FO = FO^3 for Linear Orders with Monotone Binary Relations

We show that over the class of linear orders with additional binary relations satisfying some monotonicity conditions, monadic first-order logic has the three-variable property. This generalizes (and gives a new proof of) several known results, including the fact that monadic first-order logic has the three-variable property over linear orders, as well as over (R,<,+1), and answers some open questions mentioned in a paper from Antonopoulos, Hunter, Raza and Worrell [FoSSaCS 2015]. Our proof is based on a translation of monadic first-order logic formulas into formulas of a star-free variant of Propositional Dynamic Logic, which are in turn easily expressible in monadic first-order logic with three variables

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

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

XML data exchange under expressive mappings

Data Exchange is the problem of transforming data in one format (the source schema) into data in another format (the target schema). Its core component is a schema mapping, which is a high level specification of how such transformation should be done. Relational data exchange has been extensively studied, but exchanging XML data have been paid much less attention. The goal of this thesis is to develop a theory of XML data exchange with expressive schema mappings, extending a previous work using restricted mappings. Our mapping language is based on tree patterns that can use horizontal navigation and data comparison in addition to downward navigation. First we look at static analysis problems concerning a single mapping. More specif- ically, we consider consistency problems with different flavours. One such problem, for instance, asks if any tree has a solution under the given mapping. Then we turn to analyse the complexity of mapping themselves, i.e., recognising pairs of trees such that the one is mapped to the other. For both problems, we provide classifications based on sets of features used in the mappings. Second we investigate the composition of XML schema mappings. Generally it is hard, or rather simply impossible, to achieve closure under composition in XML settings unlike in relational settings. Nevertheless we identify a class of XML schema mappings that is closed under composition. Lastly we consider the problem of query answering. It is important to exchange data so that we can feasibly answer queries while it often leads to intractability. We identify the dividing line between tractable and intractable cases: answering queries with extended features is always intractable while tractability of answering simple queries can be retained in extended mappings

Succinctness and Formula Size Games

Tämä väitöskirja tutkii erilaisten logiikoiden tiiviyttä kaavan pituuspelien avulla. Logiikan tiiviys viittaa ominaisuuksien ilmaisemiseen tarvittavien kaavojen kokoon. Kaavan pituuspelit ovat hyväksi todettu menetelmä tiiviystulosten todistamiseen. Väitöskirjan kontribuutio on kaksiosainen. Ensinnäkin väitöskirjassa määritellään kaavan pituuspeli useille logiikoille ja tarjotaan näin uusia menetelmiä tulevaan tutkimukseen. Toiseksi näitä pelejä ja muita menetelmiä käytetään tiiviystulosten todistamiseen tutkituille logiikoille. Tarkemmin sanottuna väitöskirjassa määritellään uudet parametrisoidut kaavan pituuspelit perusmodaalilogiikalle, modaaliselle μ-kalkyylille, tiimilauselogiikalle ja yleistetyille säännöllisille lausekkeille. Yleistettyjen säännöllisten lausekkeiden pelistä esitellään myös variantit, jotka vastaavat säännöllisiä lausekkeita ja uusia “RE over star-free” -lausekkeita, joissa tähtiä ei esiinny komplementtien sisällä. Pelejä käytetään useiden tiiviystulosten todistamiseen. Predikaattilogiikan näytetään olevan epäelementaarisesti tiiviimpi kuin perusmodaalilogiikka ja modaalinen μ-kalkyyli. Tiimilauselogiikassa tutkitaan systemaattisesti yleisten riippuvuuksia ilmaisevien atomien määrittelemisen tiiviyttä. Klassinen epäelementaarinen tiiviysero predikaattilogiikan ja säännöllisten lausekkeiden välillä osoitetaan uudelleen yksinkertaisemmalla tavalla ja saadaan tähtien lukumäärälle “RE over star-free” -lausekkeissa hierarkia ilmaisuvoiman suhteen. Monissa yllämainituista tuloksista hyödynnetään eksplisiittisiä kaavoja peliargumenttien lisäksi. Tällaisia kaavoja ja tyyppien laskemista hyödyntäen saadaan epäelementaarisia ala- ja ylärajoja yksittäisten sanojen määrittelemisen tiiviydelle predikaattilogiikassa ja monadisessa toisen kertaluvun logiikassa.This thesis studies the succinctness of various logics using formula size games. The succinctness of a logic refers to the size of formulas required to express properties. Formula size games are some of the most successful methods of proof for results on succinctness. The contribution of the thesis is twofold. Firstly, we define formula size games for several logics, providing methods for future research. Secondly, we use these games and other methods to prove results on the succinctness of the studied logics. More precisely, we develop new parameterized formula size games for basic modal logic, modal μ-calculus, propositional team logic and generalized regular expressions. For the generalized regular expression game we introduce variants that correspond to regular expressions and the newly defined RE over star-free expressions, where stars do not occur inside complements. We use the games to prove a number of succinctness results. We show that first-order logic is non-elementarily more succinct than both basic modal logic and modal μ-calculus. We conduct a systematic study of the succinctness of defining common atoms of dependency in propositional team logic. We reprove a classic non-elementary succinctness gap between first-order logic and regular expressions in a much simpler way and establish a hierarchy of expressive power for the number of stars in RE over star-free expressions. Many of the above results utilize explicit formulas in addition to game arguments. We use such formulas and a type counting argument to obtain non-elementary lower and upper bounds for the succinctness of defining single words in first-order logic and monadic second-order logic