637 research outputs found
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
Semantics of Higher-Order Recursion Schemes
Higher-order recursion schemes are recursive equations defining new
operations from given ones called "terminals". Every such recursion scheme is
proved to have a least interpreted semantics in every Scott's model of
\lambda-calculus in which the terminals are interpreted as continuous
operations. For the uninterpreted semantics based on infinite \lambda-terms we
follow the idea of Fiore, Plotkin and Turi and work in the category of sets in
context, which are presheaves on the category of finite sets. Fiore et al
showed how to capture the type of variable binding in \lambda-calculus by an
endofunctor H\lambda and they explained simultaneous substitution of
\lambda-terms by proving that the presheaf of \lambda-terms is an initial
H\lambda-monoid. Here we work with the presheaf of rational infinite
\lambda-terms and prove that this is an initial iterative H\lambda-monoid. We
conclude that every guarded higher-order recursion scheme has a unique
uninterpreted solution in this monoid
Uniform and Bernoulli measures on the boundary of trace monoids
Trace monoids and heaps of pieces appear in various contexts in
combinatorics. They also constitute a model used in computer science to
describe the executions of asynchronous systems. The design of a natural
probabilistic layer on top of the model has been a long standing challenge. The
difficulty comes from the presence of commuting pieces and from the absence of
a global clock. In this paper, we introduce and study the class of Bernoulli
probability measures that we claim to be the simplest adequate probability
measures on infinite traces. For this, we strongly rely on the theory of trace
combinatorics with the M\"obius polynomial in the key role. These new measures
provide a theoretical foundation for the probabilistic study of concurrent
systems.Comment: 34 pages, 5 figures, 27 reference
Multioperator Weighted Monadic Datalog
In this thesis we will introduce multioperator weighted monadic datalog (mwmd), a formal model for specifying tree series, tree transformations, and tree languages. This model combines aspects of multioperator weighted tree automata (wmta), weighted monadic datalog (wmd), and monadic datalog tree transducers (mdtt). In order to develop a rich theory we will define multiple versions of semantics for mwmd and compare their expressiveness. We will study normal forms and decidability results of mwmd and show (by employing particular semantic domains) that the theory of mwmd subsumes the theory of both wmd and mdtt. We conclude this thesis by showing that mwmd even contain wmta as a syntactic subclass and present results concerning this subclass
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