Finite Satisfiability for Guarded Fixpoint Logic

The finite satisfiability problem for guarded fixpoint logic is decidable and complete for 2ExpTime (resp. ExpTime for formulas of bounded width)

Querying the Guarded Fragment

Evaluating a Boolean conjunctive query Q against a guarded first-order theory F is equivalent to checking whether "F and not Q" is unsatisfiable. This problem is relevant to the areas of database theory and description logic. Since Q may not be guarded, well known results about the decidability, complexity, and finite-model property of the guarded fragment do not obviously carry over to conjunctive query answering over guarded theories, and had been left open in general. By investigating finite guarded bisimilar covers of hypergraphs and relational structures, and by substantially generalising Rosati's finite chase, we prove for guarded theories F and (unions of) conjunctive queries Q that (i) Q is true in each model of F iff Q is true in each finite model of F and (ii) determining whether F implies Q is 2EXPTIME-complete. We further show the following results: (iii) the existence of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof of the finite model property of the clique-guarded fragment; (v) the small model property of the guarded fragment with optimal bounds; (vi) a polynomial-time solution to the canonisation problem modulo guarded bisimulation, which yields (vii) a capturing result for guarded bisimulation invariant PTIME.Comment: This is an improved and extended version of the paper of the same title presented at LICS 201

Cardinality and counting quantifiers on omega-automatic structures

We investigate structures that can be represented by omega-automata, so called omega-automatic structures, and prove that relations defined over such structures in first-order logic expanded by the first-order quantifiers `there exist at most $\aleph_0$ many', 'there exist finitely many' and 'there exist $k$ modulo $m$ many' are omega-regular. The proof identifies certain algebraic properties of omega-semigroups. As a consequence an omega-regular equivalence relation of countable index has an omega-regular set of representatives. This implies Blumensath's conjecture that a countable structure with an $\omega$-automatic presentation can be represented using automata on finite words. This also complements a very recent result of Hj\"orth, Khoussainov, Montalban and Nies showing that there is an omega-automatic structure which has no injective presentation

Collapsible Pushdown Automata and Recursion Schemes

International audienceWe consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed {deterministic term} rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this paper we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it and contracts its silent transitions, which leads to an infinite tree which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving an effective transformations in both directions

Automatic presentations of infinite structures

The work at hand studies the possibilities and limitations of the use of finite automata in the description of infinite structures. An automatic presentation of a countable structure consists of a labelling of the elements of the structure by finite words over a finite alphabet in a consistent way so as to allow each of the relations of the structure to be recognised, given the labelling, by a synchronous multi-tape automaton. The collection of automata involved constitutes a finite presentation of the structure up to isomorphism. More generally, one can consider presentations over finite trees or over infinite words or trees, based on the appropriate model of automata. In the latter models, uncountable structures are also representable. Automatic presentations allow for effective evaluation of first-order formulas over the represented structure in line with the strong correspondence between automata and logics. Accordingly, automatic presentations can be recast in logical terms using various notions of interpretations. The simplicity and robustness of the model coupled with the diversity of automatic structures makes automatic presentations interesting subject of investigation within the scope of algorithmic model theory. Although automata have been in use in representations of infinite structures in computational group theory, in the analysis of numeration systems and finitely generated infinite sequences as well as in the theory of term rewriting systems, a systematic investigation of automatic structures using model theoretic methods has only just begun in the last twelve years. There are two main lines of research in this field. One would like to have a classification of automatic models of certain first-order theories of common interest, such as linear orderings, trees, boolean algebras, groups, etc. Though efforts aimed at obtaining structure theorems have produced considerable advancement in recent years, this programme is still in an early stage. Even further lacking is our understanding of the richness of automatic presentations of key individual structures. A prominent result in this area is the deep theorem of Cobham and Semenov concerning numeration systems. In this style, one would like to know the degree of freedom in choosing automatic presentations of a particular structure. In this thesis we present contributions to both of these problem areas. We also study restricted notions of presentations and clarify the relationship of automatic presentations over finite and infinite words. The peculiarities of using automata to represent structures up to isomorphism introduce problems out of the range of classical automata theory. We present some new techniques developed to tackle these difficulties

Automatic presentations of infinite structures

The work at hand studies the possibilities and limitations of the use of finite automata in the description of infinite structures. An automatic presentation of a countable structure consists of a labelling of the elements of the structure by finite words over a finite alphabet in a consistent way so as to allow each of the relations of the structure to be recognised, given the labelling, by a synchronous multi-tape automaton. The collection of automata involved constitutes a finite presentation of the structure up to isomorphism. More generally, one can consider presentations over finite trees or over infinite words or trees, based on the appropriate model of automata. In the latter models, uncountable structures are also representable. Automatic presentations allow for effective evaluation of first-order formulas over the represented structure in line with the strong correspondence between automata and logics. Accordingly, automatic presentations can be recast in logical terms using various notions of interpretations. The simplicity and robustness of the model coupled with the diversity of automatic structures makes automatic presentations interesting subject of investigation within the scope of algorithmic model theory. Although automata have been in use in representations of infinite structures in computational group theory, in the analysis of numeration systems and finitely generated infinite sequences as well as in the theory of term rewriting systems, a systematic investigation of automatic structures using model theoretic methods has only just begun in the last twelve years. There are two main lines of research in this field. One would like to have a classification of automatic models of certain first-order theories of common interest, such as linear orderings, trees, boolean algebras, groups, etc. Though efforts aimed at obtaining structure theorems have produced considerable advancement in recent years, this programme is still in an early stage. Even further lacking is our understanding of the richness of automatic presentations of key individual structures. A prominent result in this area is the deep theorem of Cobham and Semenov concerning numeration systems. In this style, one would like to know the degree of freedom in choosing automatic presentations of a particular structure. In this thesis we present contributions to both of these problem areas. We also study restricted notions of presentations and clarify the relationship of automatic presentations over finite and infinite words. The peculiarities of using automata to represent structures up to isomorphism introduce problems out of the range of classical automata theory. We present some new techniques developed to tackle these difficulties

A Hierarchy of Automatic Words having a Decidable MSO Theory

We investigate automatic presentations of infinite words. Starting points of our study are the works of Rigo and Maes, and Carton and Thomas concerning the lexicographic presentation, respectively the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is canonical in a certain sense. We then go on to generalize our techniques to a hierarchy of classes of infinite words enjoying the above mentioned properties. We introduce k-lexicographic presentations, and morphisms of level k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under restricted MSO interpretations, e.g. closure under deterministic sequential mappings. The classes of k-lexicographic words are shown to form an infinite hierarchy.

A Hierarchy of Automatic omegaomega-Words having a Decidable MSO Theory

We investigate automatic presentations of ω-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSOinterpretability in algebraic trees, and the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is in a certain sense canonical. We then generalize our techniques to a hierarchy of classes of ω-words enjoying the above mentioned definability and decidability properties. We introduce k-lexicographic presentations, and morphisms of level k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mappings. The classes of k-lexicographic words are shown to constitute an infinite hierarchy