21,935 research outputs found
On Varieties of Automata Enriched with an Algebraic Structure (Extended Abstract)
Eilenberg correspondence, based on the concept of syntactic monoids, relates
varieties of regular languages with pseudovarieties of finite monoids. Various
modifications of this correspondence related more general classes of regular
languages with classes of more complex algebraic objects. Such generalized
varieties also have natural counterparts formed by classes of finite automata
equipped with a certain additional algebraic structure. In this survey, we
overview several variants of such varieties of enriched automata.Comment: In Proceedings AFL 2014, arXiv:1405.527
Finitary languages
The class of omega-regular languages provides a robust specification language
in verification. Every omega-regular condition can be decomposed into a safety
part and a liveness part. The liveness part ensures that something good happens
"eventually". Finitary liveness was proposed by Alur and Henzinger as a
stronger formulation of liveness. It requires that there exists an unknown,
fixed bound b such that something good happens within b transitions. In this
work we consider automata with finitary acceptance conditions defined by
finitary Buchi, parity and Streett languages. We study languages expressible by
such automata: we give their topological complexity and present a
regular-expression characterization. We compare the expressive power of
finitary automata and give optimal algorithms for classical decisions
questions. We show that the finitary languages are Sigma 2-complete; we present
a complete picture of the expressive power of various classes of automata with
finitary and infinitary acceptance conditions; we show that the languages
defined by finitary parity automata exactly characterize the star-free fragment
of omega B-regular languages; and we show that emptiness is NLOGSPACE-complete
and universality as well as language inclusion are PSPACE-complete for finitary
parity and Streett automata
Separation for dot-depth two
The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free
languages of finite words. By a theorem of McNaughton and Papert, these are
also the first-order definable languages. The dot-depth rose to prominence
following the work of Thomas, who proved an exact correspondence with the
quantifier alternation hierarchy of first-order logic: each level in the
dot-depth hierarchy consists of all languages that can be defined with a
prescribed number of quantifier blocks. One of the most famous open problems in
automata theory is to settle whether the membership problem is decidable for
each level: is it possible to decide whether an input regular language belongs
to this level?
Despite a significant research effort, membership by itself has only been
solved for low levels. A recent breakthrough was achieved by replacing
membership with a more general problem: separation. Given two input languages,
one has to decide whether there exists a third language in the investigated
level containing the first language and disjoint from the second. The
motivation is that: (1) while more difficult, separation is more rewarding (2)
it provides a more convenient framework (3) all recent membership algorithms
are reductions to separation for lower levels.
We present a separation algorithm for dot-depth two. While this is our most
prominent application, our result is more general. We consider a family of
hierarchies that includes the dot-depth: concatenation hierarchies. They are
built via a generic construction process. One first chooses an initial class,
the basis, which is the lowest level in the hierarchy. Further levels are built
by applying generic operations. Our main theorem states that for any
concatenation hierarchy whose basis is finite, separation is decidable for
level one. In the special case of the dot-depth, this can be lifted to level
two using previously known results
Separating regular languages with two quantifier alternations
We investigate a famous decision problem in automata theory: separation.
Given a class of language C, the separation problem for C takes as input two
regular languages and asks whether there exists a third one which belongs to C,
includes the first one and is disjoint from the second. Typically, obtaining an
algorithm for separation yields a deep understanding of the investigated class
C. This explains why a lot of effort has been devoted to finding algorithms for
the most prominent classes.
Here, we are interested in classes within concatenation hierarchies. Such
hierarchies are built using a generic construction process: one starts from an
initial class called the basis and builds new levels by applying generic
operations. The most famous one, the dot-depth hierarchy of Brzozowski and
Cohen, classifies the languages definable in first-order logic. Moreover, it
was shown by Thomas that it corresponds to the quantifier alternation hierarchy
of first-order logic: each level in the dot-depth corresponds to the languages
that can be defined with a prescribed number of quantifier blocks. Finding
separation algorithms for all levels in this hierarchy is among the most famous
open problems in automata theory.
Our main theorem is generic: we show that separation is decidable for the
level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in
the special case of the dot-depth, we push this result to the level 5/2. In
logical terms, this solves separation for : first-order sentences
having at most three quantifier blocks starting with an existential one
Three hierarchies of transducers
Composition of top-down tree transducers yields a proper hierarchy of transductions and of output languages. The same is true for ETOL systems (viewed as transducers) and for two-way generalized sequential machines
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