1,660 research outputs found
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
Using Regular Languages to Explore the Representational Capacity of Recurrent Neural Architectures
The presence of Long Distance Dependencies (LDDs) in sequential data poses
significant challenges for computational models. Various recurrent neural
architectures have been designed to mitigate this issue. In order to test these
state-of-the-art architectures, there is growing need for rich benchmarking
datasets. However, one of the drawbacks of existing datasets is the lack of
experimental control with regards to the presence and/or degree of LDDs. This
lack of control limits the analysis of model performance in relation to the
specific challenge posed by LDDs. One way to address this is to use synthetic
data having the properties of subregular languages. The degree of LDDs within
the generated data can be controlled through the k parameter, length of the
generated strings, and by choosing appropriate forbidden strings. In this
paper, we explore the capacity of different RNN extensions to model LDDs, by
evaluating these models on a sequence of SPk synthesized datasets, where each
subsequent dataset exhibits a longer degree of LDD. Even though SPk are simple
languages, the presence of LDDs does have significant impact on the performance
of recurrent neural architectures, thus making them prime candidate in
benchmarking tasks.Comment: International Conference of Artificial Neural Networks (ICANN) 201
A Characterization for Decidable Separability by Piecewise Testable Languages
The separability problem for word languages of a class by
languages of a class asks, for two given languages and
from , whether there exists a language from that
includes and excludes , that is, and . In this work, we assume some mild closure properties for
and study for which such classes separability by a piecewise
testable language (PTL) is decidable. We characterize these classes in terms of
decidability of (two variants of) an unboundedness problem. From this, we
deduce that separability by PTL is decidable for a number of language classes,
such as the context-free languages and languages of labeled vector addition
systems. Furthermore, it follows that separability by PTL is decidable if and
only if one can compute for any language of the class its downward closure wrt.
the scattered substring ordering (i.e., if the set of scattered substrings of
any language of the class is effectively regular).
The obtained decidability results contrast some undecidability results. In
fact, for all (non-regular) language classes that we present as examples with
decidable separability, it is undecidable whether a given language is a PTL
itself.
Our characterization involves a result of independent interest, which states
that for any kind of languages and , non-separability by PTL is
equivalent to the existence of common patterns in and
Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
The purpose of this paper is to provide efficient algorithms that decide
membership for classes of several Boolean hierarchies for which efficiency (or
even decidability) were previously not known. We develop new forbidden-chain
characterizations for the single levels of these hierarchies and obtain the
following results: - The classes of the Boolean hierarchy over level
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
Fragments of first-order logic over infinite words
We give topological and algebraic characterizations as well as language
theoretic descriptions of the following subclasses of first-order logic FO[<]
for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and
Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These
descriptions extend the respective results for finite words. In particular, we
relate the above fragments to language classes of certain (unambiguous)
polynomials. An immediate consequence is the decidability of the membership
problem of these classes, but this was shown before by Wilke and Bojanczyk and
is therefore not our main focus. The paper is about the interplay of algebraic,
topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on
Theoretical Aspects of Computer Science, STACS 200
On FO2 quantifier alternation over words
We show that each level of the quantifier alternation hierarchy within
FO^2[<] -- the 2-variable fragment of the first order logic of order on words
-- is a variety of languages. We then use the notion of condensed rankers, a
refinement of the rankers defined by Weis and Immerman, to produce a decidable
hierarchy of varieties which is interwoven with the quantifier alternation
hierarchy -- and conjecturally equal to it. It follows that the latter
hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one
can effectively compute an integer m such that alpha is equivalent to a formula
with at most m+1 alternating blocks of quantifiers, but not to a formula with
only m-1 blocks. This is a much more precise result than what is known about
the quantifier alternation hierarchy within FO[<], where no decidability result
is known beyond the very first levels
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