Complete First-Order Axiomatization of Finite or Infinite M-extended Trees

We present in this paper an axiomatization of the structure of finite or infinite $M$-extended trees. Having a structure $M=(D_M,F_M,R_M)$, we define the structure of finite or infinite $M$-extended trees $Ext_M=(D,F,R)$ whose domain $D$ consists of trees labelled by elements of $D_M\cup F$, where $F$ is a set of function symbols containing $F_M$ and another infinite set of function symbols disjoint from $F_M$. For each $n$-ary function symbol $f\in F$, the operation $f(a_1,..,a_n)$ is evaluated in $M$ and produces an element of $D_M$ if $f\in F_M$ and all the $a_i$ are elements of $D_M$, or is a tree whose root is labelled by $f$ and whose immediate children are $a_1,..,a_n$ otherwise. The set of relations $R$ contains $R_M$ and another relation which distinguishes the elements of $D_M$ from the others. Using a first-order axiomatization $T$ of $M$, we give a first-order axiomatization $\cal{T}$ of the structure $Ext_M$ and show that if $T$ is {\em flexible} then $\cal{T}$ is {\em complete}. The flexible theories are particular theories whose function and relation symbols have some elegant properties which enable us to handle formulae more easily

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

Multi-Agent Only Knowing

Levesque introduced a notion of ``only knowing'', with the goal of capturing certain types of nonmonotonic reasoning. Levesque's logic dealt with only the case of a single agent. Recently, both Halpern and Lakemeyer independently attempted to extend Levesque's logic to the multi-agent case. Although there are a number of similarities in their approaches, there are some significant differences. In this paper, we reexamine the notion of only knowing, going back to first principles. In the process, we simplify Levesque's completeness proof, and point out some problems with the earlier definitions. This leads us to reconsider what the properties of only knowing ought to be. We provide an axiom system that captures our desiderata, and show that it has a semantics that corresponds to it. The axiom system has an added feature of interest: it includes a modal operator for satisfiability, and thus provides a complete axiomatization for satisfiability in the logic K45.Comment: To appear, Journal of Logic and Computatio

Decomposable Theories

We present in this paper a general algorithm for solving first-order formulas in particular theories called "decomposable theories". First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory "T". The algorithm is given in the form of five rewriting rules. It transforms a first-order formula "P", which can possibly contain free variables, into a conjunction "Q" of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if "P" has no free variables then "Q" is either the formula "true" or "false". The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory "Tr" of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in "Tr" with more than 160 nested alternated quantifiers