10,689 research outputs found
A characterization of power-free morphisms
AbstractA word is called kth power-free if it does not contain any non-empty factor uk. A morphism is kth power-free if it preserves kth power-free words. A morphism is power-free if it is kth power-free for every K > 1.We show that it is decidable whether a morphism is power-free; more precisely, we prove that a morphism h is power-free iff: h is a square-free morphism and, for each letter a, the image h(a2) is cube-free
The Relative Power of Composite Loop Agreement Tasks
Loop agreement is a family of wait-free tasks that includes set agreement and
simplex agreement, and was used to prove the undecidability of wait-free
solvability of distributed tasks by read/write memory. Herlihy and Rajsbaum
defined the algebraic signature of a loop agreement task, which consists of a
group and a distinguished element. They used the algebraic signature to
characterize the relative power of loop agreement tasks. In particular, they
showed that one task implements another exactly when there is a homomorphism
between their respective signatures sending one distinguished element to the
other. In this paper, we extend the previous result by defining the composition
of multiple loop agreement tasks to create a new one with the same combined
power. We generalize the original algebraic characterization of relative power
to compositions of tasks. In this way, we can think of loop agreement tasks in
terms of their basic building blocks. We also investigate a category-theoretic
perspective of loop agreement by defining a category of loops, showing that the
algebraic signature is a functor, and proving that our definition of task
composition is the "correct" one, in a categorical sense.Comment: 18 page
A Universal Characterization of the Double Powerlocale
This is a version from 29 Sept 2003 of the paper published under the same name in Theoretical Computer Science 316 (2004) 297{321.
The double powerlocale P(X) (found by composing, in either order,the upper and lower powerlocale constructions PU and PL) is shown to be isomorphic in [Locop; Set] to the double exponential SSX where S is the Sierpinski locale. Further PU(X) and PL(X) are shown to be the subobjects P(X) comprising, respectively, the meet semilattice and join
semilattice homomorphisms. A key lemma shows that, for any locales X and Y , natural transformations from SX (the presheaf Loc
Topological modular forms and conformal nets
We describe the role conformal nets, a mathematical model for conformal field
theory, could play in a geometric definition of the generalized cohomology
theory TMF of topological modular forms. Inspired by work of Segal and
Stolz-Teichner, we speculate that bundles of boundary conditions for the net of
free fermions will be the basic underlying objects representing TMF-cohomology
classes. String structures, which are the fundamental orientations for
TMF-cohomology, can be encoded by defects between free fermions, and we
construct the bundle of fermionic boundary conditions for the TMF-Euler class
of a string vector bundle. We conjecture that the free fermion net exhibits an
algebraic periodicity corresponding to the 576-fold cohomological periodicity
of TMF; using a homotopy-theoretic invariant of invertible conformal nets, we
establish a lower bound of 24 on this periodicity of the free fermions
Constructive version of Boolean algebra
The notion of overlap algebra introduced by G. Sambin provides a constructive
version of complete Boolean algebra. Here we first show some properties
concerning overlap algebras: we prove that the notion of overlap morphism
corresponds classically to that of map preserving arbitrary joins; we provide a
description of atomic set-based overlap algebras in the language of formal
topology, thus giving a predicative characterization of discrete locales; we
show that the power-collection of a set is the free overlap algebra
join-generated from the set. Then, we generalize the concept of overlap algebra
and overlap morphism in various ways to provide constructive versions of the
category of Boolean algebras with maps preserving arbitrary existing joins.Comment: 22 page
On Quasiperiodic Morphisms
Weakly and strongly quasiperiodic morphisms are tools introduced to study
quasiperiodic words. Formally they map respectively at least one or any
non-quasiperiodic word to a quasiperiodic word. Considering them both on finite
and infinite words, we get four families of morphisms between which we study
relations. We provide algorithms to decide whether a morphism is strongly
quasiperiodic on finite words or on infinite words.Comment: 12 page
Algebraic recognizability of regular tree languages
We propose a new algebraic framework to discuss and classify recognizable
tree languages, and to characterize interesting classes of such languages. Our
algebraic tool, called preclones, encompasses the classical notion of syntactic
Sigma-algebra or minimal tree automaton, but adds new expressivity to it. The
main result in this paper is a variety theorem \`{a} la Eilenberg, but we also
discuss important examples of logically defined classes of recognizable tree
languages, whose characterization and decidability was established in recent
papers (by Benedikt and S\'{e}goufin, and by Bojanczyk and Walukiewicz) and can
be naturally formulated in terms of pseudovarieties of preclones. Finally, this
paper constitutes the foundation for another paper by the same authors, where
first-order definable tree languages receive an algebraic characterization
Proper Functors and Fixed Points for Finite Behaviour
The rational fixed point of a set functor is well-known to capture the
behaviour of finite coalgebras. In this paper we consider functors on algebraic
categories. For them the rational fixed point may no longer be fully abstract,
i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's
notion of a proper semiring, we introduce the notion of a proper functor. We
show that for proper functors the rational fixed point is determined as the
colimit of all coalgebras with a free finitely generated algebra as carrier and
it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor
is proper if and only if that colimit is a subcoalgebra of the final coalgebra.
These results serve as technical tools for soundness and completeness proofs
for coalgebraic regular expression calculi, e.g. for weighted automata
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