. Finally, we have the search computable functions, denoted by SC(A), and also introduced by Moschovakis [54]. Gordon [34] proved the equivalence of search computability with Montague’s approach and with computability in admissible sets. Prime computability has a deterministic (sequential) character. REDS is nondeterministic (parallel) and allows searches on the set of natural numbers. Search computability is also nondeterministic, however here one is allowed to perform a