Infinite and Bi-infinite Words with Decidable Monadic Theories

We study word structures of the form $(D,<,P)$ where $D$ is either $\mathbb{N}$ or $\mathbb{Z}$, $<$ is the natural linear ordering on $D$ and $P\subseteq D$ is a predicate on $D$. In particular we show: (a) The set of recursive $\omega$-words with decidable monadic second order theories is $\Sigma_3$-complete. (b) Known characterisations of the $\omega$-words with decidable monadic second order theories are transfered to the corresponding question for bi-infinite words. (c) We show that such "tame" predicates $P$ exist in every Turing degree. (d) We determine, for $P\subseteq\mathbb{Z}$, the number of predicates $Q\subseteq\mathbb{Z}$ such that $(\mathbb{Z},\le,P)$ and $(\mathbb{Z},\le,Q)$ are indistinguishable. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words

Infinite and bi-infinite words with decidable monadic theories

We study word structures of the form (D,<,P) where D is either the naturals or the integers with the natural linear order < and P is a predicate on D. In particular we show: The set of recursive infinite words with decidable monadic second order theories is Sigma_3-complete. We characterise those sets P of integers that yield bi-infinite words with decidable monadic second order theories. We show that such "tame" predicates P exist in every Turing degree. We determine, for a set of integers P, the number of indistinguishable biinfinite words. Through these results we demonstrate similarities and differences between logical properties of infinite and bi-infinite words

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

On Decidability Properties of Local Sentences

Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences. We prove here several additional results on local sentences. The first one is a new decidability result in the case of local sentences whose function symbols are at most unary: one can decide, for every regular cardinal k whether a local sentence phi has a model of order type k. Secondly we show that this result can not be extended to the general case. Assuming the consistency of an inaccessible cardinal we prove that the set of local sentences having a model of order type omega_2 is not determined by the axiomatic system ZFC + GCH, where GCH is the generalized continuum hypothesi

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

Model Checking Synchronized Products of Infinite Transition Systems

Formal verification using the model checking paradigm has to deal with two aspects: The system models are structured, often as products of components, and the specification logic has to be expressive enough to allow the formalization of reachability properties. The present paper is a study on what can be achieved for infinite transition systems under these premises. As models we consider products of infinite transition systems with different synchronization constraints. We introduce finitely synchronized transition systems, i.e. product systems which contain only finitely many (parameterized) synchronized transitions, and show that the decidability of FO(R), first-order logic extended by reachability predicates, of the product system can be reduced to the decidability of FO(R) of the components. This result is optimal in the following sense: (1) If we allow semifinite synchronization, i.e. just in one component infinitely many transitions are synchronized, the FO(R)-theory of the product system is in general undecidable. (2) We cannot extend the expressive power of the logic under consideration. Already a weak extension of first-order logic with transitive closure, where we restrict the transitive closure operators to arity one and nesting depth two, is undecidable for an asynchronous (and hence finitely synchronized) product, namely for the infinite grid.Comment: 18 page

Computabilities of Validity and Satisfiability in Probability Logics over Finite and Countable Models

The $\epsilon$-logic (which is called $\epsilon$E-logic in this paper) of Kuyper and Terwijn is a variant of first order logic with the same syntax, in which the models are equipped with probability measures and in which the $\forall x$ quantifier is interpreted as "there exists a set $A$ of measure $\ge 1 - \epsilon$ such that for each $x \in A$, ...." Previously, Kuyper and Terwijn proved that the general satisfiability and validity problems for this logic are, i) for rational $\epsilon \in (0, 1)$, respectively $\Sigma^1_1$-complete and $\Pi^1_1$-hard, and ii) for $\epsilon = 0$, respectively decidable and $\Sigma^0_1$-complete. The adjective "general" here means "uniformly over all languages." We extend these results in the scenario of finite models. In particular, we show that the problems of satisfiability by and validity over finite models in $\epsilon$E-logic are, i) for rational $\epsilon \in (0, 1)$, respectively $\Sigma^0_1$- and $\Pi^0_1$-complete, and ii) for $\epsilon = 0$, respectively decidable and $\Pi^0_1$-complete. Although partial results toward the countable case are also achieved, the computability of $\epsilon$E-logic over countable models still remains largely unsolved. In addition, most of the results, of this paper and of Kuyper and Terwijn, do not apply to individual languages with a finite number of unary predicates. Reducing this requirement continues to be a major point of research. On the positive side, we derive the decidability of the corresponding problems for monadic relational languages --- equality- and function-free languages with finitely many unary and zero other predicates. This result holds for all three of the unrestricted, the countable, and the finite model cases. Applications in computational learning theory, weighted graphs, and neural networks are discussed in the context of these decidability and undecidability results.Comment: 47 pages, 4 tables. Comments welcome. Fixed errors found by Rutger Kuype

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