Spectra of Monadic Second-Order Formulas with One Unary Function

We establish the eventual periodicity of the spectrum of any monadic second-order formula where: (i) all relation symbols, except equality, are unary, and (ii) there is only one function symbol and that symbol is unary

A Uniform Method for Proving Lower Bounds of the Computational Complexity of Logical Theories

https://deepblue.lib.umich.edu/bitstream/2027.42/154178/1/39015100081655.pd

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

We investigate the relation between the theory of the itera- tions 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 cer- tain 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

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

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

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones

An Algebraic Approach to Mso-Definability on Countable Linear Orderings

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field