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

Regular Cost Functions, Part I: Logic and Algebra over Words

The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values "inside" and "outside". This theory is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be -- as in the classical theory of regular languages -- equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. These equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.Comment: 47 page

The MSO+U theory of (N, <) is undecidable

We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the logic is undecidable on infinite words, i.e. the MSO+U theory of (N,<) is undecidable. This settles an open problem about the logic, and improves a previous undecidability result, which used infinite trees and additional axioms from set theory.Comment: 9 pages, with 2 figure

In the Maze of Data Languages

In data languages the positions of strings and trees carry a label from a finite alphabet and a data value from an infinite alphabet. Extensions of automata and logics over finite alphabets have been defined to recognize data languages, both in the string and tree cases. In this paper we describe and compare the complexity and expressiveness of such models to understand which ones are better candidates as regular models

Weak MSO+U with Path Quantifiers over Infinite Trees

This paper shows that over infinite trees, satisfiability is decidable for weak monadic second-order logic extended by the unbounding quantifier U and quantification over infinite paths. The proof is by reduction to emptiness for a certain automaton model, while emptiness for the automaton model is decided using profinite trees.Comment: version of an ICALP 2014 paper with appendice

Satisfiability of CTL* with constraints

We show that satisfiability for CTL* with equality-, order-, and modulo-constraints over Z is decidable. Previously, decidability was only known for certain fragments of CTL*, e.g., the existential and positive fragments and EF.Comment: To appear at Concur 201

Cost Automata, Safe Schemes, and Downward Closures

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes

Index problems for game automata

For a given regular language of infinite trees, one can ask about the minimal number of priorities needed to recognize this language with a non-deterministic, alternating, or weak alternating parity automaton. These questions are known as, respectively, the non-deterministic, alternating, and weak Rabin-Mostowski index problems. Whether they can be answered effectively is a long-standing open problem, solved so far only for languages recognizable by deterministic automata (the alternating variant trivializes). We investigate a wider class of regular languages, recognizable by so-called game automata, which can be seen as the closure of deterministic ones under complementation and composition. Game automata are known to recognize languages arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is, the alternating index problem does not trivialize any more. Our main contribution is that all three index problems are decidable for languages recognizable by game automata. Additionally, we show that it is decidable whether a given regular language can be recognized by a game automaton

A Unified Approach to Boundedness Properties in MSO

In the past years, extensions of monadic second-order logic (MSO) that can specify boundedness properties by the use of operators referring to the sizes of sets have been considered. In particular, the logics costMSO introduced by T. Colcombet and MSO+U by M. Bojanczyk were analyzed and connections to automaton models have been established to obtain decision procedures for these logics. In this work, we propose the logic quantitative counting MSO (qcMSO for short), which combines aspects from both costMSO and MSO+U. We show that both logics can be embedded into qcMSO in a natural way. Moreover, we provide a decidability proof for the theory of its weak variant (quantification only over finite sets) for the natural numbers with order and the infinite binary tree. These decidability results are obtained using a regular cost function extension of automatic structures called resource-automatic structures

Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

We consider a specific class of tree structures that can represent basic structures in linguistics and computer science such as XML documents, parse trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We present axiomatizations of the monadic second-order logic (MSO), monadic transitive closure logic (FO(TC1)) and monadic least fixed-point logic (FO(LFP1)) theories of this class of structures. These logics can express important properties such as reachability. Using model-theoretic techniques, we show by a uniform argument that these axiomatizations are complete, i.e., each formula that is valid on all finite trees is provable using our axioms. As a backdrop to our positive results, on arbitrary structures, the logics that we study are known to be non-recursively axiomatizable