15,356 research outputs found
An Upper Bound on the Complexity of Recognizable Tree Languages
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular
tree language of infinite trees is in a class
for some natural number , where is the game quantifier. We
first give a detailed exposition of this result. Next, using an embedding of
the Wadge hierarchy of non self-dual Borel subsets of the Cantor space
into the class , and the notions of Wadge degree
and Veblen function, we argue that this upper bound on the topological
complexity of regular tree languages is much better than the usual
On Descriptive Complexity, Language Complexity, and GB
We introduce , a monadic second-order language for reasoning about
trees which characterizes the strongly Context-Free Languages in the sense that
a set of finite trees is definable in iff it is (modulo a
projection) a Local Set---the set of derivation trees generated by a CFG. This
provides a flexible approach to establishing language-theoretic complexity
results for formalisms that are based on systems of well-formedness constraints
on trees. We demonstrate this technique by sketching two such results for
Government and Binding Theory. First, we show that {\em free-indexation\/}, the
mechanism assumed to mediate a variety of agreement and binding relationships
in GB, is not definable in and therefore not enforcible by CFGs.
Second, we show how, in spite of this limitation, a reasonably complete GB
account of English can be defined in . Consequently, the language
licensed by that account is strongly context-free. We illustrate some of the
issues involved in establishing this result by looking at the definition, in
, of chains. The limitations of this definition provide some insight
into the types of natural linguistic principles that correspond to higher
levels of language complexity. We close with some speculation on the possible
significance of these results for generative linguistics.Comment: To appear in Specifying Syntactic Structures, papers from the Logic,
Structures, and Syntax workshop, Amsterdam, Sept. 1994. LaTeX source with
nine included postscript figure
Model-Checking of Ordered Multi-Pushdown Automata
We address the verification problem of ordered multi-pushdown automata: A
multi-stack extension of pushdown automata that comes with a constraint on
stack transitions such that a pop can only be performed on the first non-empty
stack. First, we show that the emptiness problem for ordered multi-pushdown
automata is in 2ETIME. Then, we prove that, for an ordered multi-pushdown
automata, the set of all predecessors of a regular set of configurations is an
effectively constructible regular set. We exploit this result to solve the
global model-checking which consists in computing the set of all configurations
of an ordered multi-pushdown automaton that satisfy a given w-regular property
(expressible in linear-time temporal logics or the linear-time \mu-calculus).
As an immediate consequence, we obtain an 2ETIME upper bound for the
model-checking problem of w-regular properties for ordered multi-pushdown
automata (matching its lower-bound).Comment: 31 page
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Hyper-Minimization for Deterministic Weighted Tree Automata
Hyper-minimization is a state reduction technique that allows a finite change
in the semantics. The theory for hyper-minimization of deterministic weighted
tree automata is provided. The presence of weights slightly complicates the
situation in comparison to the unweighted case. In addition, the first
hyper-minimization algorithm for deterministic weighted tree automata, weighted
over commutative semifields, is provided together with some implementation
remarks that enable an efficient implementation. In fact, the same run-time O(m
log n) as in the unweighted case is obtained, where m is the size of the
deterministic weighted tree automaton and n is its number of states.Comment: In Proceedings AFL 2014, arXiv:1405.527
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
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