109,111 research outputs found
Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation
In this second paper, we study the case of substitution tilings of R^d. The
substitution on tiles induces substitutions on the faces of the tiles of all
dimensions j=0, ..., d-1. We reconstruct the tiling's equivalence relation in a
purely combinatorial way using the AF-relations given by the lower dimensional
substitutions. We define a Bratteli multi-diagram B which is made of the
Bratteli diagrams B^j, j=0, ..., d, of all those substitutions. The set of
infinite paths in B^d is identified with the canonical transversal Xi of the
tiling. Any such path has a "border", which is a set of tails in B^j for some j
less than or equal to d, and this corresponds to a natural notion of border for
its associated tiling. We define an etale equivalence relation R_B on B by
saying that two infinite paths are equivalent if they have borders which are
tail equivalent in B^j for some j less than or equal to d. We show that R_B is
homeomorphic to the tiling's equivalence relation R_Xi.Comment: 34 pages, 14 figure
Tiling groupoids and Bratteli diagrams
Let T be an aperiodic and repetitive tiling of R^d with finite local
complexity. Let O be its tiling space with canonical transversal X. The tiling
equivalence relation R_X is the set of pairs of tilings in X which are
translates of each others, with a certain (etale) topology. In this paper R_X
is reconstructed as a generalized "tail equivalence" on a Bratteli diagram,
with its standard AF-relation as a subequivalence relation.
Using a generalization of the Anderson-Putnam complex, O is identified with
the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is
built from this sequence, and its set of infinite paths dB is homeomorphic to
X. The diagram B is endowed with a horizontal structure: additional edges that
encode the adjacencies of patches in T. This allows to define an etale
equivalence relation R_B on dB which is homeomorphic to R_X, and contains the
AF-relation of "tail equivalence".Comment: 34 pages, 4 figure
NIP omega-categorical structures: the rank 1 case
We classify primitive, rank 1, omega-categorical structures having
polynomially many types over finite sets. For a fixed number of 4-types, we
show that there are only finitely many such structures and that all are built
out of finitely many linear orders interacting in a restricted number of ways.
As an example of application, we deduce the classification of primitive
structures homogeneous in a language consisting of n linear orders as well as
all reducts of such structures.Comment: Substantial changes made to the presentation, especially in sections
3 and
Complexity of equivalence relations and preorders from computability theory
We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .Comment: To appear in J. Symb. Logi
Confluence Reduction for Probabilistic Systems (extended version)
This paper presents a novel technique for state space reduction of probabilistic specifications, based on a newly developed notion of confluence for probabilistic automata. We prove that this reduction preserves branching probabilistic bisimulation and can be applied on-the-fly. To support the technique, we introduce a method for detecting confluent transitions in the context of a probabilistic process algebra with data, facilitated by an earlier defined linear format. A case study demonstrates that significant reductions can be obtained
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