10 research outputs found
Logics for Unranked Trees: An Overview
Labeled unranked trees are used as a model of XML documents, and logical
languages for them have been studied actively over the past several years. Such
logics have different purposes: some are better suited for extracting data,
some for expressing navigational properties, and some make it easy to relate
complex properties of trees to the existence of tree automata for those
properties. Furthermore, logics differ significantly in their model-checking
properties, their automata models, and their behavior on ordered and unordered
trees. In this paper we present a survey of logics for unranked trees
Wreath Products of Forest Algebras, with Applications to Tree Logics
We use the recently developed theory of forest algebras to find algebraic
characterizations of the languages of unranked trees and forests definable in
various logics. These include the temporal logics CTL and EF, and first-order
logic over the ancestor relation. While the characterizations are in general
non-effective, we are able to use them to formulate necessary conditions for
definability and provide new proofs that a number of languages are not
definable in these logics
Quantified CTL: Expressiveness and Complexity
While it was defined long ago, the extension of CTL with quantification over
atomic propositions has never been studied extensively. Considering two
different semantics (depending whether propositional quantification refers to
the Kripke structure or to its unwinding tree), we study its expressiveness
(showing in particular that QCTL coincides with Monadic Second-Order Logic for
both semantics) and characterise the complexity of its model-checking and
satisfiability problems, depending on the number of nested propositional
quantifiers (showing that the structure semantics populates the polynomial
hierarchy while the tree semantics populates the exponential hierarchy)
Bisimulation equivalence and regularity for real-time one-counter automata
A one-counter automaton is a pushdown automaton with a singleton stack alphabet, where stack emptiness can be tested; it is a real-time automaton if it contains no ε -transitions. We study the computational complexity of the problems of equivalence and regularity (i.e. semantic finiteness) on real-time one-counter automata. The first main result shows PSPACEPSPACE-completeness of bisimulation equivalence; this closes the complexity gap between decidability [23] and PSPACEPSPACE-hardness [25]. The second main result shows NLNL-completeness of language equivalence of deterministic real-time one-counter automata; this improves the known PSPACEPSPACE upper bound (indirectly shown by Valiant and Paterson [27]). Finally we prove PP-completeness of the problem if a given one-counter automaton is bisimulation equivalent to a finite system, and NLNL-completeness of the problem if the language accepted by a given deterministic real-time one-counter automaton is regular.Web of Science80474372
Proving that a Tree Language is not First-Order Definable
We explore from an algebraic viewpoint the properties of the tree languages
definable with a first-order formula involving the ancestor predicate, using
the description of these languages as those recognized by iterated block
products of forest algebras defined from finite counter monoids. Proofs of
nondefinability are infinite sequences of sets of forests, one for each level
of the hierarchy of quantification levels that defines the corresponding
variety of languages. The forests at a given level are built recursively by
inserting forests from previous level at the ports of a suitable set of
multicontexts. We show that a recursive proof exists for the syntactic algebra
of every non-definable language. We also investigate certain types of uniform
recursive proofs. For this purpose, we define from a forest algebra an algebra
of mappings and an extended algebra, which we also use to redefine the notion
of aperiodicity in a way that generalizes the existing ones
Counting on CTL*: on the expressive power of monadic path logic
Monadic second-order logic (MSOL) provides a general framework for expressing properties of reactive systems as modelled by trees. Monadic path logic (MPL) is obtained by restricting second-order quantification to paths reflecting computation sequences. In this paper we show that the expressive power of MPL over trees coincides with the usual branching time logic CTL # embellished with a simple form of counting. As a corollary, we derive an earlier result that CTL # coincides with the bisimulation-invariant properties of MPL. In order to prove the main result, we first prove a new Composition Theorem for trees