The Church Problem for Countable Ordinals

A fundamental theorem of Buchi and Landweber shows that the Church synthesis problem is computable. Buchi and Landweber reduced the Church Problem to problems about &#969;-games and used the determinacy of such games as one of the main tools to show its computability. We consider a natural generalization of the Church problem to countable ordinals and investigate games of arbitrary countable length. We prove that determinacy and decidability parts of the Bu}chi and Landweber theorem hold for all countable ordinals and that its full extension holds for all ordinals < \omega\^\omega

Inverse monoids of higher-dimensional strings

International audienceHalfway between graph transformation theory and inverse semigroup theory, we define higher dimensional strings as bi-deterministic graphs with distinguished sets of input roots and output roots. We show that these generalized strings can be equipped with an associative product so that the resulting algebraic structure is an inverse semigroup. Its natural order is shown to capture existence of root preserving graph mor-phism. A simple set of generators is characterized. As a subsemigroup example, we show how all finite grids are finitely generated. Last, simple additional restrictions on products lead to the definition of subclasses with decidable Monadic Second Order (MSO) language theory

FO-definable transformations of infinite strings

The theory of regular and aperiodic transformations of finite strings has recently received a lot of interest. These classes can be equivalently defined using logic (Monadic second-order logic and first-order logic), two-way machines (regular two-way and aperiodic two-way transducers), and one-way register machines (regular streaming string and aperiodic streaming string transducers). These classes are known to be closed under operations such as sequential composition and regular (star-free) choice; and problems such as functional equivalence and type checking, are decidable for these classes. On the other hand, for infinite strings these results are only known for $\omega$-regular transformations: Alur, Filiot, and Trivedi studied transformations of infinite strings and introduced an extension of streaming string transducers over $\omega$-strings and showed that they capture monadic second-order definable transformations for infinite strings. In this paper we extend their work to recover connection for infinite strings among first-order logic definable transformations, aperiodic two-way transducers, and aperiodic streaming string transducers

Ehrenfeucht Games, the Composition Method, and the Monadic Theory of Ordinal Words

. When Ehrenfeucht introduced his game theoretic characterization of elementary equivalence in 1961, the first application of these &quot;Ehrenfeucht games&quot; was to show that certain ordinals (considered as orderings) are indistinguishable in first-order logic and weak monadic second-order logic. Here we review Shelah&apos;s extension of the method, the &quot;composition of monadic theories&quot;, explain it in the example of the monadic theory of the ordinal ordering (!; !), and compare it with the automata theoretic approach due to Buchi. We also consider the expansion of ordinals by recursive unary predicates (which gives &quot;recursive ordinal words&quot;). It is shown that the monadic theory of a recursive ! n - word belongs to the 2n-th level of the arithmetical hierarchy, and that in general this bound cannot be improved. 1 Introduction One of the most successful tools of mathematical logic, in particular of those parts of logic which are relevant to computer science, is the method of &quot;Ehrenfeucht games&quot;,..