Compatibility of Shelah and Stupp's and Muchnik's iteration with fragments of monadic second order logic

TL;DR

This paper explores how Shelah-Stupp and Muchnik iterations relate to monadic second order logic fragments, showing reductions for Shelah-Stupp but not for Muchnik, thus clarifying their logical compatibilities.

Contribution

It demonstrates that Shelah-Stupp iteration theories can be reduced to base structures for certain MSO fragments, while Muchnik's iteration cannot, revealing their differing logical properties.

Findings

Shelah-Stupp iteration theories reduce to base structure theories for certain MSO fragments.

Muchnik's iteration theory does not reduce to base structure theories.

Clarifies the logical compatibility of these iterations with MSO fragments.

Abstract

We investigate the relation between the theory of the iterations in the sense of Shelah-Stupp and of Muchnik, resp., and the theory of the base structure for several logics. These logics are obtained from the restriction of set quantification in monadic second order logic to certain subsets like, e.g., finite sets, chains, and finite unions of chains. We show that these theories of the Shelah-Stupp iteration can be reduced to corresponding theories of the base structure. This fails for Muchnik's iteration.