152 research outputs found
Order-Invariant First-Order Logic over Hollow Trees
We show that the expressive power of order-invariant first-order logic collapses to first-order logic over hollow trees. A hollow tree is an unranked ordered tree where every non leaf node has at most four adjacent nodes: two siblings (left and right) and its first and last children. In particular there is no predicate for the linear order among siblings nor for the descendant relation. Moreover only the first and last nodes of a siblinghood are linked to their parent node, and the parent-child relation cannot be completely reconstructed in first-order
Deciding definability in FO2(<h,<v) on trees
We provide a decidable characterization of regular forest languages definable
in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first
order logic built from the descendant relation and the following sibling
relation. In terms of expressive power it corresponds to a fragment of the
navigational core of XPath that contains modalities for going up to some
ancestor, down to some descendant, left to some preceding sibling, and right to
some following sibling. We also show that our techniques can be applied to
other two variable first-order logics having exactly the same vertical
modalities as FO2(<h,<v) but having different horizontal modalities
Tree Languages Defined in First-Order Logic with One Quantifier Alternation
We study tree languages that can be defined in \Delta_2 . These are tree
languages definable by a first-order formula whose quantifier prefix is forall
exists, and simultaneously by a first-order formula whose quantifier prefix is
. For the quantifier free part we consider two signatures, either the
descendant relation alone or together with the lexicographical order relation
on nodes. We provide an effective characterization of tree and forest languages
definable in \Delta_2 . This characterization is in terms of algebraic
equations. Over words, the class of word languages definable in \Delta_2 forms
a robust class, which was given an effective algebraic characterization by Pin
and Weil
Bottom-up automata on data trees and vertical XPath
A data tree is a finite tree whose every node carries a label from a finite
alphabet and a datum from some infinite domain. We introduce a new model of
automata over unranked data trees with a decidable emptiness problem. It is
essentially a bottom-up alternating automaton with one register that can store
one data value and can be used to perform equality tests with the data values
occurring within the subtree of the current node. We show that it captures the
expressive power of the vertical fragment of XPath - containing the child,
descendant, parent and ancestor axes - obtaining thus a decision procedure for
its satisfiability problem
FO2(<,+1,~) on data trees, data tree automata and branching vector addition systems
A data tree is an unranked ordered tree where each node carries a label from
a finite alphabet and a datum from some infinite domain. We consider the two
variable first order logic FO2(<,+1,~) over data trees. Here +1 refers to the
child and the next sibling relations while < refers to the descendant and
following sibling relations. Moreover, ~ is a binary predicate testing data
equality. We exhibit an automata model, denoted DAD# that is more expressive
than FO2(<,+1,~) but such that emptiness of DAD# and satisfiability of
FO2(<,+1,~) are inter-reducible. This is proved via a model of counter tree
automata, denoted EBVASS, that extends Branching Vector Addition Systems with
States (BVASS) with extra features for merging counters. We show that, as
decision problems, reachability for EBVASS, satisfiability of FO2(<,+1,~) and
emptiness of DAD# are equivalent
Automata based verification over linearly ordered data domains
In this paper we work over linearly ordered data domains equipped with finitely many unary predicates and constants. We consider nondeterministic automata processing words and storing finitely many variables ranging over the domain. During a transition, these automata can compare the data values of the current configuration with those of the previous configuration using the linear order, the unary predicates and the constants.
We show that emptiness for such automata is decidable, both over finite and infinite words, under reasonable computability assumptions on the linear order.
Finally, we show how our automata model can be used for verifying properties of workflow specifications in the presence of an underlying database
Piecewise testable tree languages
This paper presents a decidable characterization of tree languages that can
be defined by a boolean combination of Sigma_1 sentences. This is a tree
extension of the Simon theorem, which says that a string language can be
defined by a boolean combination of Sigma_1 sentences if and only if its
syntactic monoid is J-trivial
First-order queries on classes of structures with bounded expansion
We consider the evaluation of first-order queries over classes of databases
with bounded expansion. The notion of bounded expansion is fairly broad and
generalizes bounded degree, bounded treewidth and exclusion of at least one
minor. It was known that over a class of databases with bounded expansion,
first-order sentences could be evaluated in time linear in the size of the
database. We give a different proof of this result. Moreover, we show that
answers to first-order queries can be enumerated with constant delay after a
linear time preprocessing. We also show that counting the number of answers to
a query can be done in time linear in the size of the database
Nondeterminism and an abstract formulation of Ne\v{c}iporuk's lower bound method
A formulation of "Ne\v{c}iporuk's lower bound method" slightly more inclusive
than the usual complexity-measure-specific formulation is presented. Using this
general formulation, limitations to lower bounds achievable by the method are
obtained for several computation models, such as branching programs and Boolean
formulas having access to a sublinear number of nondeterministic bits. In
particular, it is shown that any lower bound achievable by the method of
Ne\v{c}iporuk for the size of nondeterministic and parity branching programs is
at most
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