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

A Model Theoretic Proof of Completeness of an Axiomatization of Monadic Second-Order Logic on Streams

International audienceWe discuss the completeness of an axiomatization of Monadic Second- Order Logic (MSO) on infinite words (or streams). By using model-theoretic tools, we give an alternative proof of D. Siefkes' result that a fragment with full comprehension and induction of second-order Peano's arithmetic is com- plete w.r.t. the validity of MSO-formulas on streams. We rely on Feferman- Vaught Theorems and the Ehrenfeucht-Fra ̈ıss ́e method for Henkin models of second-order arithmetic. Our main technical contribution is an infinitary Feferman-Vaught Fusion of such models. We show it using Ramseyan fac- torizations similar to those for standard infinite words. We also discuss a Ramsey's theorem for MSO-definable colorings, and show that in linearly ordered Henkin models, Ramsey's theorem for additive MSO-definable col- orings implies Ramsey's theorem for all MSO-definable colorings

Transitive closure logic, nested tree walking automata, and XPath

International audienceWe study FO(MTC), first-order logic with monadic transitive closure, a logical formalism in between FO and MSO on trees. We characterize the expressive power of FO(MTC) in terms of nested tree-walking automata. Using the latter, we show that FO(MTC) is strictly less expressive than MSO, solving an open problem. We also present a temporal logic on trees that is expressively complete for FO(MTC), in the form of an extension of the XML document navigation language XPath with two operators: the Kleene star for taking the transitive closure of path expressions, and a subtree relativisation operator, allowing one to restrict attention to a specific subtree while evaluating a subexpression. We show that the expressive power of this XPath dialect equals that of FO(MTC) for Boolean, unary and binary queries. We also investigate the complexity of the automata model as well as the XPath dialect. We show that query evaluation be done in polynomial time (combined complexity), but that emptiness (or, satisfiability) is 2ExpTime-complete

Complete axiomatizations of MSO, FO(TC1) and FO(LFP1) on finite trees

We propose axiomatizations of monadic second-order logic (MSO), monadic transitive closure logic (FO(TC1)) and monadic least fixpoint logic (FO(LFP1)) on finite node-labeled sibling-ordered trees. We show by a uniform argument, that our axiomatizations are complete, i.e., in each of our logics, every formula which is valid on the class of finite trees is provable using our axioms. We are interested in this class of structures because it allows to represent basic structures of computer science such as XML documents, linguistic parse trees and treebanks. The logics we consider are rich enough to express interesting properties such as reachability. On arbitrary structures, they are well known to be not recursively axiomatizable

Logical theories of trees

Summary Trees occur naturally in many mathematical settings as important partial orders yet no systematic study of their rst-order theories exists. We investigate some of the rst-order theories of trees. The two problems which motivate the thesis are (i) the rst-order de nability of sets within a given tree, and (ii) the rst-order de nability and axiomatisability of particular classes of trees. Of particular interest is the correspondence between the rst-order theory of a tree and the rst-order theory of the class of linear orders which comprise the paths in the tree. For every class C of linear orders we introduce eight classes of trees collectively called the C-classes of trees, the paths of which are related in various natural ways to the linear orders in C. We completely establish both the set-theoretical relationships between these eight classes of trees as well as the relationships between the rst-order theories of these eight classes of trees. We also investigate some of the properties of these rst-order theories. A special case is where the class C consists of a single ordinal with < !! since such ordinals are nitely axiomatisable. We obtain the rstorder theory of the class of trees where every path is isomorphic to the ordinal for any nite ordinal and also for the case where = !. The remaining cases are more di cult because of the existence of unde nable paths in the tree. For the cases where ! < < !! we introduce the notion of an almost -tree and show that every almost -tree can be elementarily extended in a natural way to a tree of which every path, de nable or unde nable, satis es the rst-order theory of . We also examine what this elementary extension of the almost -tree looks like for the case where = ! + 1. Throughout the thesis we also investigate various rst-order properties and theories of trees and establish some results in this regard