15,788 research outputs found
Regular Cost Functions, Part I: Logic and Algebra over Words
The theory of regular cost functions is a quantitative extension to the
classical notion of regularity. A cost function associates to each input a
non-negative integer value (or infinity), as opposed to languages which only
associate to each input the two values "inside" and "outside". This theory is a
continuation of the works on distance automata and similar models. These models
of automata have been successfully used for solving the star-height problem,
the finite power property, the finite substitution problem, the relative
inclusion star-height problem and the boundedness problem for monadic-second
order logic over words. Our notion of regularity can be -- as in the classical
theory of regular languages -- equivalently defined in terms of automata,
expressions, algebraic recognisability, and by a variant of the monadic
second-order logic. These equivalences are strict extensions of the
corresponding classical results. The present paper introduces the cost monadic
logic, the quantitative extension to the notion of monadic second-order logic
we use, and show that some problems of existence of bounds are decidable for
this logic. This is achieved by introducing the corresponding algebraic
formalism: stabilisation monoids.Comment: 47 page
Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests
We first propose algorithms for checking language equivalence of finite
automata over a large alphabet. We use symbolic automata, where the transition
function is compactly represented using a (multi-terminal) binary decision
diagrams (BDD). The key idea consists in computing a bisimulation by exploring
reachable pairs symbolically, so as to avoid redundancies. This idea can be
combined with already existing optimisations, and we show in particular a nice
integration with the disjoint sets forest data-structure from Hopcroft and
Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an
algebraic theory that can be used for verification in various domains ranging
from compiler optimisation to network programming analysis. This theory is
decidable by reduction to language equivalence of automata on guarded strings,
a particular kind of automata that have exponentially large alphabets. We
propose several methods allowing to construct symbolic automata out of KAT
expressions, based either on Brzozowski's derivatives or standard automata
constructions. All in all, this results in efficient algorithms for deciding
equivalence of KAT expressions
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
An introduction to finite automata and their connection to logic
This is a tutorial on finite automata. We present the standard material on
determinization and minimization, as well as an account of the equivalence of
finite automata and monadic second-order logic. We conclude with an
introduction to the syntactic monoid, and as an application give a proof of the
equivalence of first-order definability and aperiodicity
On the minimal ranks of matrix pencils and the existence of a best approximate block-term tensor decomposition
Under the action of the general linear group with tensor structure, the ranks
of matrices and forming an pencil can
change, but in a restricted manner. Specifically, with every pencil one can
associate a pair of minimal ranks, which is unique up to a permutation. This
notion can be defined for matrix pencils and, more generally, also for matrix
polynomials of arbitrary degree. In this paper, we provide a formal definition
of the minimal ranks, discuss its properties and the natural hierarchy it
induces in a pencil space. Then, we show how the minimal ranks of a pencil can
be determined from its Kronecker canonical form. For illustration, we classify
the orbits according to their minimal ranks (under the action of the general
linear group) in the case of real pencils with . Subsequently, we
show that real regular pencils having only complex-valued
eigenvalues, which form an open positive-volume set, do not admit a best
approximation (in the norm topology) on the set of real pencils whose minimal
ranks are bounded by . Our results can be interpreted from a tensor
viewpoint, where the minimal ranks of a degree- matrix polynomial
characterize the minimal ranks of matrices constituting a block-term
decomposition of an tensor into a sum of matrix-vector
tensor products.Comment: This work was supported by the European Research Council under the
European Programme FP7/2007-2013, Grant AdG-2013-320594 "DECODA.
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
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