292 research outputs found
On factorisation forests
The theorem of factorisation forests shows the existence of nested
factorisations -- a la Ramsey -- for finite words. This theorem has important
applications in semigroup theory, and beyond. The purpose of this paper is to
illustrate the importance of this approach in the context of automata over
infinite words and trees. We extend the theorem of factorisation forest in two
directions: we show that it is still valid for any word indexed by a linear
ordering; and we show that it admits a deterministic variant for words indexed
by well-orderings. A byproduct of this work is also an improvement on the known
bounds for the original result. We apply the first variant for giving a
simplified proof of the closure under complementation of rational sets of words
indexed by countable scattered linear orderings. We apply the second variant in
the analysis of monadic second-order logic over trees, yielding new results on
monadic interpretations over trees. Consequences of it are new caracterisations
of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page
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
Model Theoretic Complexity of Automatic Structures
We study the complexity of automatic structures via well-established concepts
from both logic and model theory, including ordinal heights (of well-founded
relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees).
We prove the following results: 1) The ordinal height of any automatic well-
founded partial order is bounded by \omega^\omega ; 2) The ordinal heights of
automatic well-founded relations are unbounded below the first non-computable
ordinal; 3) For any computable ordinal there is an automatic structure of Scott
rank at least that ordinal. Moreover, there are automatic structures of Scott
rank the first non-computable ordinal and its successor; 4) For any computable
ordinal, there is an automatic successor tree of Cantor-Bendixson rank that
ordinal.Comment: 23 pages. Extended abstract appeared in Proceedings of TAMC '08, LNCS
4978 pp 514-52
Monadic theory of order and topology in ZFC
True first-order arithmetic is interpreted in the monadic theories of certain chains and topological spaces including the real line and the Cantor Discontinuum. It was known that existence of such interpretations in consistent with ZFC.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23778/1/0000016.pd
Satisfiability of ECTL* with tree constraints
Recently, we have shown that satisfiability for with
constraints over is decidable using a new technique. This approach
reduces the satisfiability problem of with constraints over
some structure A (or class of structures) to the problem whether A has a
certain model theoretic property that we called EHD (for "existence of
homomorphisms is decidable"). Here we apply this approach to concrete domains
that are tree-like and obtain several results. We show that satisfiability of
with constraints is decidable over (i) semi-linear orders
(i.e., tree-like structures where branches form arbitrary linear orders), (ii)
ordinal trees (semi-linear orders where the branches form ordinals), and (iii)
infinitely branching trees of height h for each fixed . We
prove that all these classes of structures have the property EHD. In contrast,
we introduce Ehrenfeucht-Fraisse-games for (weak
with the bounding quantifier) and use them to show that the
infinite (order) tree does not have property EHD. As a consequence, a different
approach has to be taken in order to settle the question whether satisfiability
of (or even ) with constraints over the
infinite (order) tree is decidable
Compatibility of Shelah and Stupp's and of Muchnik's iteration with fragments of monadic second order logic
We investigate the relation between the theory of the itera- tions in the sense of Shelah-Stupp and of Muchnik, resp., and the theory of the base structure for several logics. These logics are obtained from the restriction of set quantification in monadic second order logic to cer- tain subsets like, e.g., finite sets, chains, and finite unions of chains. We show that these theories of the Shelah-Stupp iteration can be reduced to corresponding theories of the base structure. This fails for Muchnik's iteration
Decidability of Querying First-Order Theories via Countermodels of Finite Width
We propose a generic framework for establishing the decidability of a wide
range of logical entailment problems (briefly called querying), based on the
existence of countermodels that are structurally simple, gauged by certain
types of width measures (with treewidth and cliquewidth as popular examples).
As an important special case of our framework, we identify logics exhibiting
width-finite finitely universal model sets, warranting decidable entailment for
a wide range of homomorphism-closed queries, subsuming a diverse set of
practically relevant query languages. As a particularly powerful width measure,
we propose Blumensath's partitionwidth, which subsumes various other commonly
considered width measures and exhibits highly favorable computational and
structural properties. Focusing on the formalism of existential rules as a
popular showcase, we explain how finite partitionwidth sets of rules subsume
other known abstract decidable classes but -- leveraging existing notions of
stratification -- also cover a wide range of new rulesets. We expose natural
limitations for fitting the class of finite unification sets into our picture
and provide several options for remedy
Compatibility of Shelah and Stupp's and Muchnik's iteration with fragments of monadic second order logic
We investigate the relation between the theory of the iterations in the sense
of Shelah-Stupp and of Muchnik, resp., and the theory of the base structure for
several logics. These logics are obtained from the restriction of set
quantification in monadic second order logic to certain subsets like, e.g.,
finite sets, chains, and finite unions of chains. We show that these theories
of the Shelah-Stupp iteration can be reduced to corresponding theories of the
base structure. This fails for Muchnik's iteration
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