6,449 research outputs found
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
Boundedness in languages of infinite words
We define a new class of languages of -words, strictly extending
-regular languages.
One way to present this new class is by a type of regular expressions. The
new expressions are an extension of -regular expressions where two new
variants of the Kleene star are added: and . These new
exponents are used to say that parts of the input word have bounded size, and
that parts of the input can have arbitrarily large sizes, respectively. For
instance, the expression represents the language of infinite
words over the letters where there is a common bound on the number of
consecutive letters . The expression represents a similar
language, but this time the distance between consecutive 's is required to
tend toward the infinite.
We develop a theory for these languages, with a focus on decidability and
closure. We define an equivalent automaton model, extending B\"uchi automata.
The main technical result is a complementation lemma that works for languages
where only one type of exponent---either or ---is used.
We use the closure and decidability results to obtain partial decidability
results for the logic MSOLB, a logic obtained by extending monadic second-order
logic with new quantifiers that speak about the size of sets
A Crevice on the Crane Beach: Finite-Degree Predicates
First-order logic (FO) over words is shown to be equiexpressive with FO
equipped with a restricted set of numerical predicates, namely the order, a
binary predicate MSB, and the finite-degree predicates: FO[Arb] = FO[<,
MSB, Fin].
The Crane Beach Property (CBP), introduced more than a decade ago, is true of
a logic if all the expressible languages admitting a neutral letter are
regular.
Although it is known that FO[Arb] does not have the CBP, it is shown here
that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB].
Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express
a wide variety of languages, while being one simple predicate away from the
expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as
an application.Comment: Submitte
Tightening the Complexity of Equivalence Problems for Commutative Grammars
We show that the language equivalence problem for regular and context-free
commutative grammars is coNEXP-complete. In addition, our lower bound
immediately yields further coNEXP-completeness results for equivalence problems
for communication-free Petri nets and reversal-bounded counter automata.
Moreover, we improve both lower and upper bounds for language equivalence for
exponent-sensitive commutative grammars.Comment: 21 page
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
On the Expressive Power of 2-Stack Visibly Pushdown Automata
Visibly pushdown automata are input-driven pushdown automata that recognize
some non-regular context-free languages while preserving the nice closure and
decidability properties of finite automata. Visibly pushdown automata with
multiple stacks have been considered recently by La Torre, Madhusudan, and
Parlato, who exploit the concept of visibility further to obtain a rich
automata class that can even express properties beyond the class of
context-free languages. At the same time, their automata are closed under
boolean operations, have a decidable emptiness and inclusion problem, and enjoy
a logical characterization in terms of a monadic second-order logic over words
with an additional nesting structure. These results require a restricted
version of visibly pushdown automata with multiple stacks whose behavior can be
split up into a fixed number of phases. In this paper, we consider 2-stack
visibly pushdown automata (i.e., visibly pushdown automata with two stacks) in
their unrestricted form. We show that they are expressively equivalent to the
existential fragment of monadic second-order logic. Furthermore, it turns out
that monadic second-order quantifier alternation forms an infinite hierarchy
wrt words with multiple nestings. Combining these results, we conclude that
2-stack visibly pushdown automata are not closed under complementation.
Finally, we discuss the expressive power of B\"{u}chi 2-stack visibly pushdown
automata running on infinite (nested) words. Extending the logic by an infinity
quantifier, we can likewise establish equivalence to existential monadic
second-order logic
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