We consider the language of ∆0-formulas with list terms interpreted over hereditarily finite list superstructures. We study the complexity of reasoning in extensions of the language of ∆0-formulas with non-standard list terms, which represent bounded list search, bounded iteration, and bounded recursion. We prove a number of results on the complexity of model checking and satisfiability for these formulas. In particular, we show that the set of ∆0-formulas with bounded recursive terms true in a given list superstructure HW (M) is non-elementary (it contains the class kExpTime, for all k 1). For ∆0-formulas with restrictions on the usage of iterative and recursive terms, we show lower complexity.

A numbering of a countable family S is a surjective map from the set of natural numbers ω onto S. A numbering ν is reducible to a numbering µ if there is an effective procedure which given a ν-index of an object from S, computes a µ-index for the same object. The reducibility between numberings gives rise to a class of upper semilattices, which are usually called Rogers semilattices. The paper studies Rogers semilattices for families S ⊂ P (ω) belonging to various levels of the analytical hierarchy. We prove that for any non-zero natural numbers m = n, any non-trivial Rogers semilattice of a Π 1 m -computable family cannot be isomorphic to a Rogers semilattice of a Π 1 n -computable family. One of the key ingredients of the proof is an application of the result by Downey and Knight on degree spectra of linear orders.

No abstract