4 research outputs found
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by
omega-automata, so called omega-automatic structures, and prove
that relations defined over such structures in first-order logic
expanded by the first-order quantifiers `there exist at most
many\u27, \u27there exist finitely many\u27 and \u27there exist
modulo many\u27 are omega-regular. The proof identifies certain
algebraic properties of omega-semigroups.
As a consequence an omega-regular equivalence relation of countable
index has an omega-regular set of representatives. This implies
Blumensath\u27s conjecture that a countable structure with an
-automatic presentation can be represented using automata
on finite words. This also complements a very recent result of
Hj"orth, Khoussainov, Montalban and Nies showing that there is an
omega-automatic structure which has no injective presentation
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by omega-automata, so
called omega-automatic structures, and prove that relations defined over such
structures in first-order logic expanded by the first-order quantifiers `there
exist at most many', 'there exist finitely many' and 'there exist
modulo many' are omega-regular. The proof identifies certain algebraic
properties of omega-semigroups. As a consequence an omega-regular equivalence
relation of countable index has an omega-regular set of representatives. This
implies Blumensath's conjecture that a countable structure with an
-automatic presentation can be represented using automata on finite
words. This also complements a very recent result of Hj\"orth, Khoussainov,
Montalban and Nies showing that there is an omega-automatic structure which has
no injective presentation
Automatic structures of bounded degree revisited
The first-order theory of a string automatic structure is known to be
decidable, but there are examples of string automatic structures with
nonelementary first-order theories. We prove that the first-order theory of a
string automatic structure of bounded degree is decidable in doubly exponential
space (for injective automatic presentations, this holds even uniformly). This
result is shown to be optimal since we also present a string automatic
structure of bounded degree whose first-order theory is hard for 2EXPSPACE. We
prove similar results also for tree automatic structures. These findings close
the gaps left open in a previous paper of the second author by improving both,
the lower and the upper bounds.Comment: 26 page
First-Order Model Checking on Generalisations of Pushdown Graphs
We study the first-order model checking problem on two generalisations of
pushdown graphs. The first class is the class of nested pushdown trees. The
other is the class of collapsible pushdown graphs. Our main results are the
following. First-order logic with reachability is uniformly decidable on nested
pushdown trees. Considering first-order logic without reachability, we prove
decidability in doubly exponential alternating time with linearly many
alternations. First-order logic with regular reachability predicates is
uniformly decidable on level 2 collapsible pushdown graphs. Moreover, nested
pushdown trees are first-order interpretable in collapsible pushdown graphs of
level 2. This interpretation can be extended to an interpretation of the class
of higher-order nested pushdown trees in the collapsible pushdown graph
hierarchy. We prove that the second level of this new hierarchy of nested trees
has decidable first-order model checking. Our decidability result for
collapsible pushdown graph relies on the fact that level 2 collapsible pushdown
graphs are uniform tree-automatic. Our last result concerns tree-automatic
structures in general. We prove that first-order logic extended by Ramsey
quantifiers is decidable on all tree-automatic structures.Comment: phd thesis, 255 page