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

Covering of ordinals

The paper focuses on the structure of fundamental sequences of ordinals smaller than $\epsilon_0$. A first result is the construction of a monadic second-order formula identifying a given structure, whereas such a formula cannot exist for ordinals themselves. The structures are precisely classified in the pushdown hierarchy. Ordinals are also located in the hierarchy, and a direct presentation is given.Comment: Accepted at FSTTCS'0

The Church Synthesis Problem with Parameters

For a two-variable formula &psi;(X,Y) of Monadic Logic of Order (MLO) the Church Synthesis Problem concerns the existence and construction of an operator Y=F(X) such that &psi;(X,F(X)) is universally valid over Nat. B\"{u}chi and Landweber proved that the Church synthesis problem is decidable; moreover, they showed that if there is an operator F that solves the Church Synthesis Problem, then it can also be solved by an operator defined by a finite state automaton or equivalently by an MLO formula. We investigate a parameterized version of the Church synthesis problem. In this version &psi; might contain as a parameter a unary predicate P. We show that the Church synthesis problem for P is computable if and only if the monadic theory of is decidable. We prove that the B\"{u}chi-Landweber theorem can be extended only to ultimately periodic parameters. However, the MLO-definability part of the B\"{u}chi-Landweber theorem holds for the parameterized version of the Church synthesis problem

Synthesis of Finite-state and Definable Winning Strategies

Church\u27s Problem asks for the construction of a procedure which, given a logical specification $varphi$ on sequence pairs, realizes for any input sequence $I$ an output sequence $O$ such that $(I,O)$ satisfies $varphi$. McNaughton reduced Church\u27s Problem to a problem about two-player$omega$-games. B"uchi and Landweber gave a solution for Monadic Second-Order Logic of Order ($MLO$) specifications in terms of finite-state strategies. We consider two natural generalizations of the Church problem to countable ordinals: the first deals with finite-state strategies; the second deals with $MLO$-definable strategies. We investigate games of arbitrary countable length and prove the computability of these generalizations of Church\u27s problem

Satisfiability of ECTL* with tree constraints

Recently, we have shown that satisfiability for $\mathsf{ECTL}^*$ with constraints over $\mathbb{Z}$ is decidable using a new technique. This approach reduces the satisfiability problem of $\mathsf{ECTL}^*$ with constraints over some structure A (or class of structures) to the problem whether A has a certain model theoretic property that we called EHD (for "existence of homomorphisms is decidable"). Here we apply this approach to concrete domains that are tree-like and obtain several results. We show that satisfiability of $\mathsf{ECTL}^*$ with constraints is decidable over (i) semi-linear orders (i.e., tree-like structures where branches form arbitrary linear orders), (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching trees of height h for each fixed $h\in \mathbb{N}$. We prove that all these classes of structures have the property EHD. In contrast, we introduce Ehrenfeucht-Fraisse-games for $\mathsf{WMSO}+\mathsf{B}$ (weak $\mathsf{MSO}$ with the bounding quantifier) and use them to show that the infinite (order) tree does not have property EHD. As a consequence, a different approach has to be taken in order to settle the question whether satisfiability of $\mathsf{ECTL}^*$ (or even $\mathsf{LTL}$) with constraints over the infinite (order) tree is decidable

Graph Games on Ordinals

We consider an extension of Church\'s synthesis problem to ordinals by adding limit transitions to graph games. We consider game arenas where these limit transitions are defined using the sets of cofinal states. In a previous paper, we have shown that such games of ordinal length are determined and that the winner problem is PSPACE-complete, for a subclass of arenas where the length of plays is always smaller than \\omega^\\omega. However, \nthe proof uses a rather involved reduction to classical Muller games, and the resulting strategies need infinite memory. \n \nWe adapt the LAR reduction to prove the determinacy in the general case, and to generate strategies with finite memory, using a reduction to games where the limit transitions are defined by priorities. We provide an algorithm for computing the winning regions of both players in these games, with a complexity similar to parity games. Its analysis yields three results: determinacy without hypothesis on the length of the plays, existence of \nmemoryless strategies, and membership of the winner problem in NP and co-NP

Interpretations in Trees with Countably Many Branches

Abstract—We study the expressive power of logical interpreta-tions on the class of scattered trees, namely those with countably many infinite branches. Scattered trees can be thought of as the tree analogue of scattered linear orders. Every scattered tree has an ordinal rank that reflects the structure of its infinite branches. We prove, roughly, that trees and orders of large rank cannot be interpreted in scattered trees of small rank. We consider a quite general notion of interpretation: each element of the interpreted structure is represented by a set of tuples of subsets of the interpreting tree. Our trees are countable, not necessarily finitely branching, and may have finitely many unary predicates as labellings. We also show how to replace injective set-interpretations in (not necessarily scattered) trees by ‘finitary’ set-interpretations. Index Terms—Composition method, finite-set interpretations, infinite scattered trees, monadic second order logic. I

Multioperator Weighted Monadic Datalog

In this thesis we will introduce multioperator weighted monadic datalog (mwmd), a formal model for specifying tree series, tree transformations, and tree languages. This model combines aspects of multioperator weighted tree automata (wmta), weighted monadic datalog (wmd), and monadic datalog tree transducers (mdtt). In order to develop a rich theory we will define multiple versions of semantics for mwmd and compare their expressiveness. We will study normal forms and decidability results of mwmd and show (by employing particular semantic domains) that the theory of mwmd subsumes the theory of both wmd and mdtt. We conclude this thesis by showing that mwmd even contain wmta as a syntactic subclass and present results concerning this subclass

Logic in the Tractatus

I present a reconstruction of the logical system of the Tractatus, which differs from classical logic in two ways. It includes an account of Wittgenstein’s “form-series” device, which suffices to express some effectively generated countably infinite disjunctions. And its attendant notion of structure is relativized to the fixed underlying universe of what is named. There follow three results. First, the class of concepts definable in the system is closed under finitary induction. Second, if the universe of objects is countably infinite, then the property of being a tautology is \Pi^1_1-complete. But third, it is only granted the assumption of countability that the class of tautologies is \Sigma_1-definable in set theory. Wittgenstein famously urges that logical relationships must show themselves in the structure of signs. He also urges that the size of the universe cannot be prejudged. The results of this paper indicate that there is no single way in which logical relationships could be held to make themselves manifest in signs, which does not prejudge the number of objects