623 research outputs found
Inside the class of REGEX Languages
We study different possibilities of combining the concept of homomorphic replacement with regular expressions in order to investigate the class of languages given by extended regular expressions with backreferences (REGEX). It is shown in which regard existing and natural ways to do this fail to reach the expressive power of REGEX. Furthermore, the complexity of the membership problem for REGEX with a bounded number of backreferences is considered
Finite reflection groups and graph norms
Given a graph on vertex set and a function , define \begin{align*} \|f\|_{H}:=\left\vert\int
\prod_{ij\in E(H)}f(x_i,x_j)d\mu^{|V(H)|}\right\vert^{1/|E(H)|}, \end{align*}
where is the Lebesgue measure on . We say that is norming if
is a semi-norm. A similar notion is defined by
and is said to be weakly norming if
is a norm. Classical results show that weakly norming graphs
are necessarily bipartite. In the other direction, Hatami showed that even
cycles, complete bipartite graphs, and hypercubes are all weakly norming. We
demonstrate that any graph whose edges percolate in an appropriate way under
the action of a certain natural family of automorphisms is weakly norming. This
result includes all previously known examples of weakly norming graphs, but
also allows us to identify a much broader class arising from finite reflection
groups. We include several applications of our results. In particular, we
define and compare a number of generalisations of Gowers' octahedral norms and
we prove some new instances of Sidorenko's conjecture.Comment: 29 page
The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera
For a compact oriented surface of genus with boundary
components, the space spanned by free homotopy classes of loops
in carries the structure of a Lie bialgebra. The Lie bracket was
defined by Goldman and it is canonical. The Lie cobracket was defined by Turaev
and it depends on the framing of . The Lie bialgebra has
a natural decreasing filtration such that both the Goldman bracket and the
Turaev cobracket have degree .
In this paper, we address the following Goldman-Turaev formality problem:
construct a Lie bialgebra homomorphism from to its
associated graded such that . In
order to solve it, we define a family of higher genus Kashiwara-Vergne (KV)
problems for an element , where is a free Lie algebra. In the
case of and , the problem for is the classical KV problem from
Lie theory. For , these KV problems are new.
Our main results are as follows. On the one hand, every solution of the KV
problem induces a GT formality map. On the other hand, higher genus KV problems
admit solutions for any and . In fact, the solution reduces to two
important cases: which admits solutions by Alekseev and Torossian
and for which we construct solutions in terms of certain elliptic
associators following Enriquez. By combining these two results, we obtain a
proof of the GT formality for any and .
We also study the set of solutions of higher genus KV problems and introduce
pro-unipotent groups which act on them freely and transitively.
These groups admit graded pro-nilpotent Lie algebras . We show
that the elliptic Lie algebra contains a copy of the
Grothendieck-Teichmuller Lie algebra as well as symplectic derivations
.Comment: 61 pages, 5 figure
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
Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations
AbstractThis paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
- …