2,400 research outputs found
A first-order axiomatization of the theory of finite trees
We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes
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
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
Characterizing perfect recall using next-step temporal operators in S5 and sub-S5 Epistemic Temporal Logic
We review the notion of perfect recall in the literature on interpreted
systems, game theory, and epistemic logic. In the context of Epistemic Temporal
Logic (ETL), we give a (to our knowledge) novel frame condition for perfect
recall, which is local and can straightforwardly be translated to a defining
formula in a language that only has next-step temporal operators. This frame
condition also gives rise to a complete axiomatization for S5 ETL frames with
perfect recall. We then consider how to extend and consolidate the notion of
perfect recall in sub-S5 settings, where the various notions discussed are no
longer equivalent
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