424,137 research outputs found
On Cayley graphs of virtually free groups
In 1985, Dunwoody showed that finitely presentable groups are accessible.
Dunwoody's result was used to show that context-free groups, groups
quasi-isometric to trees or finitely presentable groups of asymptotic dimension
1 are virtually free. Using another theorem of Dunwoody of 1979, we study when
a group is virtually free in terms of its Cayley graph and we obtain new proofs
of the mentioned results and other previously depending on them
On Descriptive Complexity, Language Complexity, and GB
We introduce , a monadic second-order language for reasoning about
trees which characterizes the strongly Context-Free Languages in the sense that
a set of finite trees is definable in iff it is (modulo a
projection) a Local Set---the set of derivation trees generated by a CFG. This
provides a flexible approach to establishing language-theoretic complexity
results for formalisms that are based on systems of well-formedness constraints
on trees. We demonstrate this technique by sketching two such results for
Government and Binding Theory. First, we show that {\em free-indexation\/}, the
mechanism assumed to mediate a variety of agreement and binding relationships
in GB, is not definable in and therefore not enforcible by CFGs.
Second, we show how, in spite of this limitation, a reasonably complete GB
account of English can be defined in . Consequently, the language
licensed by that account is strongly context-free. We illustrate some of the
issues involved in establishing this result by looking at the definition, in
, of chains. The limitations of this definition provide some insight
into the types of natural linguistic principles that correspond to higher
levels of language complexity. We close with some speculation on the possible
significance of these results for generative linguistics.Comment: To appear in Specifying Syntactic Structures, papers from the Logic,
Structures, and Syntax workshop, Amsterdam, Sept. 1994. LaTeX source with
nine included postscript figure
Invisible pushdown languages
Context free languages allow one to express data with hierarchical structure,
at the cost of losing some of the useful properties of languages recognized by
finite automata on words. However, it is possible to restore some of these
properties by making the structure of the tree visible, such as is done by
visibly pushdown languages, or finite automata on trees. In this paper, we show
that the structure given by such approaches remains invisible when it is read
by a finite automaton (on word). In particular, we show that separability with
a regular language is undecidable for visibly pushdown languages, just as it is
undecidable for general context free languages
Recursive Program Schemes and Context-Free Monads
AbstractSolutions of recursive program schemes over a given signature Σ were characterized by Bruno Courcelle as precisely the context-free (or algebraic) Σ-trees. These are the finite and infinite Σ-trees yielding, via labelling of paths, context-free languages. Our aim is to generalize this to finitary endofunctors H of general categories: we construct a monad CH “generated” by solutions of recursive program schemes of type H, and prove that this monad is ideal. In case of polynomial endofunctors of Set our construction precisely yields the monad of context-free Σ-trees of Courcelle. Our result builds on a result by N. Ghani et al on solutions of algebraic systems
The Ramsey Theory of Henson graphs
Analogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte
Counting, generating and sampling tree alignments
Pairwise ordered tree alignment are combinatorial objects that appear in RNA
secondary structure comparison. However, the usual representation of tree
alignments as supertrees is ambiguous, i.e. two distinct supertrees may induce
identical sets of matches between identical pairs of trees. This ambiguity is
uninformative, and detrimental to any probabilistic analysis.In this work, we
consider tree alignments up to equivalence. Our first result is a precise
asymptotic enumeration of tree alignments, obtained from a context-free grammar
by mean of basic analytic combinatorics. Our second result focuses on
alignments between two given ordered trees and . By refining our grammar
to align specific trees, we obtain a decomposition scheme for the space of
alignments, and use it to design an efficient dynamic programming algorithm for
sampling alignments under the Gibbs-Boltzmann probability distribution. This
generalizes existing tree alignment algorithms, and opens the door for a
probabilistic analysis of the space of suboptimal RNA secondary structures
alignments.Comment: ALCOB - 3rd International Conference on Algorithms for Computational
Biology - 2016, Jun 2016, Trujillo, Spain. 201
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